Numerical Solution of Two-Dimensional Linear Fuzzy...
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Research Article Numerical Solution of Two-Dimensional Linear Fuzzy Fredholm Integral Equations by the Fuzzy Lagrange Interpolation H. Nouriani and R. Ezzati Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran Correspondence should be addressed to R. Ezzati; [email protected] Received 26 October 2017; Accepted 19 May 2018; Published 2 December 2018 Academic Editor: Oscar Castillo Copyright © 2018 H. Nouriani and R. Ezzati. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this study, at first, we propose a new approach based on the two-dimensional fuzzy Lagrange interpolation and iterative method to approximate the solution of two-dimensional linear fuzzy Fredholm integral equation (2DLFFIE). en, we prove convergence analysis and numerical stability analysis for the proposed numerical algorithm by two theorems. Finally, by some examples, we show the efficiency of the proposed method. 1. Introduction Recently many authors proposed various numerical methods for solving one-dimensional fuzzy integral equations [1– 9]. Also, two-dimensional fuzzy integral equations have been noticed by a lot of researchers because of their broad applications in engineering sciences. Some of the most important papers in this area are trapezoidal quadrature rule and iterative method [10–12], triangular functions [13], quadrature iterative [14], Bernstein polynomials [15], collo- cation fuzzy wavelet like operator [16], homotopy analysis method (HAM) [17], open fuzzy cubature rule [18], kernel iterative method [19], modified homotopy pertubation [20], block-pulse functions [21], optimal fuzzy quadrature for- mula [22], and finally, iterative method and fuzzy bivariate block-pulse functions [23]. Also, some researchers have solved one-dimensional fuzzy Fredholm integral equations by using fuzzy interpolation via iterative method such as: iterative interpolation method [9], Lagrange interpolation based on the extension principle [5], and spline interpolation [7]. As we know interpolation is one of the most substantial and the most applicable methods in numerical analysis. So, in this paper, we want to solve 2DLFFIEs by applying two-dimensional fuzzy Lagrange interpolation and iterative method. First of all an approximate solution of integral by applying Lagrange interpolation and iterative method is provided. en, convergence analysis and numerical stability analysis of the proposed method in eorems 11 and 12 are proved. e paper is organized as follows: Some notations and theorems about the structure of fuzzy sets are reviewed in Section 2. In Section 3, firstly, we introduce two- dimensional fuzzy Lagrange interpolation. Also, we propose two-dimensional fuzzy Lagrange interpolation and iterative method for solving 2DLFFIEs. In Section 4, we verify con- vergence analysis for proposed method. Also, in Section 5, we prove numerical stability analysis for the method. Two numerical examples are presented in Section 6. 2. Preliminaries At first, we review some basic definitions and necessary results about fuzzy set theory. Definition (see [24]). A fuzzy number is a function : R → [0,1] with the following properties: (1) ∃ 0 ∈ R such that ( 0 )=1. (2) (+(1−)) ≥ min{ (), ()}, ∀, ∈ R, ∀ ∈ [0, 1]. (3) ∀ 0 ∈ R and ∀ > 0, ∃ neighborhood ( 0 ): () ≤ ( 0 ) + , ∀ ∈ ( 0 ). Hindawi Advances in Fuzzy Systems Volume 2018, Article ID 5405124, 8 pages https://doi.org/10.1155/2018/5405124
Transcript of Numerical Solution of Two-Dimensional Linear Fuzzy...
Research ArticleNumerical Solution of Two-Dimensional Linear Fuzzy FredholmIntegral Equations by the Fuzzy Lagrange Interpolation
H Nouriani and R Ezzati
Department of Mathematics Karaj Branch Islamic Azad University Karaj Iran
Correspondence should be addressed to R Ezzati ezatikiauacir
Received 26 October 2017 Accepted 19 May 2018 Published 2 December 2018
Academic Editor Oscar Castillo
Copyright copy 2018 H Nouriani and R Ezzati This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
In this study at first we propose a new approach based on the two-dimensional fuzzy Lagrange interpolation and iterative methodto approximate the solution of two-dimensional linear fuzzy Fredholm integral equation (2DLFFIE) Then we prove convergenceanalysis and numerical stability analysis for the proposed numerical algorithm by two theorems Finally by some examples weshow the efficiency of the proposed method
1 Introduction
Recently many authors proposed various numerical methodsfor solving one-dimensional fuzzy integral equations [1ndash9] Also two-dimensional fuzzy integral equations havebeen noticed by a lot of researchers because of their broadapplications in engineering sciences Some of the mostimportant papers in this area are trapezoidal quadraturerule and iterative method [10ndash12] triangular functions [13]quadrature iterative [14] Bernstein polynomials [15] collo-cation fuzzy wavelet like operator [16] homotopy analysismethod (HAM) [17] open fuzzy cubature rule [18] kerneliterative method [19] modified homotopy pertubation [20]block-pulse functions [21] optimal fuzzy quadrature for-mula [22] and finally iterative method and fuzzy bivariateblock-pulse functions [23] Also some researchers havesolved one-dimensional fuzzy Fredholm integral equationsby using fuzzy interpolation via iterative method such asiterative interpolation method [9] Lagrange interpolationbased on the extension principle [5] and spline interpolation[7]
As we know interpolation is one of the most substantialand the most applicable methods in numerical analysisSo in this paper we want to solve 2DLFFIEs by applyingtwo-dimensional fuzzy Lagrange interpolation and iterativemethod First of all an approximate solution of integralby applying Lagrange interpolation and iterative method is
providedThen convergence analysis and numerical stabilityanalysis of the proposed method in Theorems 11 and 12 areproved
The paper is organized as follows Some notations andtheorems about the structure of fuzzy sets are reviewedin Section 2 In Section 3 firstly we introduce two-dimensional fuzzy Lagrange interpolation Also we proposetwo-dimensional fuzzy Lagrange interpolation and iterativemethod for solving 2DLFFIEs In Section 4 we verify con-vergence analysis for proposed method Also in Section 5we prove numerical stability analysis for the method Twonumerical examples are presented in Section 6
2 Preliminaries
At first we review some basic definitions and necessaryresults about fuzzy set theory
Definition 1 (see [24]) A fuzzy number is a function 119891 R 997888rarr [0 1] with the following properties
(1) exist1199090 isin R such that 119891(1199090) = 1(2) 119891(120578119909+(1minus120578)119910) ge min119891(119909) 119891(119910) forall119909 119910 isin R forall120578 isin[0 1](3) forall1199090 isin R and forall120598 gt 0 exist neighborhood 119880(1199090) 119891(119909) le119891(1199090) + 120598 forall119909 isin 119880(1199090)
HindawiAdvances in Fuzzy SystemsVolume 2018 Article ID 5405124 8 pageshttpsdoiorg10115520185405124
2 Advances in Fuzzy Systems
(4) In R the set 119904119906119901119901(119891) is compact
The set of all fuzzy numbers is denoted byR119865
Definition 2 (see [24]) For 119891 isin R119865 and 0 lt 120572 le 1 define[119891]0 fl 119909 isin R 119891(119909) gt 0 and
[119891]120572 fl 119909 isin R 119891 (119909) ge 120572 (1)
Then it is well known that for each 120572 isin [0 1] [119891]120572 is abounded and closed interval of R We define uniquely thesum 119891oplus 119892 and the product 120583 ⊙ 119891 for 119891 119892 isin R119865 and 120583 isin R by
[119891 oplus 119892]120572 = [119891]120572 + [119892]120572 [120583 ⊙ 119891]120572 = 120583 [119891]120572
forall120572 isin [0 1] (2)
where [119891]120572 + [119892]120572 means the usual addition of two intervals(as subsets ofR) and 120583[119891]120572means the usual product betweena scalar and a subset ofR Notice 1⊙119891 = 119891 and it holds119891oplus119892 =119892 oplus 119891 120583 ⊙ 119891 = 119891 ⊙ 120583 If 0 le 1205721 le 1205722 le 1 then [119891]1205722 sube [119891]1205721 Actually [119891]120572 = [119891(120572)minus 119891(120572)+ ] where119891(120572)minus le 119891(120572)+ 119891(120572)minus 119891(120572)+ isin Rforall120572 isin [0 1] For 120583 gt 0 one has 120583119891(120572)plusmn = (120583⊙119891)(120572)plusmn respectively
Definition 3 (see [24]) Define 119863 R119865 times R119865 997888rarr R+ by
119863(119891 119892) fl sup120572isin[01]
max 10038161003816100381610038161003816119891(120572)minus minus 119892(120572)minus 10038161003816100381610038161003816 10038161003816100381610038161003816119891(120572)+ minus 119892(120572)+ 10038161003816100381610038161003816= sup120572isin[01]
Hausdorff distance ([119891]120572 [119892]120572) (3)
where [119892]120572 = [119892(120572)minus 119892(120572)+ ] 119891 119892 isin R119865 Clearly 119863 is a metricon R119865 Also (R119865 119863) is a complete metric space with thefollowing properties [24]
119863(119891 oplus ℎ 119892 oplus ℎ) = 119863 (119891 119892) forall119891 119892 ℎ isin R119865119863 (1198961015840 ⊙ 119891 1198961015840 ⊙ 119892) = 10038161003816100381610038161003816119896101584010038161003816100381610038161003816 119863 (119891 119892)
forall119891 119892 isin R119865 forall1198961015840 isin R119863 (119891 oplus 119892 ℎ oplus 119890) le 119863 (119891 ℎ) + 119863 (119892 119890)
forall119891 119892 ℎ 119890 isin R119865
(4)
Definition 4 (see [24]) Suppose 119891 119892 R 997888rarr R119865 be fuzzynumber valued functions then the distance between 119891 and 119892is defined by
119863lowast (119891 119892) fl sup119909isinR
119863(119891 (119909) 119892 (119909)) (5)
Lemma 5 (see [24])
(1) If we denote 0 fl 1205940 then forall119891 isin R119865 119891 oplus 0 = 0 oplus 119891 =119891
(2) With respect to 0 none of 119891 isin R119865 119891 = 0 has oppositein R119865
(3) Let 120572 120573 isin R 120572120573 ge 0 and any 119891 isin R119865 we have(120572 + 120573) ⊙ 119891 = 120572 ⊙ 119891oplus 120573 ⊙119891 Notice that for general 120572120573 isin R the above property is false
(4) For any 120574 isin R and any119891 119892 isin R119865 we have 120574⊙(119891oplus119892) =120574 ⊙ 119891 oplus 120574 ⊙ 119892(5) For any 120574 120578 isin R and any119891 isin R119865 we have 120574⊙(120578⊙119891) =(120574 ⊙ 120578) ⊙ 119891
If we denote 119891119865 fl 119863(119891 0) forall119891 isin R119865 then sdot 119865 has theproperties of a usual norm on R119865 ie1003817100381710038171003817100381711989110038171003817100381710038171003817119865 = 0 iff 119891 = 0
10038171003817100381710038171003817120583 ⊙ 11989110038171003817100381710038171003817119865 = 10038161003816100381610038161205831003816100381610038161003816 sdot 1003817100381710038171003817100381711989110038171003817100381710038171003817119865 10038171003817100381710038171003817119891 oplus 11989210038171003817100381710038171003817119865 le 1003817100381710038171003817100381711989110038171003817100381710038171003817119865 + 10038171003817100381710038171198921003817100381710038171003817119865 1003817100381710038171003817100381711989110038171003817100381710038171003817119865 minus 10038171003817100381710038171198921003817100381710038171003817119865 le 119863 (119891 119892) (6)
Notice that (R119865 oplus ⊙) is not a linear space over R andconsequently (R119865 sdot119865) is not a normed space Heresumlowast denotesthe fuzzy summation
Definition 6 (see [24]) A fuzzy valued function 119891 [119886 119887] 997888rarrR119865 is said to be continuous at 1199090 isin [119886 119887] if for each 120598 gt 0there exists 120575 gt 0 such that 119863(119891(119909) 119891(1199090)) lt 120598 whenever119909 isin [119886 119887] and |119909 minus 1199090| lt 120575 We say that 119891 is fuzzy continuouson [119886 119887] if 119891 is continuous at each 1199090 isin [119886 119887] and denotes thespace of all such functions by 119862119865[119886 119887]Definition 7 (see [11]) Suppose that 119891 [119886 119887] times [119888 119889] 997888rarr R119891
is a bounded mapping The function 120596[119886119887]times[119888119889](119891 sdot) R+ cup0 997888rarr R+ defined by
120596[119886119887]times[119888119889] (119891 120575) = sup119863(119891 (119909 119910) 119891 (119904 119905)) 119909 119904isin [119886 119887] 119910 119905 isin [119888 119889] radic(119909 minus 119904)2 + (119910 minus 119905)2 le 120575
(7)
is called modules of oscillation of 119891 on [119886 119887] times [119888 119889] Also if119891 isin 119862119865([119886 119887] times [119888 119889]) then 120596[119886119887]times[119888119889](119891 120575) is called uniformmodules of continuity of 119891Theorem 8 (see [11]) e following properties hold
(1) 119863(119891(119909 119910) 119891(119904119905)) le 120596[119886119887]times[119888119889](119891radic(119909minus119904)2 + (119910minus119905)2)forall119909 119904 isin [119886 119887] 119910 119905 isin [119888 119889](2) 120596[119886119887]times[119888119889](119891 120575) is a nondecreasing mapping in 120575(3) 120596[119886119887]times[119888119889](119891 0) = 0(4) 120596[119886119887]times[119888119889](119891 1205751+1205752) le 120596[119886119887]times[119888119889](119891 1205751)+120596[119886119887]times[119888119889](1198911205752) forall1205751 1205752 ge 0
Advances in Fuzzy Systems 3
(5) 120596[119886119887]times[119888119889](119891 119899120575) le 119899120596[119886119887]times[119888119889](119891 120575) forall120575 ge 0 119899 isin N
(6) 120596[119886119887]times[119888119889](119891 120583120575) le (120583 + 1)120596[119886119887]times[119888119889](119891 120575) forall120583 120575 ge 0Theorem 9 (see [11]) If 119891 and 119892 are Henstock integrablemapping on [119886 119887] times [119888 119889] and if 119863(119891(119904 119905) 119892(119904 119905)) is Lebesgueintegrable then
119863((119865119867)int119889119888int119887119886119891 (119904 119905) 119889119904119889119905 (119865119867)int119889
119888int119887119886119892 (119904 119905) 119889119904119889119905)
le (119871) int119889119888int119887119886119863(119891 (119904 119905) 119892 (119904 119905)) 119889119904 119889119905
(8)
3 The Main Result
In this section first we introduce two-dimensional fuzzyLagrange interpolation Then we propose two-dimensionalfuzzy Lagrange interpolation and iterative method for solving(9)
Consider 2DLFFIE as follows
119866 (119904 119905)= 119892 (119904 119905) oplus 120583
⊙ (119865119877) int119889119888(119865119877) int119887
119886K (119909 119910 119904 119905) ⊙ 119866 (119909 119910) 119889119909 119889119910
(9)
where 120583 gt 0K(119909 119910 119904 119905) is an arbitrary positive function on[119886 119887] times [119888 119889] times [119886 119887] times [119888 119889] and 119892 [119886 119887] times [119888 119889] 997888rarr R119865 Weassume that K is continuous and therefore it is uniformlycontinuous with respect to (119904 119905) So there existsM gt 0 suchthatM = max119904119909isin[119886119887]119910119905isin[119888119889]|K(119909 119910 119904 119905)|
Two-dimensional Lagrange polynomials ℓ119894119895(119904 119905) aredefined as follows
ℓ119894119895 (119904 119905) = ℓ119894 (119904) otimes ℓ119895 (119905) 0 le 119894 119895 le 119899 (10)
where ℓ119894(119904) is the Lagrange polynomial and is defined asfollows
ℓ119894 (119904) = 119899prod119903=0119903 =119894
(119904 minus 119904119903)(119904119894 minus 119904119903) (11)
therefore
ℓ119894119895 (119904119903 1199051199031015840) = 1 if 119894 = 119903 119895 = 11990310158400 otherwise (12)
So the two-dimensional interpolation in the Lagrange formis (see [25])
119901 (119909 119910) = 119899sum119895=0
119899sum119894=0
119891 (119904119894 119905119895) ⊙ ℓ119894119895 (119909 119910) (13)
where the coefficients 119891(119904119894 119905119895) are the fuzzy numbers
Here we consider the two-dimensional fuzzy Lagrangeinterpolation in the given points 119886 = 1199040 lt 1199041 lt sdot sdot sdot lt 119904119899 = 119887and 119888 = 1199050 lt 1199051 lt sdot sdot sdot lt 119905119899 = 119889 such that
K (119909 119910 119904 119905) ⊙ 119866 (119909 119910)asymp 119899sum119895=0
119899sum119894=0
ℓ119894119895 (119909 119910) ⊙K (119904119894 119905119895 119904 119905) ⊙ 119866 (119904119894 119905119895) (14)
Now we propose a numerical method to solve (9) To do thiswe suppose the following iterative procedure to approximatethe solution of (9) in point (119904 119905)
0 (119904 119905) = 119892 (119904 119905) 119896 (119904 119905) = 119892 (119904 119905) oplus 120583
⊙ 119899sum119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895) (15)
where
119862119894119895 = int119889119888int119887119886ℓ119894119895 (119909 119910) 119889119909119889119910 (16)
In Theorem 10 authors of [11] proved the existence anduniqueness solution of (9) by using the Banach fixed pointtheorem
Theorem 10 (see [11]) Let the function K(119909 119910 119904 119905) be con-tinuous and positive for 119909 119904 isin [119886 119887] and 119910 119905 isin [119888 119889] and let119892 [119886 119887] times [119888 119889] 997888rarr R119865 be continuous on [119886 119887] times [119888 119889] If119861 = 120583M(119889 minus 119888)(119887 minus 119886) lt 1 then the fuzzy integral equation (9)has a unique solution 119866lowast isin 119883 where
119883 = 119892 [119886 119887] times [119888 119889] 997888rarr R119865 119892 is continous (17)
be the space of two-dimensional fuzzy continuous functionswith the metric 119863lowast and it can be obtained by the followingsuccessive approximations method
1198660 (119904 119905) = 119892 (119904 119905) 119866119896 (119904 119905) = 119892 (119904 119905) oplus 120583 ⊙ (119865119877)
sdot int119889119888(119865119877) int119887
119886K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910
forall119896 ge 1(18)
Moreover the sequence of successive approximations (119866119896)119896ge1converges to the solution 119866lowast Furthermore the following errorbound holds
119863lowast (119866lowast 119866119896) le 119861119896+11 minus 119861M1 forall119896 ge 1 (19)
whereM1 = sup119904isin[119886119887]119905isin[119888119889]119866(119904 119905)
4 Advances in Fuzzy Systems
4 Convergence Analysis
In this section we obtain an error estimate between the exactsolution and the approximate solution of 2DLFFIE (9)
Theorem 11 Under the hypotheses of eorem 10 and 120583 gt 0the iterative procedure (15) converges to the unique solution of(9) 119866lowast and its error estimate is as follows
119863lowast (119866lowast 119896)le 119861119896+1(1 minus 119861)M1
+ 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897) (20)
Proof Clearly we have
119863(119866lowast (119904 119905) 119896 (119904 119905)) le 119863 (119866lowast (119904 119905) 119866119896 (119904 119905))+ 119863 (119866119896 (119904 119905) 119896 (119904 119905)) (21)
From (18) and (15) we conclude that
119863(119866119896 (119904 119905) 119896 (119904 119905)) = 119863(119892 (119904 119905) oplus 120583 ⊙ (119865119877)int119889119888(119865119877)int119887
119886K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909119889119910 119892 (119904 119905) oplus 120583 ⊙ 119899sum
119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895)) = 120583119863((119865119877) int119889119888(119865119877)
sdot int119887119886K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910 119899sum
119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895)) = 120583119863( 119899sum119895=1
119899sum119894=1
(119865119877) int119905119895119905119895minus1
(119865119877) int119904119894119904119894minus1
K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909119889119910119899sum119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895)) = 120583119863( 119899sum119895=1
119899sum119894=1
(119865119877) int119905119895119905119895minus1
(119865119877) int119904119894119904119894minus1
K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909119889119910 119899sum119895=1
119899sum119894=1
(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1) 119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905)
⊙ 119896minus1 (119904119894 119905119895) oplus 11986200K (1199040 1199050 119904 119905) ⊙ 119896minus1 (1199040 1199050))
le 120583 119899sum119895=1
119899sum119894=1
119863((119865119877) int119905119895119905119895minus1
(119865119877) int119904119894119904119894minus1
K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909119889119910 (119865119877) int119905119895119905119895minus1
(119865119877) int119904119894119904119894minus1
119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895) 119889119909 119889119910) oplus 120583119863(11986200K (1199040 1199050 119904 119905)⊙ 119896minus1 (1199040 1199050) 0)le 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
[119863(K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) K (119909 119910 119904 119905) ⊙ 119896minus1 (119904119894 119905119895)) + 119863(K (119909 119910 119904 119905) ⊙ 119896minus1 (119904119894 119905119895) 119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895))]119889119909 119889119910+ 120583119863 (11986200K (1199040 1199050 119904 119905) ⊙ 119896minus1 (1199040 1199050) 0)le 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
(1003816100381610038161003816K (119909 119910 119904 119905)1003816100381610038161003816 119863 (119866119896minus1 (119909 119910) 119896minus1 (119904119894 119905119895)) + 1003816100381610038161003816K (119909 119910 119904 119905)1003816100381610038161003816 119863 (119896minus1 (119904119894 119905119895) 0) +10038161003816100381610038161003816100381610038161003816100381610038161003816
119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905)10038161003816100381610038161003816100381610038161003816100381610038161003816 119863 (119896minus1 (119904119894 119905119895) 0))119889119909 119889119910
+ 120583119863 (11986200K (1199040 1199050 119904 119905) ⊙ 119896minus1 (1199040 1199050) 0)
(22)
By supposing119898119896minus1 = sup(119904119905)isin[119886119887]times[119888119889]119896minus1(119904 119905) we get119863(119866119896 (119904 119905) 119896 (119904 119905))
le 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
(M119863(119866119896minus1 (119909 119910) 119896minus1 (119904119894 119905119895)) +M119898119896minus1 +10038161003816100381610038161003816100381610038161003816100381610038161003816
119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905)10038161003816100381610038161003816100381610038161003816100381610038161003816 119898119896minus1)119889119909 119889119910
+ 120583 100381610038161003816100381611986200K (1199040 1199050 119904 119905)1003816100381610038161003816 119898119896minus1(23)
Also we have
119863(119866119896 (119904 119905) 119896 (119904 119905)) le 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
(M119863lowast (119866119896minus1 119896minus1)) 119889119909 119889119910 + 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
M119898119896minus1119889119909 119889119910
+ 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
10038161003816100381610038161003816100381610038161003816100381610038161003816119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905)
10038161003816100381610038161003816100381610038161003816100381610038161003816 119898119896minus1119889119909119889119910 + 120583 100381610038161003816100381611986200K (1199040 1199050 119904 119905)1003816100381610038161003816 119898119896minus1(24)
Advances in Fuzzy Systems 5
Therefore
119863(119866119896 (119904 119905) 119896 (119904 119905))le 120583M (119889 minus 119888) (119887 minus 119886)119863lowast (119866119896minus1 119896minus1)
+ 120583M (119889 minus 119888) (119887 minus 119886)119898119896minus1+ 120583M (119889 minus 119888) (119887 minus 119886)119898119896minus1 119862(119889 minus 119888) (119887 minus 119886)+ 120583M119862119898119896minus1
le 119861119863lowast (119866119896minus1 119896minus1) + 119861119898119896minus1+ 2119861119898119896minus1 119862(119889 minus 119888) (119887 minus 119886)
= 119861119863lowast (119866119896minus1 119896minus1)+ 119861119898119896minus1(1 + 2119862(119889 minus 119888) (119887 minus 119886))
(25)
where 119862 = max|119862119894119895| 119894 119895 = 0 sdot sdot sdot 119899 Hence we conclude that119863lowast (119866119896minus1 119896minus1) le 119861119863lowast (119866119896minus2 119896minus2)
+ 119861119898119896minus2(1 + 2119862(119889 minus 119888) (119887 minus 119886))
119863lowast (1198661 1) le 119861119863lowast (1198660 0)+ 1198611198980(1 + 2119862(119889 minus 119888) (119887 minus 119886))
(26)
So
119863lowast (119866119896 119896) le (1 + 2119862(119889 minus 119888) (119887 minus 119886))sdot (119861119898119896minus1 + 1198612119898119896minus2 + sdot sdot sdot + 1198611198961198980)
(27)
If119898119897 = max1198980 119898119896minus1 then we obtain
119863lowast (119866119896 119896) lt 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897) (28)
therefore
119863lowast (119866lowast ) le 119861119896+11 minus 1198611198721+ 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897)
(29)
5 Numerical Stability Analysis
To show the numerical stability analysis of the proposedmethod in previous section we consider another startingapproximation 119891(119904 119905) = 0(119904 119905) such that exist120598 gt 0 for which119863(1198660(119904 119905) 0(119904 119905)) lt 120598 forall119904 119905 isin [119886 119887] times [119888 119889] The obtainedsequence of successive approximations is
119896 (119904 119905) = 119891 (119904 119905) oplus 120583 ⊙ (119865119877)sdot int119889119888(119865119877)int119887
119886K (119909 119910 119904 119905) ⊙ 119896minus1 (119909 119910) 119889119909 119889119910 (30)
and using the same iterative method the terms of producedsequence are
V0 (119904 119905) = 0 (119904 119905) = 119891 (119904 119905) V119896 (119904 119905) = 119891 (119904 119905) oplus 120583
⊙ 119899sum119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ V119896minus1 (119904119894 119905119895) (31)
Theorem 12 e proposed method (15) under the assump-tions of eorem 11 is numerically stable with respect to thechoice of the first iteration
Proof At first we obtain that
119863(119896 (119904 119905) V119896 (119904 119905))le 119863 (119896 (119904 119905) 119866119896 (119904 119905)) + 119863 (119866119896 (119904 119905) 119896 (119904 119905))
+ 119863 (119896 (119904 119905) V119896 (119904 119905))le 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897)
+ 119863 (119866119896 (119904 119905) 119896 (119904 119905))+ 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897)
(32)
However
119863(119866119896 (119904 119905) 119896 (119904 119905)) = 119863(119892 (119904 119905) oplus 120583 ⊙ (119865119877)sdot int119889119888(119865119877)
sdot int119887119886K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910 119891 (119904 119905) oplus 120583
⊙ (119865119877) int119889119888(119865119877) int119887
119886K (119909 119910 119904 119905)
⊙ 119896minus1 (119909 119910) 119889119909 119889119910) le 119863 (119892 (119904 119905) 119891 (119904 119905))+ 120583119863((119865119877) int119889
119888(119865119877) int119887
119886K (119909 119910 119904 119905)
6 Advances in Fuzzy Systems
Table 1 The absolute errors on the level sets with 119899 = 2 119896 = 5 for Example 13 by using the proposed method (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 123485119890 minus 501 666819119890 minus 7 1141 119890 minus 502 13501 119890 minus 6 105703119890 minus 503 204985119890 minus 6 982941119890 minus 604 276606119890 minus 6 918729119890 minus 605 349874119890 minus 6 864395119890 minus 606 424788119890 minus 6 819941119890 minus 607 501349119890 minus 6 785365119890 minus 608 579556119890 minus 6 760668119890 minus 609 65941 119890 minus 6 745849119890 minus 610 74091 119890 minus 6 74091 119890 minus 6
⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910 (119865119877) int119889119888(119865119877)
sdot int119887119886K (119909 119910 119904 119905) ⊙ 119896minus1 (119909 119910) 119889119909 119889119910)
le 120598 + 120583int119889119888int119887119886
1003816100381610038161003816K (119909 119910 119904 119905)1003816100381610038161003816sdot 119863 (119866119896minus1 (119909 119910) 119896minus1 (119909 119910)) 119889119909 119889119910
(33)
We conclude that
119863lowast (119866119896 119896) le 120598 + 120583int119889119888int119887119886M119863lowast (119866119896minus1 119896minus1) 119889119909 119889119910
= 120598 + 119861119863lowast (119866119896minus1 119896minus1) (34)
and thus
119863lowast (119866119896 119896) le 120598 + 119861119863lowast (119866119896minus1 119896minus1)119863lowast (119866119896minus1 119896minus1) le 120598 + 119861119863lowast (119866119896minus2 119896minus2)
119863lowast (1198661 1) le 120598 + 119861119863lowast (1198660 0)
(35)
So
119863lowast (119866119896 119896) le 120598 + 119861 (120598 + 119861119863lowast (119866119896minus2 119896minus2))le 120598 + 119861120598 + 1198612 (120598 + 119861119863lowast (119866119896minus3 119896minus3))
le 120598 + 119861120598 + 1198612120598 + 1198613120598 + sdot sdot sdot + 119861119896119863lowast (1198660 0)le 120598 (1 + 119861 + 1198612 + 1198613 + sdot sdot sdot + 119861119896) le 1205981 minus 119861
(36)
Therefore
119863lowast (119896 V119896)le 11 minus 119861 (2119898119897 (1 + 2119862(119889 minus 119888) (119887 minus 119886)) + 120598) (37)
6 Numerical Examples
In this section we use the proposed method in two-dimensional linear fuzzy Fredholm integral equations forsolving two examples By using the proposed method for 119899 =2 119896 = 5 8 16 and 119903 isin 00 01 02 03 04 05 06 07 0809 10 in (119904 119905) = (05 05) we present the absolute errors inTables 1ndash4
Example 13 (see [11]) Consider the linear integral equation
119866 (119904 119905)= 119892 (119904 119905)
oplus (119865119877) int10(119865119877) int1
0K (119909 119910 119904 119905) ⊙ 119866 (119909 119910) 119889119909 119889119910
(38)
with
K (119909 119910 119904 119905) = 119909119910119904119905 (119904 119905 119903) = 119903 (13119903 + 83)(1 + 119904 + 119905 minus 712119904119905) 119892 (119904 119905 119903) = (21199032 minus 4119903 + 5) (1 + 119904 + 119905 minus 712119904119905)
(39)
The exact solution for 2DLFFIE (38) is
lowast (119904 119905 119903) = 119903 (13119903 + 83) (119904 + 119905 + 1) 119866lowast (119904 119905 119903) = (21199032 minus 4119903 + 5) (119904 + 119905 + 1)
(40)
Advances in Fuzzy Systems 7
Table 2 The absolute errors on the level sets with 119899 = 2 119896 = 16 for Example 13 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 177636119890 minus 1501 111022119890 minus 16 177636119890 minus 1502 0 177636119890 minus 1503 222045119890 minus 16 888178119890 minus 1604 444089119890 minus 16 888178119890 minus 1605 444089119890 minus 16 888178119890 minus 1606 444089119890 minus 16 007 888178119890 minus 16 008 0 009 0 010 0 0
Table 3 The absolute errors on the level sets with 119899 = 2 119896 = 5 for Example 14 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 960565119890 minus 1301 480276119890 minus 14 912548119890 minus 1302 960551119890 minus 14 864531119890 minus 1303 144079119890 minus 13 816458119890 minus 1304 19211 119890 minus 13 768441119890 minus 1305 240141119890 minus 13 720479119890 minus 1306 288158119890 minus 13 672407119890 minus 1307 336203119890 minus 13 624334119890 minus 1308 38422 119890 minus 13 576317119890 minus 1309 432265119890 minus 13 5283 119890 minus 1310 480282119890 minus 13 480282119890 minus 13
Table 4 The absolute errors on the level sets with 119899 = 2 119896 = 8 for Example 14 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 001 0 002 0 555112119890 minus 1703 0 004 0 005 0 555112119890 minus 1706 0 007 0 008 0 009 277556119890 minus 17 010 0 0
Example 14 (see [13]) Consider the following fuzzy Fred-holm integral equation (38) with
(119904 119905 119903) = 119903 (119904119905 + 1676 (1199042 + 1199052 minus 2)) 119892 (119904 119905 119903) = (2 minus 119903) (119904119905 + 1676 (1199042 + 1199052 minus 2)) (41)
and kernel
K (119909 119910 119904 119905) = 1169 (1199092 + 1199102 minus 2) (1199042 + 1199052 minus 2) 0 le 119904 119905 119909 119910 le 1 (42)
The exact solution is
lowast (119904 119905 119903) = 119903119904119905
8 Advances in Fuzzy Systems
119866lowast (119904 119905 119903) = (2 minus 119903) 119904119905(43)
7 Conclusion
The 2DLFFIE is solved by utilizing two-dimensional fuzzyLagrange interpolation and iterative method As it wasexpected the method used to approximate the integral inthis equation is a suitable one since convergence analysis andstability analysis have been proved and also absolute errorin examples is nearly zero As a result considering the factthat the proposed method does not lead to solve fuzzy linearsystem it can be utilized as an efficient method to solve thistype of equations As future researches we can use finite anddivided differences methods fuzzy spline interpolation andfuzzy quasi-interpolation for solving two-dimensional linearor nonlinear fuzzy Fredholm integral equations
Conflicts of Interest
The authors declare that there are no conflicts of interestrelated to this paper
References
[1] E Babolian H Sadeghi Goghary and S Abbasbandy ldquoNumer-ical solution of linear Fredholm fuzzy integral equations of thesecond kind by Adomian methodrdquo Applied Mathematics andComputation vol 161 no 3 pp 733ndash744 2005
[2] A M Bica ldquoError estimation in the approximation of thesolution of nonlinear fuzzy Fredholm integral equationsrdquo Infor-mation Sciences vol 178 no 5 pp 1279ndash1292 2008
[3] A M Bica and C Popescu ldquoApproximating the solution ofnonlinear Hammerstein fuzzy integral equationsrdquo Fuzzy Setsand Systems vol 245 pp 1ndash17 2014
[4] M A Fariborzi Araghi and G Kazemi Gelian ldquoSolving fuzzyFredholm linear integral equations using Sinc method anddouble exponential transformationrdquo So Computing vol 19 no4 pp 1063ndash1070 2015
[5] M A Faborzi Araghi and N Parandin ldquoNumerical solution offuzzy Fredholm integral equations by the Lagrange interpola-tion based on the extension principlerdquo So Computing vol 15no 12 pp 2449ndash2456 2011
[6] H Hosseini Fadravi R Buzhabadi and H Saberi Nik ldquoSolvinglinear Fredholm fuzzy integral equations of the second kind byartificial neural networksrdquo Alexandria Engineering Journal vol53 no 1 pp 249ndash257 2014
[7] Y Jafarzadeh ldquoNumerical solution for fuzzy Fredholm integralequations with upper-bound on error by splines interpolationrdquoFuzzy Information and Engineering vol 4 no 3 pp 339ndash3472012
[8] A Molabahrami A Shidfar and A Ghyasi ldquoAn analyticalmethod for solving linear Fredholm fuzzy integral equations ofthe second kindrdquo Computers amp Mathematics with Applicationsvol 61 no 9 pp 2754ndash2761 2011
[9] N Parandin and M A Fariborzi Araghi ldquoThe approximatesolution of linear fuzzy Fredholm integral equations of thesecond kind by using iterative interpolationrdquoWorld Academy ofScience Engineering and Technology vol 49 pp 978ndash984 2009
[10] A M Bica and S Ziari ldquoIterative numerical method for fuzzyVolterra linear integral equations in two dimensionsrdquo SoComputing vol 21 no 5 pp 1097ndash1108 2017
[11] SM Sadatrasoul andR Ezzati ldquoQuadrature Rules and IterativeMethod for Numerical Solution of Two-Dimensional FuzzyIntegral Equationsrdquo Abstract and Applied Analysis vol 2014Article ID 413570 18 pages 2014
[12] SM Sadatrasoul andR Ezzati ldquoIterativemethod for numericalsolution of two-dimensional nonlinear fuzzy integral equa-tionsrdquo Fuzzy Sets and Systems vol 280 pp 91ndash106 2015
[13] F Mirzaee M K Yari and E Hadadiyan ldquoNumerical solutionof two-dimensional fuzzy Fredholm integral equations of thesecond kind using triangular functionsrdquo Beni-Suef UniversityJournal of Basic and Applied Sciences vol 4 no 2 pp 109ndash1182015
[14] H Nouriani and R Ezzati ldquoQuadrature iterative method fornumerical solution of two-dimensional linear fuzzy Fredholmintegral equationsrdquoMathematical Sciences vol 11 no 1 pp 63ndash72 2017
[15] R Ezzati and S Ziari ldquoNumerical solution of two-dimensionalfuzzy Fredholm integral equations of the second kind usingfuzzy bivariate Bernstein polynomialsrdquo International Journal ofFuzzy Systems vol 15 no 1 pp 84ndash89 2013
[16] N Hassasi and R Ezzati ldquoNumerical solution of two-dimensional fuzzy Fredholm integral equations using collo-cation fuzzy wavelet like operatorrdquo International Journal ofIndustrial Mathematics vol 7 p 11 2015
[17] M A Vali M J Agheli and S Gohari Nezhad ldquoHomotopyanalysis method to solve two-dimensional fuzzy Fredholmintegral equationrdquo General Mathematics Notes vol 22 pp 31ndash43 2014
[18] A M Bica and S Ziari ldquoOpen fuzzy cubature rule withapplication tononlinear fuzzyVolterra integral equations in twodimensionsrdquo Fuzzy Sets and Systems In press Corrected Proof
[19] A Rivaz and F Yousefi ldquoKernel iterative method for solvingtwo-dimensional fuzzy Fredholm integral equations of thesecond kindrdquo Journal of Fuzzy Set Valued Analysis vol 2013 p9 2013
[20] A Rivaz and F Yousefi ldquoModified homotopy perturbationmethod for solving two-dimensional fuzzy Fredholm integralequationrdquo International Journal of AppliedMathematics vol 25no 4 pp 591ndash602 2012
[21] A Rivaz F Yousefi and H Salehinejad ldquoUsing block pulsefunctions for solving two-dimensional fuzzy Fredholm integralequations of the second kindrdquo International Journal of AppliedMathematics vol 25 no 4 pp 571ndash582 2012
[22] S M Sadatrasoul and R Ezzati ldquoNumerical solution of two-dimensional nonlinear Hammerstein fuzzy integral equationsbased on optimal fuzzy quadrature formulardquo Journal of Compu-tational and Applied Mathematics vol 292 pp 430ndash446 2016
[23] S Ziari ldquoIterative method for solving two-dimensional non-linear fuzzy integral equations using fuzzy bivariate block-pulse functions with error estimationrdquo Iranian Journal of FuzzySystems vol 15 no 1 pp 55ndash76 183 2018
[24] G A Anastassiou Fuzzy Mathematics Approximation eoryvol 251 Springer 2010
[25] A R BozorgmaneshMOtadi A A Safe Kordi et al ldquoBarkhor-dari Ahmadi Lagrange two-dimensional interpolationmethodfor modeling nanoparticle formation during RESS processrdquoInternational Journal of Industrial Mathematics vol 1 pp 175ndash181 2009
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2 Advances in Fuzzy Systems
(4) In R the set 119904119906119901119901(119891) is compact
The set of all fuzzy numbers is denoted byR119865
Definition 2 (see [24]) For 119891 isin R119865 and 0 lt 120572 le 1 define[119891]0 fl 119909 isin R 119891(119909) gt 0 and
[119891]120572 fl 119909 isin R 119891 (119909) ge 120572 (1)
Then it is well known that for each 120572 isin [0 1] [119891]120572 is abounded and closed interval of R We define uniquely thesum 119891oplus 119892 and the product 120583 ⊙ 119891 for 119891 119892 isin R119865 and 120583 isin R by
[119891 oplus 119892]120572 = [119891]120572 + [119892]120572 [120583 ⊙ 119891]120572 = 120583 [119891]120572
forall120572 isin [0 1] (2)
where [119891]120572 + [119892]120572 means the usual addition of two intervals(as subsets ofR) and 120583[119891]120572means the usual product betweena scalar and a subset ofR Notice 1⊙119891 = 119891 and it holds119891oplus119892 =119892 oplus 119891 120583 ⊙ 119891 = 119891 ⊙ 120583 If 0 le 1205721 le 1205722 le 1 then [119891]1205722 sube [119891]1205721 Actually [119891]120572 = [119891(120572)minus 119891(120572)+ ] where119891(120572)minus le 119891(120572)+ 119891(120572)minus 119891(120572)+ isin Rforall120572 isin [0 1] For 120583 gt 0 one has 120583119891(120572)plusmn = (120583⊙119891)(120572)plusmn respectively
Definition 3 (see [24]) Define 119863 R119865 times R119865 997888rarr R+ by
119863(119891 119892) fl sup120572isin[01]
max 10038161003816100381610038161003816119891(120572)minus minus 119892(120572)minus 10038161003816100381610038161003816 10038161003816100381610038161003816119891(120572)+ minus 119892(120572)+ 10038161003816100381610038161003816= sup120572isin[01]
Hausdorff distance ([119891]120572 [119892]120572) (3)
where [119892]120572 = [119892(120572)minus 119892(120572)+ ] 119891 119892 isin R119865 Clearly 119863 is a metricon R119865 Also (R119865 119863) is a complete metric space with thefollowing properties [24]
119863(119891 oplus ℎ 119892 oplus ℎ) = 119863 (119891 119892) forall119891 119892 ℎ isin R119865119863 (1198961015840 ⊙ 119891 1198961015840 ⊙ 119892) = 10038161003816100381610038161003816119896101584010038161003816100381610038161003816 119863 (119891 119892)
forall119891 119892 isin R119865 forall1198961015840 isin R119863 (119891 oplus 119892 ℎ oplus 119890) le 119863 (119891 ℎ) + 119863 (119892 119890)
forall119891 119892 ℎ 119890 isin R119865
(4)
Definition 4 (see [24]) Suppose 119891 119892 R 997888rarr R119865 be fuzzynumber valued functions then the distance between 119891 and 119892is defined by
119863lowast (119891 119892) fl sup119909isinR
119863(119891 (119909) 119892 (119909)) (5)
Lemma 5 (see [24])
(1) If we denote 0 fl 1205940 then forall119891 isin R119865 119891 oplus 0 = 0 oplus 119891 =119891
(2) With respect to 0 none of 119891 isin R119865 119891 = 0 has oppositein R119865
(3) Let 120572 120573 isin R 120572120573 ge 0 and any 119891 isin R119865 we have(120572 + 120573) ⊙ 119891 = 120572 ⊙ 119891oplus 120573 ⊙119891 Notice that for general 120572120573 isin R the above property is false
(4) For any 120574 isin R and any119891 119892 isin R119865 we have 120574⊙(119891oplus119892) =120574 ⊙ 119891 oplus 120574 ⊙ 119892(5) For any 120574 120578 isin R and any119891 isin R119865 we have 120574⊙(120578⊙119891) =(120574 ⊙ 120578) ⊙ 119891
If we denote 119891119865 fl 119863(119891 0) forall119891 isin R119865 then sdot 119865 has theproperties of a usual norm on R119865 ie1003817100381710038171003817100381711989110038171003817100381710038171003817119865 = 0 iff 119891 = 0
10038171003817100381710038171003817120583 ⊙ 11989110038171003817100381710038171003817119865 = 10038161003816100381610038161205831003816100381610038161003816 sdot 1003817100381710038171003817100381711989110038171003817100381710038171003817119865 10038171003817100381710038171003817119891 oplus 11989210038171003817100381710038171003817119865 le 1003817100381710038171003817100381711989110038171003817100381710038171003817119865 + 10038171003817100381710038171198921003817100381710038171003817119865 1003817100381710038171003817100381711989110038171003817100381710038171003817119865 minus 10038171003817100381710038171198921003817100381710038171003817119865 le 119863 (119891 119892) (6)
Notice that (R119865 oplus ⊙) is not a linear space over R andconsequently (R119865 sdot119865) is not a normed space Heresumlowast denotesthe fuzzy summation
Definition 6 (see [24]) A fuzzy valued function 119891 [119886 119887] 997888rarrR119865 is said to be continuous at 1199090 isin [119886 119887] if for each 120598 gt 0there exists 120575 gt 0 such that 119863(119891(119909) 119891(1199090)) lt 120598 whenever119909 isin [119886 119887] and |119909 minus 1199090| lt 120575 We say that 119891 is fuzzy continuouson [119886 119887] if 119891 is continuous at each 1199090 isin [119886 119887] and denotes thespace of all such functions by 119862119865[119886 119887]Definition 7 (see [11]) Suppose that 119891 [119886 119887] times [119888 119889] 997888rarr R119891
is a bounded mapping The function 120596[119886119887]times[119888119889](119891 sdot) R+ cup0 997888rarr R+ defined by
120596[119886119887]times[119888119889] (119891 120575) = sup119863(119891 (119909 119910) 119891 (119904 119905)) 119909 119904isin [119886 119887] 119910 119905 isin [119888 119889] radic(119909 minus 119904)2 + (119910 minus 119905)2 le 120575
(7)
is called modules of oscillation of 119891 on [119886 119887] times [119888 119889] Also if119891 isin 119862119865([119886 119887] times [119888 119889]) then 120596[119886119887]times[119888119889](119891 120575) is called uniformmodules of continuity of 119891Theorem 8 (see [11]) e following properties hold
(1) 119863(119891(119909 119910) 119891(119904119905)) le 120596[119886119887]times[119888119889](119891radic(119909minus119904)2 + (119910minus119905)2)forall119909 119904 isin [119886 119887] 119910 119905 isin [119888 119889](2) 120596[119886119887]times[119888119889](119891 120575) is a nondecreasing mapping in 120575(3) 120596[119886119887]times[119888119889](119891 0) = 0(4) 120596[119886119887]times[119888119889](119891 1205751+1205752) le 120596[119886119887]times[119888119889](119891 1205751)+120596[119886119887]times[119888119889](1198911205752) forall1205751 1205752 ge 0
Advances in Fuzzy Systems 3
(5) 120596[119886119887]times[119888119889](119891 119899120575) le 119899120596[119886119887]times[119888119889](119891 120575) forall120575 ge 0 119899 isin N
(6) 120596[119886119887]times[119888119889](119891 120583120575) le (120583 + 1)120596[119886119887]times[119888119889](119891 120575) forall120583 120575 ge 0Theorem 9 (see [11]) If 119891 and 119892 are Henstock integrablemapping on [119886 119887] times [119888 119889] and if 119863(119891(119904 119905) 119892(119904 119905)) is Lebesgueintegrable then
119863((119865119867)int119889119888int119887119886119891 (119904 119905) 119889119904119889119905 (119865119867)int119889
119888int119887119886119892 (119904 119905) 119889119904119889119905)
le (119871) int119889119888int119887119886119863(119891 (119904 119905) 119892 (119904 119905)) 119889119904 119889119905
(8)
3 The Main Result
In this section first we introduce two-dimensional fuzzyLagrange interpolation Then we propose two-dimensionalfuzzy Lagrange interpolation and iterative method for solving(9)
Consider 2DLFFIE as follows
119866 (119904 119905)= 119892 (119904 119905) oplus 120583
⊙ (119865119877) int119889119888(119865119877) int119887
119886K (119909 119910 119904 119905) ⊙ 119866 (119909 119910) 119889119909 119889119910
(9)
where 120583 gt 0K(119909 119910 119904 119905) is an arbitrary positive function on[119886 119887] times [119888 119889] times [119886 119887] times [119888 119889] and 119892 [119886 119887] times [119888 119889] 997888rarr R119865 Weassume that K is continuous and therefore it is uniformlycontinuous with respect to (119904 119905) So there existsM gt 0 suchthatM = max119904119909isin[119886119887]119910119905isin[119888119889]|K(119909 119910 119904 119905)|
Two-dimensional Lagrange polynomials ℓ119894119895(119904 119905) aredefined as follows
ℓ119894119895 (119904 119905) = ℓ119894 (119904) otimes ℓ119895 (119905) 0 le 119894 119895 le 119899 (10)
where ℓ119894(119904) is the Lagrange polynomial and is defined asfollows
ℓ119894 (119904) = 119899prod119903=0119903 =119894
(119904 minus 119904119903)(119904119894 minus 119904119903) (11)
therefore
ℓ119894119895 (119904119903 1199051199031015840) = 1 if 119894 = 119903 119895 = 11990310158400 otherwise (12)
So the two-dimensional interpolation in the Lagrange formis (see [25])
119901 (119909 119910) = 119899sum119895=0
119899sum119894=0
119891 (119904119894 119905119895) ⊙ ℓ119894119895 (119909 119910) (13)
where the coefficients 119891(119904119894 119905119895) are the fuzzy numbers
Here we consider the two-dimensional fuzzy Lagrangeinterpolation in the given points 119886 = 1199040 lt 1199041 lt sdot sdot sdot lt 119904119899 = 119887and 119888 = 1199050 lt 1199051 lt sdot sdot sdot lt 119905119899 = 119889 such that
K (119909 119910 119904 119905) ⊙ 119866 (119909 119910)asymp 119899sum119895=0
119899sum119894=0
ℓ119894119895 (119909 119910) ⊙K (119904119894 119905119895 119904 119905) ⊙ 119866 (119904119894 119905119895) (14)
Now we propose a numerical method to solve (9) To do thiswe suppose the following iterative procedure to approximatethe solution of (9) in point (119904 119905)
0 (119904 119905) = 119892 (119904 119905) 119896 (119904 119905) = 119892 (119904 119905) oplus 120583
⊙ 119899sum119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895) (15)
where
119862119894119895 = int119889119888int119887119886ℓ119894119895 (119909 119910) 119889119909119889119910 (16)
In Theorem 10 authors of [11] proved the existence anduniqueness solution of (9) by using the Banach fixed pointtheorem
Theorem 10 (see [11]) Let the function K(119909 119910 119904 119905) be con-tinuous and positive for 119909 119904 isin [119886 119887] and 119910 119905 isin [119888 119889] and let119892 [119886 119887] times [119888 119889] 997888rarr R119865 be continuous on [119886 119887] times [119888 119889] If119861 = 120583M(119889 minus 119888)(119887 minus 119886) lt 1 then the fuzzy integral equation (9)has a unique solution 119866lowast isin 119883 where
119883 = 119892 [119886 119887] times [119888 119889] 997888rarr R119865 119892 is continous (17)
be the space of two-dimensional fuzzy continuous functionswith the metric 119863lowast and it can be obtained by the followingsuccessive approximations method
1198660 (119904 119905) = 119892 (119904 119905) 119866119896 (119904 119905) = 119892 (119904 119905) oplus 120583 ⊙ (119865119877)
sdot int119889119888(119865119877) int119887
119886K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910
forall119896 ge 1(18)
Moreover the sequence of successive approximations (119866119896)119896ge1converges to the solution 119866lowast Furthermore the following errorbound holds
119863lowast (119866lowast 119866119896) le 119861119896+11 minus 119861M1 forall119896 ge 1 (19)
whereM1 = sup119904isin[119886119887]119905isin[119888119889]119866(119904 119905)
4 Advances in Fuzzy Systems
4 Convergence Analysis
In this section we obtain an error estimate between the exactsolution and the approximate solution of 2DLFFIE (9)
Theorem 11 Under the hypotheses of eorem 10 and 120583 gt 0the iterative procedure (15) converges to the unique solution of(9) 119866lowast and its error estimate is as follows
119863lowast (119866lowast 119896)le 119861119896+1(1 minus 119861)M1
+ 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897) (20)
Proof Clearly we have
119863(119866lowast (119904 119905) 119896 (119904 119905)) le 119863 (119866lowast (119904 119905) 119866119896 (119904 119905))+ 119863 (119866119896 (119904 119905) 119896 (119904 119905)) (21)
From (18) and (15) we conclude that
119863(119866119896 (119904 119905) 119896 (119904 119905)) = 119863(119892 (119904 119905) oplus 120583 ⊙ (119865119877)int119889119888(119865119877)int119887
119886K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909119889119910 119892 (119904 119905) oplus 120583 ⊙ 119899sum
119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895)) = 120583119863((119865119877) int119889119888(119865119877)
sdot int119887119886K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910 119899sum
119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895)) = 120583119863( 119899sum119895=1
119899sum119894=1
(119865119877) int119905119895119905119895minus1
(119865119877) int119904119894119904119894minus1
K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909119889119910119899sum119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895)) = 120583119863( 119899sum119895=1
119899sum119894=1
(119865119877) int119905119895119905119895minus1
(119865119877) int119904119894119904119894minus1
K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909119889119910 119899sum119895=1
119899sum119894=1
(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1) 119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905)
⊙ 119896minus1 (119904119894 119905119895) oplus 11986200K (1199040 1199050 119904 119905) ⊙ 119896minus1 (1199040 1199050))
le 120583 119899sum119895=1
119899sum119894=1
119863((119865119877) int119905119895119905119895minus1
(119865119877) int119904119894119904119894minus1
K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909119889119910 (119865119877) int119905119895119905119895minus1
(119865119877) int119904119894119904119894minus1
119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895) 119889119909 119889119910) oplus 120583119863(11986200K (1199040 1199050 119904 119905)⊙ 119896minus1 (1199040 1199050) 0)le 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
[119863(K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) K (119909 119910 119904 119905) ⊙ 119896minus1 (119904119894 119905119895)) + 119863(K (119909 119910 119904 119905) ⊙ 119896minus1 (119904119894 119905119895) 119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895))]119889119909 119889119910+ 120583119863 (11986200K (1199040 1199050 119904 119905) ⊙ 119896minus1 (1199040 1199050) 0)le 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
(1003816100381610038161003816K (119909 119910 119904 119905)1003816100381610038161003816 119863 (119866119896minus1 (119909 119910) 119896minus1 (119904119894 119905119895)) + 1003816100381610038161003816K (119909 119910 119904 119905)1003816100381610038161003816 119863 (119896minus1 (119904119894 119905119895) 0) +10038161003816100381610038161003816100381610038161003816100381610038161003816
119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905)10038161003816100381610038161003816100381610038161003816100381610038161003816 119863 (119896minus1 (119904119894 119905119895) 0))119889119909 119889119910
+ 120583119863 (11986200K (1199040 1199050 119904 119905) ⊙ 119896minus1 (1199040 1199050) 0)
(22)
By supposing119898119896minus1 = sup(119904119905)isin[119886119887]times[119888119889]119896minus1(119904 119905) we get119863(119866119896 (119904 119905) 119896 (119904 119905))
le 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
(M119863(119866119896minus1 (119909 119910) 119896minus1 (119904119894 119905119895)) +M119898119896minus1 +10038161003816100381610038161003816100381610038161003816100381610038161003816
119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905)10038161003816100381610038161003816100381610038161003816100381610038161003816 119898119896minus1)119889119909 119889119910
+ 120583 100381610038161003816100381611986200K (1199040 1199050 119904 119905)1003816100381610038161003816 119898119896minus1(23)
Also we have
119863(119866119896 (119904 119905) 119896 (119904 119905)) le 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
(M119863lowast (119866119896minus1 119896minus1)) 119889119909 119889119910 + 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
M119898119896minus1119889119909 119889119910
+ 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
10038161003816100381610038161003816100381610038161003816100381610038161003816119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905)
10038161003816100381610038161003816100381610038161003816100381610038161003816 119898119896minus1119889119909119889119910 + 120583 100381610038161003816100381611986200K (1199040 1199050 119904 119905)1003816100381610038161003816 119898119896minus1(24)
Advances in Fuzzy Systems 5
Therefore
119863(119866119896 (119904 119905) 119896 (119904 119905))le 120583M (119889 minus 119888) (119887 minus 119886)119863lowast (119866119896minus1 119896minus1)
+ 120583M (119889 minus 119888) (119887 minus 119886)119898119896minus1+ 120583M (119889 minus 119888) (119887 minus 119886)119898119896minus1 119862(119889 minus 119888) (119887 minus 119886)+ 120583M119862119898119896minus1
le 119861119863lowast (119866119896minus1 119896minus1) + 119861119898119896minus1+ 2119861119898119896minus1 119862(119889 minus 119888) (119887 minus 119886)
= 119861119863lowast (119866119896minus1 119896minus1)+ 119861119898119896minus1(1 + 2119862(119889 minus 119888) (119887 minus 119886))
(25)
where 119862 = max|119862119894119895| 119894 119895 = 0 sdot sdot sdot 119899 Hence we conclude that119863lowast (119866119896minus1 119896minus1) le 119861119863lowast (119866119896minus2 119896minus2)
+ 119861119898119896minus2(1 + 2119862(119889 minus 119888) (119887 minus 119886))
119863lowast (1198661 1) le 119861119863lowast (1198660 0)+ 1198611198980(1 + 2119862(119889 minus 119888) (119887 minus 119886))
(26)
So
119863lowast (119866119896 119896) le (1 + 2119862(119889 minus 119888) (119887 minus 119886))sdot (119861119898119896minus1 + 1198612119898119896minus2 + sdot sdot sdot + 1198611198961198980)
(27)
If119898119897 = max1198980 119898119896minus1 then we obtain
119863lowast (119866119896 119896) lt 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897) (28)
therefore
119863lowast (119866lowast ) le 119861119896+11 minus 1198611198721+ 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897)
(29)
5 Numerical Stability Analysis
To show the numerical stability analysis of the proposedmethod in previous section we consider another startingapproximation 119891(119904 119905) = 0(119904 119905) such that exist120598 gt 0 for which119863(1198660(119904 119905) 0(119904 119905)) lt 120598 forall119904 119905 isin [119886 119887] times [119888 119889] The obtainedsequence of successive approximations is
119896 (119904 119905) = 119891 (119904 119905) oplus 120583 ⊙ (119865119877)sdot int119889119888(119865119877)int119887
119886K (119909 119910 119904 119905) ⊙ 119896minus1 (119909 119910) 119889119909 119889119910 (30)
and using the same iterative method the terms of producedsequence are
V0 (119904 119905) = 0 (119904 119905) = 119891 (119904 119905) V119896 (119904 119905) = 119891 (119904 119905) oplus 120583
⊙ 119899sum119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ V119896minus1 (119904119894 119905119895) (31)
Theorem 12 e proposed method (15) under the assump-tions of eorem 11 is numerically stable with respect to thechoice of the first iteration
Proof At first we obtain that
119863(119896 (119904 119905) V119896 (119904 119905))le 119863 (119896 (119904 119905) 119866119896 (119904 119905)) + 119863 (119866119896 (119904 119905) 119896 (119904 119905))
+ 119863 (119896 (119904 119905) V119896 (119904 119905))le 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897)
+ 119863 (119866119896 (119904 119905) 119896 (119904 119905))+ 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897)
(32)
However
119863(119866119896 (119904 119905) 119896 (119904 119905)) = 119863(119892 (119904 119905) oplus 120583 ⊙ (119865119877)sdot int119889119888(119865119877)
sdot int119887119886K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910 119891 (119904 119905) oplus 120583
⊙ (119865119877) int119889119888(119865119877) int119887
119886K (119909 119910 119904 119905)
⊙ 119896minus1 (119909 119910) 119889119909 119889119910) le 119863 (119892 (119904 119905) 119891 (119904 119905))+ 120583119863((119865119877) int119889
119888(119865119877) int119887
119886K (119909 119910 119904 119905)
6 Advances in Fuzzy Systems
Table 1 The absolute errors on the level sets with 119899 = 2 119896 = 5 for Example 13 by using the proposed method (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 123485119890 minus 501 666819119890 minus 7 1141 119890 minus 502 13501 119890 minus 6 105703119890 minus 503 204985119890 minus 6 982941119890 minus 604 276606119890 minus 6 918729119890 minus 605 349874119890 minus 6 864395119890 minus 606 424788119890 minus 6 819941119890 minus 607 501349119890 minus 6 785365119890 minus 608 579556119890 minus 6 760668119890 minus 609 65941 119890 minus 6 745849119890 minus 610 74091 119890 minus 6 74091 119890 minus 6
⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910 (119865119877) int119889119888(119865119877)
sdot int119887119886K (119909 119910 119904 119905) ⊙ 119896minus1 (119909 119910) 119889119909 119889119910)
le 120598 + 120583int119889119888int119887119886
1003816100381610038161003816K (119909 119910 119904 119905)1003816100381610038161003816sdot 119863 (119866119896minus1 (119909 119910) 119896minus1 (119909 119910)) 119889119909 119889119910
(33)
We conclude that
119863lowast (119866119896 119896) le 120598 + 120583int119889119888int119887119886M119863lowast (119866119896minus1 119896minus1) 119889119909 119889119910
= 120598 + 119861119863lowast (119866119896minus1 119896minus1) (34)
and thus
119863lowast (119866119896 119896) le 120598 + 119861119863lowast (119866119896minus1 119896minus1)119863lowast (119866119896minus1 119896minus1) le 120598 + 119861119863lowast (119866119896minus2 119896minus2)
119863lowast (1198661 1) le 120598 + 119861119863lowast (1198660 0)
(35)
So
119863lowast (119866119896 119896) le 120598 + 119861 (120598 + 119861119863lowast (119866119896minus2 119896minus2))le 120598 + 119861120598 + 1198612 (120598 + 119861119863lowast (119866119896minus3 119896minus3))
le 120598 + 119861120598 + 1198612120598 + 1198613120598 + sdot sdot sdot + 119861119896119863lowast (1198660 0)le 120598 (1 + 119861 + 1198612 + 1198613 + sdot sdot sdot + 119861119896) le 1205981 minus 119861
(36)
Therefore
119863lowast (119896 V119896)le 11 minus 119861 (2119898119897 (1 + 2119862(119889 minus 119888) (119887 minus 119886)) + 120598) (37)
6 Numerical Examples
In this section we use the proposed method in two-dimensional linear fuzzy Fredholm integral equations forsolving two examples By using the proposed method for 119899 =2 119896 = 5 8 16 and 119903 isin 00 01 02 03 04 05 06 07 0809 10 in (119904 119905) = (05 05) we present the absolute errors inTables 1ndash4
Example 13 (see [11]) Consider the linear integral equation
119866 (119904 119905)= 119892 (119904 119905)
oplus (119865119877) int10(119865119877) int1
0K (119909 119910 119904 119905) ⊙ 119866 (119909 119910) 119889119909 119889119910
(38)
with
K (119909 119910 119904 119905) = 119909119910119904119905 (119904 119905 119903) = 119903 (13119903 + 83)(1 + 119904 + 119905 minus 712119904119905) 119892 (119904 119905 119903) = (21199032 minus 4119903 + 5) (1 + 119904 + 119905 minus 712119904119905)
(39)
The exact solution for 2DLFFIE (38) is
lowast (119904 119905 119903) = 119903 (13119903 + 83) (119904 + 119905 + 1) 119866lowast (119904 119905 119903) = (21199032 minus 4119903 + 5) (119904 + 119905 + 1)
(40)
Advances in Fuzzy Systems 7
Table 2 The absolute errors on the level sets with 119899 = 2 119896 = 16 for Example 13 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 177636119890 minus 1501 111022119890 minus 16 177636119890 minus 1502 0 177636119890 minus 1503 222045119890 minus 16 888178119890 minus 1604 444089119890 minus 16 888178119890 minus 1605 444089119890 minus 16 888178119890 minus 1606 444089119890 minus 16 007 888178119890 minus 16 008 0 009 0 010 0 0
Table 3 The absolute errors on the level sets with 119899 = 2 119896 = 5 for Example 14 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 960565119890 minus 1301 480276119890 minus 14 912548119890 minus 1302 960551119890 minus 14 864531119890 minus 1303 144079119890 minus 13 816458119890 minus 1304 19211 119890 minus 13 768441119890 minus 1305 240141119890 minus 13 720479119890 minus 1306 288158119890 minus 13 672407119890 minus 1307 336203119890 minus 13 624334119890 minus 1308 38422 119890 minus 13 576317119890 minus 1309 432265119890 minus 13 5283 119890 minus 1310 480282119890 minus 13 480282119890 minus 13
Table 4 The absolute errors on the level sets with 119899 = 2 119896 = 8 for Example 14 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 001 0 002 0 555112119890 minus 1703 0 004 0 005 0 555112119890 minus 1706 0 007 0 008 0 009 277556119890 minus 17 010 0 0
Example 14 (see [13]) Consider the following fuzzy Fred-holm integral equation (38) with
(119904 119905 119903) = 119903 (119904119905 + 1676 (1199042 + 1199052 minus 2)) 119892 (119904 119905 119903) = (2 minus 119903) (119904119905 + 1676 (1199042 + 1199052 minus 2)) (41)
and kernel
K (119909 119910 119904 119905) = 1169 (1199092 + 1199102 minus 2) (1199042 + 1199052 minus 2) 0 le 119904 119905 119909 119910 le 1 (42)
The exact solution is
lowast (119904 119905 119903) = 119903119904119905
8 Advances in Fuzzy Systems
119866lowast (119904 119905 119903) = (2 minus 119903) 119904119905(43)
7 Conclusion
The 2DLFFIE is solved by utilizing two-dimensional fuzzyLagrange interpolation and iterative method As it wasexpected the method used to approximate the integral inthis equation is a suitable one since convergence analysis andstability analysis have been proved and also absolute errorin examples is nearly zero As a result considering the factthat the proposed method does not lead to solve fuzzy linearsystem it can be utilized as an efficient method to solve thistype of equations As future researches we can use finite anddivided differences methods fuzzy spline interpolation andfuzzy quasi-interpolation for solving two-dimensional linearor nonlinear fuzzy Fredholm integral equations
Conflicts of Interest
The authors declare that there are no conflicts of interestrelated to this paper
References
[1] E Babolian H Sadeghi Goghary and S Abbasbandy ldquoNumer-ical solution of linear Fredholm fuzzy integral equations of thesecond kind by Adomian methodrdquo Applied Mathematics andComputation vol 161 no 3 pp 733ndash744 2005
[2] A M Bica ldquoError estimation in the approximation of thesolution of nonlinear fuzzy Fredholm integral equationsrdquo Infor-mation Sciences vol 178 no 5 pp 1279ndash1292 2008
[3] A M Bica and C Popescu ldquoApproximating the solution ofnonlinear Hammerstein fuzzy integral equationsrdquo Fuzzy Setsand Systems vol 245 pp 1ndash17 2014
[4] M A Fariborzi Araghi and G Kazemi Gelian ldquoSolving fuzzyFredholm linear integral equations using Sinc method anddouble exponential transformationrdquo So Computing vol 19 no4 pp 1063ndash1070 2015
[5] M A Faborzi Araghi and N Parandin ldquoNumerical solution offuzzy Fredholm integral equations by the Lagrange interpola-tion based on the extension principlerdquo So Computing vol 15no 12 pp 2449ndash2456 2011
[6] H Hosseini Fadravi R Buzhabadi and H Saberi Nik ldquoSolvinglinear Fredholm fuzzy integral equations of the second kind byartificial neural networksrdquo Alexandria Engineering Journal vol53 no 1 pp 249ndash257 2014
[7] Y Jafarzadeh ldquoNumerical solution for fuzzy Fredholm integralequations with upper-bound on error by splines interpolationrdquoFuzzy Information and Engineering vol 4 no 3 pp 339ndash3472012
[8] A Molabahrami A Shidfar and A Ghyasi ldquoAn analyticalmethod for solving linear Fredholm fuzzy integral equations ofthe second kindrdquo Computers amp Mathematics with Applicationsvol 61 no 9 pp 2754ndash2761 2011
[9] N Parandin and M A Fariborzi Araghi ldquoThe approximatesolution of linear fuzzy Fredholm integral equations of thesecond kind by using iterative interpolationrdquoWorld Academy ofScience Engineering and Technology vol 49 pp 978ndash984 2009
[10] A M Bica and S Ziari ldquoIterative numerical method for fuzzyVolterra linear integral equations in two dimensionsrdquo SoComputing vol 21 no 5 pp 1097ndash1108 2017
[11] SM Sadatrasoul andR Ezzati ldquoQuadrature Rules and IterativeMethod for Numerical Solution of Two-Dimensional FuzzyIntegral Equationsrdquo Abstract and Applied Analysis vol 2014Article ID 413570 18 pages 2014
[12] SM Sadatrasoul andR Ezzati ldquoIterativemethod for numericalsolution of two-dimensional nonlinear fuzzy integral equa-tionsrdquo Fuzzy Sets and Systems vol 280 pp 91ndash106 2015
[13] F Mirzaee M K Yari and E Hadadiyan ldquoNumerical solutionof two-dimensional fuzzy Fredholm integral equations of thesecond kind using triangular functionsrdquo Beni-Suef UniversityJournal of Basic and Applied Sciences vol 4 no 2 pp 109ndash1182015
[14] H Nouriani and R Ezzati ldquoQuadrature iterative method fornumerical solution of two-dimensional linear fuzzy Fredholmintegral equationsrdquoMathematical Sciences vol 11 no 1 pp 63ndash72 2017
[15] R Ezzati and S Ziari ldquoNumerical solution of two-dimensionalfuzzy Fredholm integral equations of the second kind usingfuzzy bivariate Bernstein polynomialsrdquo International Journal ofFuzzy Systems vol 15 no 1 pp 84ndash89 2013
[16] N Hassasi and R Ezzati ldquoNumerical solution of two-dimensional fuzzy Fredholm integral equations using collo-cation fuzzy wavelet like operatorrdquo International Journal ofIndustrial Mathematics vol 7 p 11 2015
[17] M A Vali M J Agheli and S Gohari Nezhad ldquoHomotopyanalysis method to solve two-dimensional fuzzy Fredholmintegral equationrdquo General Mathematics Notes vol 22 pp 31ndash43 2014
[18] A M Bica and S Ziari ldquoOpen fuzzy cubature rule withapplication tononlinear fuzzyVolterra integral equations in twodimensionsrdquo Fuzzy Sets and Systems In press Corrected Proof
[19] A Rivaz and F Yousefi ldquoKernel iterative method for solvingtwo-dimensional fuzzy Fredholm integral equations of thesecond kindrdquo Journal of Fuzzy Set Valued Analysis vol 2013 p9 2013
[20] A Rivaz and F Yousefi ldquoModified homotopy perturbationmethod for solving two-dimensional fuzzy Fredholm integralequationrdquo International Journal of AppliedMathematics vol 25no 4 pp 591ndash602 2012
[21] A Rivaz F Yousefi and H Salehinejad ldquoUsing block pulsefunctions for solving two-dimensional fuzzy Fredholm integralequations of the second kindrdquo International Journal of AppliedMathematics vol 25 no 4 pp 571ndash582 2012
[22] S M Sadatrasoul and R Ezzati ldquoNumerical solution of two-dimensional nonlinear Hammerstein fuzzy integral equationsbased on optimal fuzzy quadrature formulardquo Journal of Compu-tational and Applied Mathematics vol 292 pp 430ndash446 2016
[23] S Ziari ldquoIterative method for solving two-dimensional non-linear fuzzy integral equations using fuzzy bivariate block-pulse functions with error estimationrdquo Iranian Journal of FuzzySystems vol 15 no 1 pp 55ndash76 183 2018
[24] G A Anastassiou Fuzzy Mathematics Approximation eoryvol 251 Springer 2010
[25] A R BozorgmaneshMOtadi A A Safe Kordi et al ldquoBarkhor-dari Ahmadi Lagrange two-dimensional interpolationmethodfor modeling nanoparticle formation during RESS processrdquoInternational Journal of Industrial Mathematics vol 1 pp 175ndash181 2009
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Advances in Fuzzy Systems 3
(5) 120596[119886119887]times[119888119889](119891 119899120575) le 119899120596[119886119887]times[119888119889](119891 120575) forall120575 ge 0 119899 isin N
(6) 120596[119886119887]times[119888119889](119891 120583120575) le (120583 + 1)120596[119886119887]times[119888119889](119891 120575) forall120583 120575 ge 0Theorem 9 (see [11]) If 119891 and 119892 are Henstock integrablemapping on [119886 119887] times [119888 119889] and if 119863(119891(119904 119905) 119892(119904 119905)) is Lebesgueintegrable then
119863((119865119867)int119889119888int119887119886119891 (119904 119905) 119889119904119889119905 (119865119867)int119889
119888int119887119886119892 (119904 119905) 119889119904119889119905)
le (119871) int119889119888int119887119886119863(119891 (119904 119905) 119892 (119904 119905)) 119889119904 119889119905
(8)
3 The Main Result
In this section first we introduce two-dimensional fuzzyLagrange interpolation Then we propose two-dimensionalfuzzy Lagrange interpolation and iterative method for solving(9)
Consider 2DLFFIE as follows
119866 (119904 119905)= 119892 (119904 119905) oplus 120583
⊙ (119865119877) int119889119888(119865119877) int119887
119886K (119909 119910 119904 119905) ⊙ 119866 (119909 119910) 119889119909 119889119910
(9)
where 120583 gt 0K(119909 119910 119904 119905) is an arbitrary positive function on[119886 119887] times [119888 119889] times [119886 119887] times [119888 119889] and 119892 [119886 119887] times [119888 119889] 997888rarr R119865 Weassume that K is continuous and therefore it is uniformlycontinuous with respect to (119904 119905) So there existsM gt 0 suchthatM = max119904119909isin[119886119887]119910119905isin[119888119889]|K(119909 119910 119904 119905)|
Two-dimensional Lagrange polynomials ℓ119894119895(119904 119905) aredefined as follows
ℓ119894119895 (119904 119905) = ℓ119894 (119904) otimes ℓ119895 (119905) 0 le 119894 119895 le 119899 (10)
where ℓ119894(119904) is the Lagrange polynomial and is defined asfollows
ℓ119894 (119904) = 119899prod119903=0119903 =119894
(119904 minus 119904119903)(119904119894 minus 119904119903) (11)
therefore
ℓ119894119895 (119904119903 1199051199031015840) = 1 if 119894 = 119903 119895 = 11990310158400 otherwise (12)
So the two-dimensional interpolation in the Lagrange formis (see [25])
119901 (119909 119910) = 119899sum119895=0
119899sum119894=0
119891 (119904119894 119905119895) ⊙ ℓ119894119895 (119909 119910) (13)
where the coefficients 119891(119904119894 119905119895) are the fuzzy numbers
Here we consider the two-dimensional fuzzy Lagrangeinterpolation in the given points 119886 = 1199040 lt 1199041 lt sdot sdot sdot lt 119904119899 = 119887and 119888 = 1199050 lt 1199051 lt sdot sdot sdot lt 119905119899 = 119889 such that
K (119909 119910 119904 119905) ⊙ 119866 (119909 119910)asymp 119899sum119895=0
119899sum119894=0
ℓ119894119895 (119909 119910) ⊙K (119904119894 119905119895 119904 119905) ⊙ 119866 (119904119894 119905119895) (14)
Now we propose a numerical method to solve (9) To do thiswe suppose the following iterative procedure to approximatethe solution of (9) in point (119904 119905)
0 (119904 119905) = 119892 (119904 119905) 119896 (119904 119905) = 119892 (119904 119905) oplus 120583
⊙ 119899sum119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895) (15)
where
119862119894119895 = int119889119888int119887119886ℓ119894119895 (119909 119910) 119889119909119889119910 (16)
In Theorem 10 authors of [11] proved the existence anduniqueness solution of (9) by using the Banach fixed pointtheorem
Theorem 10 (see [11]) Let the function K(119909 119910 119904 119905) be con-tinuous and positive for 119909 119904 isin [119886 119887] and 119910 119905 isin [119888 119889] and let119892 [119886 119887] times [119888 119889] 997888rarr R119865 be continuous on [119886 119887] times [119888 119889] If119861 = 120583M(119889 minus 119888)(119887 minus 119886) lt 1 then the fuzzy integral equation (9)has a unique solution 119866lowast isin 119883 where
119883 = 119892 [119886 119887] times [119888 119889] 997888rarr R119865 119892 is continous (17)
be the space of two-dimensional fuzzy continuous functionswith the metric 119863lowast and it can be obtained by the followingsuccessive approximations method
1198660 (119904 119905) = 119892 (119904 119905) 119866119896 (119904 119905) = 119892 (119904 119905) oplus 120583 ⊙ (119865119877)
sdot int119889119888(119865119877) int119887
119886K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910
forall119896 ge 1(18)
Moreover the sequence of successive approximations (119866119896)119896ge1converges to the solution 119866lowast Furthermore the following errorbound holds
119863lowast (119866lowast 119866119896) le 119861119896+11 minus 119861M1 forall119896 ge 1 (19)
whereM1 = sup119904isin[119886119887]119905isin[119888119889]119866(119904 119905)
4 Advances in Fuzzy Systems
4 Convergence Analysis
In this section we obtain an error estimate between the exactsolution and the approximate solution of 2DLFFIE (9)
Theorem 11 Under the hypotheses of eorem 10 and 120583 gt 0the iterative procedure (15) converges to the unique solution of(9) 119866lowast and its error estimate is as follows
119863lowast (119866lowast 119896)le 119861119896+1(1 minus 119861)M1
+ 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897) (20)
Proof Clearly we have
119863(119866lowast (119904 119905) 119896 (119904 119905)) le 119863 (119866lowast (119904 119905) 119866119896 (119904 119905))+ 119863 (119866119896 (119904 119905) 119896 (119904 119905)) (21)
From (18) and (15) we conclude that
119863(119866119896 (119904 119905) 119896 (119904 119905)) = 119863(119892 (119904 119905) oplus 120583 ⊙ (119865119877)int119889119888(119865119877)int119887
119886K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909119889119910 119892 (119904 119905) oplus 120583 ⊙ 119899sum
119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895)) = 120583119863((119865119877) int119889119888(119865119877)
sdot int119887119886K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910 119899sum
119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895)) = 120583119863( 119899sum119895=1
119899sum119894=1
(119865119877) int119905119895119905119895minus1
(119865119877) int119904119894119904119894minus1
K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909119889119910119899sum119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895)) = 120583119863( 119899sum119895=1
119899sum119894=1
(119865119877) int119905119895119905119895minus1
(119865119877) int119904119894119904119894minus1
K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909119889119910 119899sum119895=1
119899sum119894=1
(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1) 119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905)
⊙ 119896minus1 (119904119894 119905119895) oplus 11986200K (1199040 1199050 119904 119905) ⊙ 119896minus1 (1199040 1199050))
le 120583 119899sum119895=1
119899sum119894=1
119863((119865119877) int119905119895119905119895minus1
(119865119877) int119904119894119904119894minus1
K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909119889119910 (119865119877) int119905119895119905119895minus1
(119865119877) int119904119894119904119894minus1
119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895) 119889119909 119889119910) oplus 120583119863(11986200K (1199040 1199050 119904 119905)⊙ 119896minus1 (1199040 1199050) 0)le 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
[119863(K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) K (119909 119910 119904 119905) ⊙ 119896minus1 (119904119894 119905119895)) + 119863(K (119909 119910 119904 119905) ⊙ 119896minus1 (119904119894 119905119895) 119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895))]119889119909 119889119910+ 120583119863 (11986200K (1199040 1199050 119904 119905) ⊙ 119896minus1 (1199040 1199050) 0)le 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
(1003816100381610038161003816K (119909 119910 119904 119905)1003816100381610038161003816 119863 (119866119896minus1 (119909 119910) 119896minus1 (119904119894 119905119895)) + 1003816100381610038161003816K (119909 119910 119904 119905)1003816100381610038161003816 119863 (119896minus1 (119904119894 119905119895) 0) +10038161003816100381610038161003816100381610038161003816100381610038161003816
119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905)10038161003816100381610038161003816100381610038161003816100381610038161003816 119863 (119896minus1 (119904119894 119905119895) 0))119889119909 119889119910
+ 120583119863 (11986200K (1199040 1199050 119904 119905) ⊙ 119896minus1 (1199040 1199050) 0)
(22)
By supposing119898119896minus1 = sup(119904119905)isin[119886119887]times[119888119889]119896minus1(119904 119905) we get119863(119866119896 (119904 119905) 119896 (119904 119905))
le 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
(M119863(119866119896minus1 (119909 119910) 119896minus1 (119904119894 119905119895)) +M119898119896minus1 +10038161003816100381610038161003816100381610038161003816100381610038161003816
119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905)10038161003816100381610038161003816100381610038161003816100381610038161003816 119898119896minus1)119889119909 119889119910
+ 120583 100381610038161003816100381611986200K (1199040 1199050 119904 119905)1003816100381610038161003816 119898119896minus1(23)
Also we have
119863(119866119896 (119904 119905) 119896 (119904 119905)) le 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
(M119863lowast (119866119896minus1 119896minus1)) 119889119909 119889119910 + 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
M119898119896minus1119889119909 119889119910
+ 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
10038161003816100381610038161003816100381610038161003816100381610038161003816119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905)
10038161003816100381610038161003816100381610038161003816100381610038161003816 119898119896minus1119889119909119889119910 + 120583 100381610038161003816100381611986200K (1199040 1199050 119904 119905)1003816100381610038161003816 119898119896minus1(24)
Advances in Fuzzy Systems 5
Therefore
119863(119866119896 (119904 119905) 119896 (119904 119905))le 120583M (119889 minus 119888) (119887 minus 119886)119863lowast (119866119896minus1 119896minus1)
+ 120583M (119889 minus 119888) (119887 minus 119886)119898119896minus1+ 120583M (119889 minus 119888) (119887 minus 119886)119898119896minus1 119862(119889 minus 119888) (119887 minus 119886)+ 120583M119862119898119896minus1
le 119861119863lowast (119866119896minus1 119896minus1) + 119861119898119896minus1+ 2119861119898119896minus1 119862(119889 minus 119888) (119887 minus 119886)
= 119861119863lowast (119866119896minus1 119896minus1)+ 119861119898119896minus1(1 + 2119862(119889 minus 119888) (119887 minus 119886))
(25)
where 119862 = max|119862119894119895| 119894 119895 = 0 sdot sdot sdot 119899 Hence we conclude that119863lowast (119866119896minus1 119896minus1) le 119861119863lowast (119866119896minus2 119896minus2)
+ 119861119898119896minus2(1 + 2119862(119889 minus 119888) (119887 minus 119886))
119863lowast (1198661 1) le 119861119863lowast (1198660 0)+ 1198611198980(1 + 2119862(119889 minus 119888) (119887 minus 119886))
(26)
So
119863lowast (119866119896 119896) le (1 + 2119862(119889 minus 119888) (119887 minus 119886))sdot (119861119898119896minus1 + 1198612119898119896minus2 + sdot sdot sdot + 1198611198961198980)
(27)
If119898119897 = max1198980 119898119896minus1 then we obtain
119863lowast (119866119896 119896) lt 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897) (28)
therefore
119863lowast (119866lowast ) le 119861119896+11 minus 1198611198721+ 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897)
(29)
5 Numerical Stability Analysis
To show the numerical stability analysis of the proposedmethod in previous section we consider another startingapproximation 119891(119904 119905) = 0(119904 119905) such that exist120598 gt 0 for which119863(1198660(119904 119905) 0(119904 119905)) lt 120598 forall119904 119905 isin [119886 119887] times [119888 119889] The obtainedsequence of successive approximations is
119896 (119904 119905) = 119891 (119904 119905) oplus 120583 ⊙ (119865119877)sdot int119889119888(119865119877)int119887
119886K (119909 119910 119904 119905) ⊙ 119896minus1 (119909 119910) 119889119909 119889119910 (30)
and using the same iterative method the terms of producedsequence are
V0 (119904 119905) = 0 (119904 119905) = 119891 (119904 119905) V119896 (119904 119905) = 119891 (119904 119905) oplus 120583
⊙ 119899sum119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ V119896minus1 (119904119894 119905119895) (31)
Theorem 12 e proposed method (15) under the assump-tions of eorem 11 is numerically stable with respect to thechoice of the first iteration
Proof At first we obtain that
119863(119896 (119904 119905) V119896 (119904 119905))le 119863 (119896 (119904 119905) 119866119896 (119904 119905)) + 119863 (119866119896 (119904 119905) 119896 (119904 119905))
+ 119863 (119896 (119904 119905) V119896 (119904 119905))le 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897)
+ 119863 (119866119896 (119904 119905) 119896 (119904 119905))+ 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897)
(32)
However
119863(119866119896 (119904 119905) 119896 (119904 119905)) = 119863(119892 (119904 119905) oplus 120583 ⊙ (119865119877)sdot int119889119888(119865119877)
sdot int119887119886K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910 119891 (119904 119905) oplus 120583
⊙ (119865119877) int119889119888(119865119877) int119887
119886K (119909 119910 119904 119905)
⊙ 119896minus1 (119909 119910) 119889119909 119889119910) le 119863 (119892 (119904 119905) 119891 (119904 119905))+ 120583119863((119865119877) int119889
119888(119865119877) int119887
119886K (119909 119910 119904 119905)
6 Advances in Fuzzy Systems
Table 1 The absolute errors on the level sets with 119899 = 2 119896 = 5 for Example 13 by using the proposed method (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 123485119890 minus 501 666819119890 minus 7 1141 119890 minus 502 13501 119890 minus 6 105703119890 minus 503 204985119890 minus 6 982941119890 minus 604 276606119890 minus 6 918729119890 minus 605 349874119890 minus 6 864395119890 minus 606 424788119890 minus 6 819941119890 minus 607 501349119890 minus 6 785365119890 minus 608 579556119890 minus 6 760668119890 minus 609 65941 119890 minus 6 745849119890 minus 610 74091 119890 minus 6 74091 119890 minus 6
⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910 (119865119877) int119889119888(119865119877)
sdot int119887119886K (119909 119910 119904 119905) ⊙ 119896minus1 (119909 119910) 119889119909 119889119910)
le 120598 + 120583int119889119888int119887119886
1003816100381610038161003816K (119909 119910 119904 119905)1003816100381610038161003816sdot 119863 (119866119896minus1 (119909 119910) 119896minus1 (119909 119910)) 119889119909 119889119910
(33)
We conclude that
119863lowast (119866119896 119896) le 120598 + 120583int119889119888int119887119886M119863lowast (119866119896minus1 119896minus1) 119889119909 119889119910
= 120598 + 119861119863lowast (119866119896minus1 119896minus1) (34)
and thus
119863lowast (119866119896 119896) le 120598 + 119861119863lowast (119866119896minus1 119896minus1)119863lowast (119866119896minus1 119896minus1) le 120598 + 119861119863lowast (119866119896minus2 119896minus2)
119863lowast (1198661 1) le 120598 + 119861119863lowast (1198660 0)
(35)
So
119863lowast (119866119896 119896) le 120598 + 119861 (120598 + 119861119863lowast (119866119896minus2 119896minus2))le 120598 + 119861120598 + 1198612 (120598 + 119861119863lowast (119866119896minus3 119896minus3))
le 120598 + 119861120598 + 1198612120598 + 1198613120598 + sdot sdot sdot + 119861119896119863lowast (1198660 0)le 120598 (1 + 119861 + 1198612 + 1198613 + sdot sdot sdot + 119861119896) le 1205981 minus 119861
(36)
Therefore
119863lowast (119896 V119896)le 11 minus 119861 (2119898119897 (1 + 2119862(119889 minus 119888) (119887 minus 119886)) + 120598) (37)
6 Numerical Examples
In this section we use the proposed method in two-dimensional linear fuzzy Fredholm integral equations forsolving two examples By using the proposed method for 119899 =2 119896 = 5 8 16 and 119903 isin 00 01 02 03 04 05 06 07 0809 10 in (119904 119905) = (05 05) we present the absolute errors inTables 1ndash4
Example 13 (see [11]) Consider the linear integral equation
119866 (119904 119905)= 119892 (119904 119905)
oplus (119865119877) int10(119865119877) int1
0K (119909 119910 119904 119905) ⊙ 119866 (119909 119910) 119889119909 119889119910
(38)
with
K (119909 119910 119904 119905) = 119909119910119904119905 (119904 119905 119903) = 119903 (13119903 + 83)(1 + 119904 + 119905 minus 712119904119905) 119892 (119904 119905 119903) = (21199032 minus 4119903 + 5) (1 + 119904 + 119905 minus 712119904119905)
(39)
The exact solution for 2DLFFIE (38) is
lowast (119904 119905 119903) = 119903 (13119903 + 83) (119904 + 119905 + 1) 119866lowast (119904 119905 119903) = (21199032 minus 4119903 + 5) (119904 + 119905 + 1)
(40)
Advances in Fuzzy Systems 7
Table 2 The absolute errors on the level sets with 119899 = 2 119896 = 16 for Example 13 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 177636119890 minus 1501 111022119890 minus 16 177636119890 minus 1502 0 177636119890 minus 1503 222045119890 minus 16 888178119890 minus 1604 444089119890 minus 16 888178119890 minus 1605 444089119890 minus 16 888178119890 minus 1606 444089119890 minus 16 007 888178119890 minus 16 008 0 009 0 010 0 0
Table 3 The absolute errors on the level sets with 119899 = 2 119896 = 5 for Example 14 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 960565119890 minus 1301 480276119890 minus 14 912548119890 minus 1302 960551119890 minus 14 864531119890 minus 1303 144079119890 minus 13 816458119890 minus 1304 19211 119890 minus 13 768441119890 minus 1305 240141119890 minus 13 720479119890 minus 1306 288158119890 minus 13 672407119890 minus 1307 336203119890 minus 13 624334119890 minus 1308 38422 119890 minus 13 576317119890 minus 1309 432265119890 minus 13 5283 119890 minus 1310 480282119890 minus 13 480282119890 minus 13
Table 4 The absolute errors on the level sets with 119899 = 2 119896 = 8 for Example 14 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 001 0 002 0 555112119890 minus 1703 0 004 0 005 0 555112119890 minus 1706 0 007 0 008 0 009 277556119890 minus 17 010 0 0
Example 14 (see [13]) Consider the following fuzzy Fred-holm integral equation (38) with
(119904 119905 119903) = 119903 (119904119905 + 1676 (1199042 + 1199052 minus 2)) 119892 (119904 119905 119903) = (2 minus 119903) (119904119905 + 1676 (1199042 + 1199052 minus 2)) (41)
and kernel
K (119909 119910 119904 119905) = 1169 (1199092 + 1199102 minus 2) (1199042 + 1199052 minus 2) 0 le 119904 119905 119909 119910 le 1 (42)
The exact solution is
lowast (119904 119905 119903) = 119903119904119905
8 Advances in Fuzzy Systems
119866lowast (119904 119905 119903) = (2 minus 119903) 119904119905(43)
7 Conclusion
The 2DLFFIE is solved by utilizing two-dimensional fuzzyLagrange interpolation and iterative method As it wasexpected the method used to approximate the integral inthis equation is a suitable one since convergence analysis andstability analysis have been proved and also absolute errorin examples is nearly zero As a result considering the factthat the proposed method does not lead to solve fuzzy linearsystem it can be utilized as an efficient method to solve thistype of equations As future researches we can use finite anddivided differences methods fuzzy spline interpolation andfuzzy quasi-interpolation for solving two-dimensional linearor nonlinear fuzzy Fredholm integral equations
Conflicts of Interest
The authors declare that there are no conflicts of interestrelated to this paper
References
[1] E Babolian H Sadeghi Goghary and S Abbasbandy ldquoNumer-ical solution of linear Fredholm fuzzy integral equations of thesecond kind by Adomian methodrdquo Applied Mathematics andComputation vol 161 no 3 pp 733ndash744 2005
[2] A M Bica ldquoError estimation in the approximation of thesolution of nonlinear fuzzy Fredholm integral equationsrdquo Infor-mation Sciences vol 178 no 5 pp 1279ndash1292 2008
[3] A M Bica and C Popescu ldquoApproximating the solution ofnonlinear Hammerstein fuzzy integral equationsrdquo Fuzzy Setsand Systems vol 245 pp 1ndash17 2014
[4] M A Fariborzi Araghi and G Kazemi Gelian ldquoSolving fuzzyFredholm linear integral equations using Sinc method anddouble exponential transformationrdquo So Computing vol 19 no4 pp 1063ndash1070 2015
[5] M A Faborzi Araghi and N Parandin ldquoNumerical solution offuzzy Fredholm integral equations by the Lagrange interpola-tion based on the extension principlerdquo So Computing vol 15no 12 pp 2449ndash2456 2011
[6] H Hosseini Fadravi R Buzhabadi and H Saberi Nik ldquoSolvinglinear Fredholm fuzzy integral equations of the second kind byartificial neural networksrdquo Alexandria Engineering Journal vol53 no 1 pp 249ndash257 2014
[7] Y Jafarzadeh ldquoNumerical solution for fuzzy Fredholm integralequations with upper-bound on error by splines interpolationrdquoFuzzy Information and Engineering vol 4 no 3 pp 339ndash3472012
[8] A Molabahrami A Shidfar and A Ghyasi ldquoAn analyticalmethod for solving linear Fredholm fuzzy integral equations ofthe second kindrdquo Computers amp Mathematics with Applicationsvol 61 no 9 pp 2754ndash2761 2011
[9] N Parandin and M A Fariborzi Araghi ldquoThe approximatesolution of linear fuzzy Fredholm integral equations of thesecond kind by using iterative interpolationrdquoWorld Academy ofScience Engineering and Technology vol 49 pp 978ndash984 2009
[10] A M Bica and S Ziari ldquoIterative numerical method for fuzzyVolterra linear integral equations in two dimensionsrdquo SoComputing vol 21 no 5 pp 1097ndash1108 2017
[11] SM Sadatrasoul andR Ezzati ldquoQuadrature Rules and IterativeMethod for Numerical Solution of Two-Dimensional FuzzyIntegral Equationsrdquo Abstract and Applied Analysis vol 2014Article ID 413570 18 pages 2014
[12] SM Sadatrasoul andR Ezzati ldquoIterativemethod for numericalsolution of two-dimensional nonlinear fuzzy integral equa-tionsrdquo Fuzzy Sets and Systems vol 280 pp 91ndash106 2015
[13] F Mirzaee M K Yari and E Hadadiyan ldquoNumerical solutionof two-dimensional fuzzy Fredholm integral equations of thesecond kind using triangular functionsrdquo Beni-Suef UniversityJournal of Basic and Applied Sciences vol 4 no 2 pp 109ndash1182015
[14] H Nouriani and R Ezzati ldquoQuadrature iterative method fornumerical solution of two-dimensional linear fuzzy Fredholmintegral equationsrdquoMathematical Sciences vol 11 no 1 pp 63ndash72 2017
[15] R Ezzati and S Ziari ldquoNumerical solution of two-dimensionalfuzzy Fredholm integral equations of the second kind usingfuzzy bivariate Bernstein polynomialsrdquo International Journal ofFuzzy Systems vol 15 no 1 pp 84ndash89 2013
[16] N Hassasi and R Ezzati ldquoNumerical solution of two-dimensional fuzzy Fredholm integral equations using collo-cation fuzzy wavelet like operatorrdquo International Journal ofIndustrial Mathematics vol 7 p 11 2015
[17] M A Vali M J Agheli and S Gohari Nezhad ldquoHomotopyanalysis method to solve two-dimensional fuzzy Fredholmintegral equationrdquo General Mathematics Notes vol 22 pp 31ndash43 2014
[18] A M Bica and S Ziari ldquoOpen fuzzy cubature rule withapplication tononlinear fuzzyVolterra integral equations in twodimensionsrdquo Fuzzy Sets and Systems In press Corrected Proof
[19] A Rivaz and F Yousefi ldquoKernel iterative method for solvingtwo-dimensional fuzzy Fredholm integral equations of thesecond kindrdquo Journal of Fuzzy Set Valued Analysis vol 2013 p9 2013
[20] A Rivaz and F Yousefi ldquoModified homotopy perturbationmethod for solving two-dimensional fuzzy Fredholm integralequationrdquo International Journal of AppliedMathematics vol 25no 4 pp 591ndash602 2012
[21] A Rivaz F Yousefi and H Salehinejad ldquoUsing block pulsefunctions for solving two-dimensional fuzzy Fredholm integralequations of the second kindrdquo International Journal of AppliedMathematics vol 25 no 4 pp 571ndash582 2012
[22] S M Sadatrasoul and R Ezzati ldquoNumerical solution of two-dimensional nonlinear Hammerstein fuzzy integral equationsbased on optimal fuzzy quadrature formulardquo Journal of Compu-tational and Applied Mathematics vol 292 pp 430ndash446 2016
[23] S Ziari ldquoIterative method for solving two-dimensional non-linear fuzzy integral equations using fuzzy bivariate block-pulse functions with error estimationrdquo Iranian Journal of FuzzySystems vol 15 no 1 pp 55ndash76 183 2018
[24] G A Anastassiou Fuzzy Mathematics Approximation eoryvol 251 Springer 2010
[25] A R BozorgmaneshMOtadi A A Safe Kordi et al ldquoBarkhor-dari Ahmadi Lagrange two-dimensional interpolationmethodfor modeling nanoparticle formation during RESS processrdquoInternational Journal of Industrial Mathematics vol 1 pp 175ndash181 2009
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4 Advances in Fuzzy Systems
4 Convergence Analysis
In this section we obtain an error estimate between the exactsolution and the approximate solution of 2DLFFIE (9)
Theorem 11 Under the hypotheses of eorem 10 and 120583 gt 0the iterative procedure (15) converges to the unique solution of(9) 119866lowast and its error estimate is as follows
119863lowast (119866lowast 119896)le 119861119896+1(1 minus 119861)M1
+ 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897) (20)
Proof Clearly we have
119863(119866lowast (119904 119905) 119896 (119904 119905)) le 119863 (119866lowast (119904 119905) 119866119896 (119904 119905))+ 119863 (119866119896 (119904 119905) 119896 (119904 119905)) (21)
From (18) and (15) we conclude that
119863(119866119896 (119904 119905) 119896 (119904 119905)) = 119863(119892 (119904 119905) oplus 120583 ⊙ (119865119877)int119889119888(119865119877)int119887
119886K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909119889119910 119892 (119904 119905) oplus 120583 ⊙ 119899sum
119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895)) = 120583119863((119865119877) int119889119888(119865119877)
sdot int119887119886K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910 119899sum
119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895)) = 120583119863( 119899sum119895=1
119899sum119894=1
(119865119877) int119905119895119905119895minus1
(119865119877) int119904119894119904119894minus1
K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909119889119910119899sum119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895)) = 120583119863( 119899sum119895=1
119899sum119894=1
(119865119877) int119905119895119905119895minus1
(119865119877) int119904119894119904119894minus1
K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909119889119910 119899sum119895=1
119899sum119894=1
(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1) 119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905)
⊙ 119896minus1 (119904119894 119905119895) oplus 11986200K (1199040 1199050 119904 119905) ⊙ 119896minus1 (1199040 1199050))
le 120583 119899sum119895=1
119899sum119894=1
119863((119865119877) int119905119895119905119895minus1
(119865119877) int119904119894119904119894minus1
K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909119889119910 (119865119877) int119905119895119905119895minus1
(119865119877) int119904119894119904119894minus1
119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895) 119889119909 119889119910) oplus 120583119863(11986200K (1199040 1199050 119904 119905)⊙ 119896minus1 (1199040 1199050) 0)le 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
[119863(K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) K (119909 119910 119904 119905) ⊙ 119896minus1 (119904119894 119905119895)) + 119863(K (119909 119910 119904 119905) ⊙ 119896minus1 (119904119894 119905119895) 119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905) ⊙ 119896minus1 (119904119894 119905119895))]119889119909 119889119910+ 120583119863 (11986200K (1199040 1199050 119904 119905) ⊙ 119896minus1 (1199040 1199050) 0)le 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
(1003816100381610038161003816K (119909 119910 119904 119905)1003816100381610038161003816 119863 (119866119896minus1 (119909 119910) 119896minus1 (119904119894 119905119895)) + 1003816100381610038161003816K (119909 119910 119904 119905)1003816100381610038161003816 119863 (119896minus1 (119904119894 119905119895) 0) +10038161003816100381610038161003816100381610038161003816100381610038161003816
119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905)10038161003816100381610038161003816100381610038161003816100381610038161003816 119863 (119896minus1 (119904119894 119905119895) 0))119889119909 119889119910
+ 120583119863 (11986200K (1199040 1199050 119904 119905) ⊙ 119896minus1 (1199040 1199050) 0)
(22)
By supposing119898119896minus1 = sup(119904119905)isin[119886119887]times[119888119889]119896minus1(119904 119905) we get119863(119866119896 (119904 119905) 119896 (119904 119905))
le 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
(M119863(119866119896minus1 (119909 119910) 119896minus1 (119904119894 119905119895)) +M119898119896minus1 +10038161003816100381610038161003816100381610038161003816100381610038161003816
119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905)10038161003816100381610038161003816100381610038161003816100381610038161003816 119898119896minus1)119889119909 119889119910
+ 120583 100381610038161003816100381611986200K (1199040 1199050 119904 119905)1003816100381610038161003816 119898119896minus1(23)
Also we have
119863(119866119896 (119904 119905) 119896 (119904 119905)) le 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
(M119863lowast (119866119896minus1 119896minus1)) 119889119909 119889119910 + 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
M119898119896minus1119889119909 119889119910
+ 120583 119899sum119895=1
119899sum119894=1
int119905119895119905119895minus1
int119904119894119904119894minus1
10038161003816100381610038161003816100381610038161003816100381610038161003816119862119894119895(119905119895 minus 119905119895minus1) (119904119894 minus 119904119894minus1)K (119904119894 119905119895 119904 119905)
10038161003816100381610038161003816100381610038161003816100381610038161003816 119898119896minus1119889119909119889119910 + 120583 100381610038161003816100381611986200K (1199040 1199050 119904 119905)1003816100381610038161003816 119898119896minus1(24)
Advances in Fuzzy Systems 5
Therefore
119863(119866119896 (119904 119905) 119896 (119904 119905))le 120583M (119889 minus 119888) (119887 minus 119886)119863lowast (119866119896minus1 119896minus1)
+ 120583M (119889 minus 119888) (119887 minus 119886)119898119896minus1+ 120583M (119889 minus 119888) (119887 minus 119886)119898119896minus1 119862(119889 minus 119888) (119887 minus 119886)+ 120583M119862119898119896minus1
le 119861119863lowast (119866119896minus1 119896minus1) + 119861119898119896minus1+ 2119861119898119896minus1 119862(119889 minus 119888) (119887 minus 119886)
= 119861119863lowast (119866119896minus1 119896minus1)+ 119861119898119896minus1(1 + 2119862(119889 minus 119888) (119887 minus 119886))
(25)
where 119862 = max|119862119894119895| 119894 119895 = 0 sdot sdot sdot 119899 Hence we conclude that119863lowast (119866119896minus1 119896minus1) le 119861119863lowast (119866119896minus2 119896minus2)
+ 119861119898119896minus2(1 + 2119862(119889 minus 119888) (119887 minus 119886))
119863lowast (1198661 1) le 119861119863lowast (1198660 0)+ 1198611198980(1 + 2119862(119889 minus 119888) (119887 minus 119886))
(26)
So
119863lowast (119866119896 119896) le (1 + 2119862(119889 minus 119888) (119887 minus 119886))sdot (119861119898119896minus1 + 1198612119898119896minus2 + sdot sdot sdot + 1198611198961198980)
(27)
If119898119897 = max1198980 119898119896minus1 then we obtain
119863lowast (119866119896 119896) lt 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897) (28)
therefore
119863lowast (119866lowast ) le 119861119896+11 minus 1198611198721+ 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897)
(29)
5 Numerical Stability Analysis
To show the numerical stability analysis of the proposedmethod in previous section we consider another startingapproximation 119891(119904 119905) = 0(119904 119905) such that exist120598 gt 0 for which119863(1198660(119904 119905) 0(119904 119905)) lt 120598 forall119904 119905 isin [119886 119887] times [119888 119889] The obtainedsequence of successive approximations is
119896 (119904 119905) = 119891 (119904 119905) oplus 120583 ⊙ (119865119877)sdot int119889119888(119865119877)int119887
119886K (119909 119910 119904 119905) ⊙ 119896minus1 (119909 119910) 119889119909 119889119910 (30)
and using the same iterative method the terms of producedsequence are
V0 (119904 119905) = 0 (119904 119905) = 119891 (119904 119905) V119896 (119904 119905) = 119891 (119904 119905) oplus 120583
⊙ 119899sum119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ V119896minus1 (119904119894 119905119895) (31)
Theorem 12 e proposed method (15) under the assump-tions of eorem 11 is numerically stable with respect to thechoice of the first iteration
Proof At first we obtain that
119863(119896 (119904 119905) V119896 (119904 119905))le 119863 (119896 (119904 119905) 119866119896 (119904 119905)) + 119863 (119866119896 (119904 119905) 119896 (119904 119905))
+ 119863 (119896 (119904 119905) V119896 (119904 119905))le 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897)
+ 119863 (119866119896 (119904 119905) 119896 (119904 119905))+ 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897)
(32)
However
119863(119866119896 (119904 119905) 119896 (119904 119905)) = 119863(119892 (119904 119905) oplus 120583 ⊙ (119865119877)sdot int119889119888(119865119877)
sdot int119887119886K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910 119891 (119904 119905) oplus 120583
⊙ (119865119877) int119889119888(119865119877) int119887
119886K (119909 119910 119904 119905)
⊙ 119896minus1 (119909 119910) 119889119909 119889119910) le 119863 (119892 (119904 119905) 119891 (119904 119905))+ 120583119863((119865119877) int119889
119888(119865119877) int119887
119886K (119909 119910 119904 119905)
6 Advances in Fuzzy Systems
Table 1 The absolute errors on the level sets with 119899 = 2 119896 = 5 for Example 13 by using the proposed method (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 123485119890 minus 501 666819119890 minus 7 1141 119890 minus 502 13501 119890 minus 6 105703119890 minus 503 204985119890 minus 6 982941119890 minus 604 276606119890 minus 6 918729119890 minus 605 349874119890 minus 6 864395119890 minus 606 424788119890 minus 6 819941119890 minus 607 501349119890 minus 6 785365119890 minus 608 579556119890 minus 6 760668119890 minus 609 65941 119890 minus 6 745849119890 minus 610 74091 119890 minus 6 74091 119890 minus 6
⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910 (119865119877) int119889119888(119865119877)
sdot int119887119886K (119909 119910 119904 119905) ⊙ 119896minus1 (119909 119910) 119889119909 119889119910)
le 120598 + 120583int119889119888int119887119886
1003816100381610038161003816K (119909 119910 119904 119905)1003816100381610038161003816sdot 119863 (119866119896minus1 (119909 119910) 119896minus1 (119909 119910)) 119889119909 119889119910
(33)
We conclude that
119863lowast (119866119896 119896) le 120598 + 120583int119889119888int119887119886M119863lowast (119866119896minus1 119896minus1) 119889119909 119889119910
= 120598 + 119861119863lowast (119866119896minus1 119896minus1) (34)
and thus
119863lowast (119866119896 119896) le 120598 + 119861119863lowast (119866119896minus1 119896minus1)119863lowast (119866119896minus1 119896minus1) le 120598 + 119861119863lowast (119866119896minus2 119896minus2)
119863lowast (1198661 1) le 120598 + 119861119863lowast (1198660 0)
(35)
So
119863lowast (119866119896 119896) le 120598 + 119861 (120598 + 119861119863lowast (119866119896minus2 119896minus2))le 120598 + 119861120598 + 1198612 (120598 + 119861119863lowast (119866119896minus3 119896minus3))
le 120598 + 119861120598 + 1198612120598 + 1198613120598 + sdot sdot sdot + 119861119896119863lowast (1198660 0)le 120598 (1 + 119861 + 1198612 + 1198613 + sdot sdot sdot + 119861119896) le 1205981 minus 119861
(36)
Therefore
119863lowast (119896 V119896)le 11 minus 119861 (2119898119897 (1 + 2119862(119889 minus 119888) (119887 minus 119886)) + 120598) (37)
6 Numerical Examples
In this section we use the proposed method in two-dimensional linear fuzzy Fredholm integral equations forsolving two examples By using the proposed method for 119899 =2 119896 = 5 8 16 and 119903 isin 00 01 02 03 04 05 06 07 0809 10 in (119904 119905) = (05 05) we present the absolute errors inTables 1ndash4
Example 13 (see [11]) Consider the linear integral equation
119866 (119904 119905)= 119892 (119904 119905)
oplus (119865119877) int10(119865119877) int1
0K (119909 119910 119904 119905) ⊙ 119866 (119909 119910) 119889119909 119889119910
(38)
with
K (119909 119910 119904 119905) = 119909119910119904119905 (119904 119905 119903) = 119903 (13119903 + 83)(1 + 119904 + 119905 minus 712119904119905) 119892 (119904 119905 119903) = (21199032 minus 4119903 + 5) (1 + 119904 + 119905 minus 712119904119905)
(39)
The exact solution for 2DLFFIE (38) is
lowast (119904 119905 119903) = 119903 (13119903 + 83) (119904 + 119905 + 1) 119866lowast (119904 119905 119903) = (21199032 minus 4119903 + 5) (119904 + 119905 + 1)
(40)
Advances in Fuzzy Systems 7
Table 2 The absolute errors on the level sets with 119899 = 2 119896 = 16 for Example 13 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 177636119890 minus 1501 111022119890 minus 16 177636119890 minus 1502 0 177636119890 minus 1503 222045119890 minus 16 888178119890 minus 1604 444089119890 minus 16 888178119890 minus 1605 444089119890 minus 16 888178119890 minus 1606 444089119890 minus 16 007 888178119890 minus 16 008 0 009 0 010 0 0
Table 3 The absolute errors on the level sets with 119899 = 2 119896 = 5 for Example 14 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 960565119890 minus 1301 480276119890 minus 14 912548119890 minus 1302 960551119890 minus 14 864531119890 minus 1303 144079119890 minus 13 816458119890 minus 1304 19211 119890 minus 13 768441119890 minus 1305 240141119890 minus 13 720479119890 minus 1306 288158119890 minus 13 672407119890 minus 1307 336203119890 minus 13 624334119890 minus 1308 38422 119890 minus 13 576317119890 minus 1309 432265119890 minus 13 5283 119890 minus 1310 480282119890 minus 13 480282119890 minus 13
Table 4 The absolute errors on the level sets with 119899 = 2 119896 = 8 for Example 14 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 001 0 002 0 555112119890 minus 1703 0 004 0 005 0 555112119890 minus 1706 0 007 0 008 0 009 277556119890 minus 17 010 0 0
Example 14 (see [13]) Consider the following fuzzy Fred-holm integral equation (38) with
(119904 119905 119903) = 119903 (119904119905 + 1676 (1199042 + 1199052 minus 2)) 119892 (119904 119905 119903) = (2 minus 119903) (119904119905 + 1676 (1199042 + 1199052 minus 2)) (41)
and kernel
K (119909 119910 119904 119905) = 1169 (1199092 + 1199102 minus 2) (1199042 + 1199052 minus 2) 0 le 119904 119905 119909 119910 le 1 (42)
The exact solution is
lowast (119904 119905 119903) = 119903119904119905
8 Advances in Fuzzy Systems
119866lowast (119904 119905 119903) = (2 minus 119903) 119904119905(43)
7 Conclusion
The 2DLFFIE is solved by utilizing two-dimensional fuzzyLagrange interpolation and iterative method As it wasexpected the method used to approximate the integral inthis equation is a suitable one since convergence analysis andstability analysis have been proved and also absolute errorin examples is nearly zero As a result considering the factthat the proposed method does not lead to solve fuzzy linearsystem it can be utilized as an efficient method to solve thistype of equations As future researches we can use finite anddivided differences methods fuzzy spline interpolation andfuzzy quasi-interpolation for solving two-dimensional linearor nonlinear fuzzy Fredholm integral equations
Conflicts of Interest
The authors declare that there are no conflicts of interestrelated to this paper
References
[1] E Babolian H Sadeghi Goghary and S Abbasbandy ldquoNumer-ical solution of linear Fredholm fuzzy integral equations of thesecond kind by Adomian methodrdquo Applied Mathematics andComputation vol 161 no 3 pp 733ndash744 2005
[2] A M Bica ldquoError estimation in the approximation of thesolution of nonlinear fuzzy Fredholm integral equationsrdquo Infor-mation Sciences vol 178 no 5 pp 1279ndash1292 2008
[3] A M Bica and C Popescu ldquoApproximating the solution ofnonlinear Hammerstein fuzzy integral equationsrdquo Fuzzy Setsand Systems vol 245 pp 1ndash17 2014
[4] M A Fariborzi Araghi and G Kazemi Gelian ldquoSolving fuzzyFredholm linear integral equations using Sinc method anddouble exponential transformationrdquo So Computing vol 19 no4 pp 1063ndash1070 2015
[5] M A Faborzi Araghi and N Parandin ldquoNumerical solution offuzzy Fredholm integral equations by the Lagrange interpola-tion based on the extension principlerdquo So Computing vol 15no 12 pp 2449ndash2456 2011
[6] H Hosseini Fadravi R Buzhabadi and H Saberi Nik ldquoSolvinglinear Fredholm fuzzy integral equations of the second kind byartificial neural networksrdquo Alexandria Engineering Journal vol53 no 1 pp 249ndash257 2014
[7] Y Jafarzadeh ldquoNumerical solution for fuzzy Fredholm integralequations with upper-bound on error by splines interpolationrdquoFuzzy Information and Engineering vol 4 no 3 pp 339ndash3472012
[8] A Molabahrami A Shidfar and A Ghyasi ldquoAn analyticalmethod for solving linear Fredholm fuzzy integral equations ofthe second kindrdquo Computers amp Mathematics with Applicationsvol 61 no 9 pp 2754ndash2761 2011
[9] N Parandin and M A Fariborzi Araghi ldquoThe approximatesolution of linear fuzzy Fredholm integral equations of thesecond kind by using iterative interpolationrdquoWorld Academy ofScience Engineering and Technology vol 49 pp 978ndash984 2009
[10] A M Bica and S Ziari ldquoIterative numerical method for fuzzyVolterra linear integral equations in two dimensionsrdquo SoComputing vol 21 no 5 pp 1097ndash1108 2017
[11] SM Sadatrasoul andR Ezzati ldquoQuadrature Rules and IterativeMethod for Numerical Solution of Two-Dimensional FuzzyIntegral Equationsrdquo Abstract and Applied Analysis vol 2014Article ID 413570 18 pages 2014
[12] SM Sadatrasoul andR Ezzati ldquoIterativemethod for numericalsolution of two-dimensional nonlinear fuzzy integral equa-tionsrdquo Fuzzy Sets and Systems vol 280 pp 91ndash106 2015
[13] F Mirzaee M K Yari and E Hadadiyan ldquoNumerical solutionof two-dimensional fuzzy Fredholm integral equations of thesecond kind using triangular functionsrdquo Beni-Suef UniversityJournal of Basic and Applied Sciences vol 4 no 2 pp 109ndash1182015
[14] H Nouriani and R Ezzati ldquoQuadrature iterative method fornumerical solution of two-dimensional linear fuzzy Fredholmintegral equationsrdquoMathematical Sciences vol 11 no 1 pp 63ndash72 2017
[15] R Ezzati and S Ziari ldquoNumerical solution of two-dimensionalfuzzy Fredholm integral equations of the second kind usingfuzzy bivariate Bernstein polynomialsrdquo International Journal ofFuzzy Systems vol 15 no 1 pp 84ndash89 2013
[16] N Hassasi and R Ezzati ldquoNumerical solution of two-dimensional fuzzy Fredholm integral equations using collo-cation fuzzy wavelet like operatorrdquo International Journal ofIndustrial Mathematics vol 7 p 11 2015
[17] M A Vali M J Agheli and S Gohari Nezhad ldquoHomotopyanalysis method to solve two-dimensional fuzzy Fredholmintegral equationrdquo General Mathematics Notes vol 22 pp 31ndash43 2014
[18] A M Bica and S Ziari ldquoOpen fuzzy cubature rule withapplication tononlinear fuzzyVolterra integral equations in twodimensionsrdquo Fuzzy Sets and Systems In press Corrected Proof
[19] A Rivaz and F Yousefi ldquoKernel iterative method for solvingtwo-dimensional fuzzy Fredholm integral equations of thesecond kindrdquo Journal of Fuzzy Set Valued Analysis vol 2013 p9 2013
[20] A Rivaz and F Yousefi ldquoModified homotopy perturbationmethod for solving two-dimensional fuzzy Fredholm integralequationrdquo International Journal of AppliedMathematics vol 25no 4 pp 591ndash602 2012
[21] A Rivaz F Yousefi and H Salehinejad ldquoUsing block pulsefunctions for solving two-dimensional fuzzy Fredholm integralequations of the second kindrdquo International Journal of AppliedMathematics vol 25 no 4 pp 571ndash582 2012
[22] S M Sadatrasoul and R Ezzati ldquoNumerical solution of two-dimensional nonlinear Hammerstein fuzzy integral equationsbased on optimal fuzzy quadrature formulardquo Journal of Compu-tational and Applied Mathematics vol 292 pp 430ndash446 2016
[23] S Ziari ldquoIterative method for solving two-dimensional non-linear fuzzy integral equations using fuzzy bivariate block-pulse functions with error estimationrdquo Iranian Journal of FuzzySystems vol 15 no 1 pp 55ndash76 183 2018
[24] G A Anastassiou Fuzzy Mathematics Approximation eoryvol 251 Springer 2010
[25] A R BozorgmaneshMOtadi A A Safe Kordi et al ldquoBarkhor-dari Ahmadi Lagrange two-dimensional interpolationmethodfor modeling nanoparticle formation during RESS processrdquoInternational Journal of Industrial Mathematics vol 1 pp 175ndash181 2009
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Advances in Fuzzy Systems 5
Therefore
119863(119866119896 (119904 119905) 119896 (119904 119905))le 120583M (119889 minus 119888) (119887 minus 119886)119863lowast (119866119896minus1 119896minus1)
+ 120583M (119889 minus 119888) (119887 minus 119886)119898119896minus1+ 120583M (119889 minus 119888) (119887 minus 119886)119898119896minus1 119862(119889 minus 119888) (119887 minus 119886)+ 120583M119862119898119896minus1
le 119861119863lowast (119866119896minus1 119896minus1) + 119861119898119896minus1+ 2119861119898119896minus1 119862(119889 minus 119888) (119887 minus 119886)
= 119861119863lowast (119866119896minus1 119896minus1)+ 119861119898119896minus1(1 + 2119862(119889 minus 119888) (119887 minus 119886))
(25)
where 119862 = max|119862119894119895| 119894 119895 = 0 sdot sdot sdot 119899 Hence we conclude that119863lowast (119866119896minus1 119896minus1) le 119861119863lowast (119866119896minus2 119896minus2)
+ 119861119898119896minus2(1 + 2119862(119889 minus 119888) (119887 minus 119886))
119863lowast (1198661 1) le 119861119863lowast (1198660 0)+ 1198611198980(1 + 2119862(119889 minus 119888) (119887 minus 119886))
(26)
So
119863lowast (119866119896 119896) le (1 + 2119862(119889 minus 119888) (119887 minus 119886))sdot (119861119898119896minus1 + 1198612119898119896minus2 + sdot sdot sdot + 1198611198961198980)
(27)
If119898119897 = max1198980 119898119896minus1 then we obtain
119863lowast (119866119896 119896) lt 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897) (28)
therefore
119863lowast (119866lowast ) le 119861119896+11 minus 1198611198721+ 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897)
(29)
5 Numerical Stability Analysis
To show the numerical stability analysis of the proposedmethod in previous section we consider another startingapproximation 119891(119904 119905) = 0(119904 119905) such that exist120598 gt 0 for which119863(1198660(119904 119905) 0(119904 119905)) lt 120598 forall119904 119905 isin [119886 119887] times [119888 119889] The obtainedsequence of successive approximations is
119896 (119904 119905) = 119891 (119904 119905) oplus 120583 ⊙ (119865119877)sdot int119889119888(119865119877)int119887
119886K (119909 119910 119904 119905) ⊙ 119896minus1 (119909 119910) 119889119909 119889119910 (30)
and using the same iterative method the terms of producedsequence are
V0 (119904 119905) = 0 (119904 119905) = 119891 (119904 119905) V119896 (119904 119905) = 119891 (119904 119905) oplus 120583
⊙ 119899sum119895=0
119899sum119894=0
119862119894119895K (119904119894 119905119895 119904 119905) ⊙ V119896minus1 (119904119894 119905119895) (31)
Theorem 12 e proposed method (15) under the assump-tions of eorem 11 is numerically stable with respect to thechoice of the first iteration
Proof At first we obtain that
119863(119896 (119904 119905) V119896 (119904 119905))le 119863 (119896 (119904 119905) 119866119896 (119904 119905)) + 119863 (119866119896 (119904 119905) 119896 (119904 119905))
+ 119863 (119896 (119904 119905) V119896 (119904 119905))le 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897)
+ 119863 (119866119896 (119904 119905) 119896 (119904 119905))+ 11 minus 119861 ((1 + 2119862(119889 minus 119888) (119887 minus 119886))119898119897)
(32)
However
119863(119866119896 (119904 119905) 119896 (119904 119905)) = 119863(119892 (119904 119905) oplus 120583 ⊙ (119865119877)sdot int119889119888(119865119877)
sdot int119887119886K (119909 119910 119904 119905) ⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910 119891 (119904 119905) oplus 120583
⊙ (119865119877) int119889119888(119865119877) int119887
119886K (119909 119910 119904 119905)
⊙ 119896minus1 (119909 119910) 119889119909 119889119910) le 119863 (119892 (119904 119905) 119891 (119904 119905))+ 120583119863((119865119877) int119889
119888(119865119877) int119887
119886K (119909 119910 119904 119905)
6 Advances in Fuzzy Systems
Table 1 The absolute errors on the level sets with 119899 = 2 119896 = 5 for Example 13 by using the proposed method (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 123485119890 minus 501 666819119890 minus 7 1141 119890 minus 502 13501 119890 minus 6 105703119890 minus 503 204985119890 minus 6 982941119890 minus 604 276606119890 minus 6 918729119890 minus 605 349874119890 minus 6 864395119890 minus 606 424788119890 minus 6 819941119890 minus 607 501349119890 minus 6 785365119890 minus 608 579556119890 minus 6 760668119890 minus 609 65941 119890 minus 6 745849119890 minus 610 74091 119890 minus 6 74091 119890 minus 6
⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910 (119865119877) int119889119888(119865119877)
sdot int119887119886K (119909 119910 119904 119905) ⊙ 119896minus1 (119909 119910) 119889119909 119889119910)
le 120598 + 120583int119889119888int119887119886
1003816100381610038161003816K (119909 119910 119904 119905)1003816100381610038161003816sdot 119863 (119866119896minus1 (119909 119910) 119896minus1 (119909 119910)) 119889119909 119889119910
(33)
We conclude that
119863lowast (119866119896 119896) le 120598 + 120583int119889119888int119887119886M119863lowast (119866119896minus1 119896minus1) 119889119909 119889119910
= 120598 + 119861119863lowast (119866119896minus1 119896minus1) (34)
and thus
119863lowast (119866119896 119896) le 120598 + 119861119863lowast (119866119896minus1 119896minus1)119863lowast (119866119896minus1 119896minus1) le 120598 + 119861119863lowast (119866119896minus2 119896minus2)
119863lowast (1198661 1) le 120598 + 119861119863lowast (1198660 0)
(35)
So
119863lowast (119866119896 119896) le 120598 + 119861 (120598 + 119861119863lowast (119866119896minus2 119896minus2))le 120598 + 119861120598 + 1198612 (120598 + 119861119863lowast (119866119896minus3 119896minus3))
le 120598 + 119861120598 + 1198612120598 + 1198613120598 + sdot sdot sdot + 119861119896119863lowast (1198660 0)le 120598 (1 + 119861 + 1198612 + 1198613 + sdot sdot sdot + 119861119896) le 1205981 minus 119861
(36)
Therefore
119863lowast (119896 V119896)le 11 minus 119861 (2119898119897 (1 + 2119862(119889 minus 119888) (119887 minus 119886)) + 120598) (37)
6 Numerical Examples
In this section we use the proposed method in two-dimensional linear fuzzy Fredholm integral equations forsolving two examples By using the proposed method for 119899 =2 119896 = 5 8 16 and 119903 isin 00 01 02 03 04 05 06 07 0809 10 in (119904 119905) = (05 05) we present the absolute errors inTables 1ndash4
Example 13 (see [11]) Consider the linear integral equation
119866 (119904 119905)= 119892 (119904 119905)
oplus (119865119877) int10(119865119877) int1
0K (119909 119910 119904 119905) ⊙ 119866 (119909 119910) 119889119909 119889119910
(38)
with
K (119909 119910 119904 119905) = 119909119910119904119905 (119904 119905 119903) = 119903 (13119903 + 83)(1 + 119904 + 119905 minus 712119904119905) 119892 (119904 119905 119903) = (21199032 minus 4119903 + 5) (1 + 119904 + 119905 minus 712119904119905)
(39)
The exact solution for 2DLFFIE (38) is
lowast (119904 119905 119903) = 119903 (13119903 + 83) (119904 + 119905 + 1) 119866lowast (119904 119905 119903) = (21199032 minus 4119903 + 5) (119904 + 119905 + 1)
(40)
Advances in Fuzzy Systems 7
Table 2 The absolute errors on the level sets with 119899 = 2 119896 = 16 for Example 13 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 177636119890 minus 1501 111022119890 minus 16 177636119890 minus 1502 0 177636119890 minus 1503 222045119890 minus 16 888178119890 minus 1604 444089119890 minus 16 888178119890 minus 1605 444089119890 minus 16 888178119890 minus 1606 444089119890 minus 16 007 888178119890 minus 16 008 0 009 0 010 0 0
Table 3 The absolute errors on the level sets with 119899 = 2 119896 = 5 for Example 14 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 960565119890 minus 1301 480276119890 minus 14 912548119890 minus 1302 960551119890 minus 14 864531119890 minus 1303 144079119890 minus 13 816458119890 minus 1304 19211 119890 minus 13 768441119890 minus 1305 240141119890 minus 13 720479119890 minus 1306 288158119890 minus 13 672407119890 minus 1307 336203119890 minus 13 624334119890 minus 1308 38422 119890 minus 13 576317119890 minus 1309 432265119890 minus 13 5283 119890 minus 1310 480282119890 minus 13 480282119890 minus 13
Table 4 The absolute errors on the level sets with 119899 = 2 119896 = 8 for Example 14 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 001 0 002 0 555112119890 minus 1703 0 004 0 005 0 555112119890 minus 1706 0 007 0 008 0 009 277556119890 minus 17 010 0 0
Example 14 (see [13]) Consider the following fuzzy Fred-holm integral equation (38) with
(119904 119905 119903) = 119903 (119904119905 + 1676 (1199042 + 1199052 minus 2)) 119892 (119904 119905 119903) = (2 minus 119903) (119904119905 + 1676 (1199042 + 1199052 minus 2)) (41)
and kernel
K (119909 119910 119904 119905) = 1169 (1199092 + 1199102 minus 2) (1199042 + 1199052 minus 2) 0 le 119904 119905 119909 119910 le 1 (42)
The exact solution is
lowast (119904 119905 119903) = 119903119904119905
8 Advances in Fuzzy Systems
119866lowast (119904 119905 119903) = (2 minus 119903) 119904119905(43)
7 Conclusion
The 2DLFFIE is solved by utilizing two-dimensional fuzzyLagrange interpolation and iterative method As it wasexpected the method used to approximate the integral inthis equation is a suitable one since convergence analysis andstability analysis have been proved and also absolute errorin examples is nearly zero As a result considering the factthat the proposed method does not lead to solve fuzzy linearsystem it can be utilized as an efficient method to solve thistype of equations As future researches we can use finite anddivided differences methods fuzzy spline interpolation andfuzzy quasi-interpolation for solving two-dimensional linearor nonlinear fuzzy Fredholm integral equations
Conflicts of Interest
The authors declare that there are no conflicts of interestrelated to this paper
References
[1] E Babolian H Sadeghi Goghary and S Abbasbandy ldquoNumer-ical solution of linear Fredholm fuzzy integral equations of thesecond kind by Adomian methodrdquo Applied Mathematics andComputation vol 161 no 3 pp 733ndash744 2005
[2] A M Bica ldquoError estimation in the approximation of thesolution of nonlinear fuzzy Fredholm integral equationsrdquo Infor-mation Sciences vol 178 no 5 pp 1279ndash1292 2008
[3] A M Bica and C Popescu ldquoApproximating the solution ofnonlinear Hammerstein fuzzy integral equationsrdquo Fuzzy Setsand Systems vol 245 pp 1ndash17 2014
[4] M A Fariborzi Araghi and G Kazemi Gelian ldquoSolving fuzzyFredholm linear integral equations using Sinc method anddouble exponential transformationrdquo So Computing vol 19 no4 pp 1063ndash1070 2015
[5] M A Faborzi Araghi and N Parandin ldquoNumerical solution offuzzy Fredholm integral equations by the Lagrange interpola-tion based on the extension principlerdquo So Computing vol 15no 12 pp 2449ndash2456 2011
[6] H Hosseini Fadravi R Buzhabadi and H Saberi Nik ldquoSolvinglinear Fredholm fuzzy integral equations of the second kind byartificial neural networksrdquo Alexandria Engineering Journal vol53 no 1 pp 249ndash257 2014
[7] Y Jafarzadeh ldquoNumerical solution for fuzzy Fredholm integralequations with upper-bound on error by splines interpolationrdquoFuzzy Information and Engineering vol 4 no 3 pp 339ndash3472012
[8] A Molabahrami A Shidfar and A Ghyasi ldquoAn analyticalmethod for solving linear Fredholm fuzzy integral equations ofthe second kindrdquo Computers amp Mathematics with Applicationsvol 61 no 9 pp 2754ndash2761 2011
[9] N Parandin and M A Fariborzi Araghi ldquoThe approximatesolution of linear fuzzy Fredholm integral equations of thesecond kind by using iterative interpolationrdquoWorld Academy ofScience Engineering and Technology vol 49 pp 978ndash984 2009
[10] A M Bica and S Ziari ldquoIterative numerical method for fuzzyVolterra linear integral equations in two dimensionsrdquo SoComputing vol 21 no 5 pp 1097ndash1108 2017
[11] SM Sadatrasoul andR Ezzati ldquoQuadrature Rules and IterativeMethod for Numerical Solution of Two-Dimensional FuzzyIntegral Equationsrdquo Abstract and Applied Analysis vol 2014Article ID 413570 18 pages 2014
[12] SM Sadatrasoul andR Ezzati ldquoIterativemethod for numericalsolution of two-dimensional nonlinear fuzzy integral equa-tionsrdquo Fuzzy Sets and Systems vol 280 pp 91ndash106 2015
[13] F Mirzaee M K Yari and E Hadadiyan ldquoNumerical solutionof two-dimensional fuzzy Fredholm integral equations of thesecond kind using triangular functionsrdquo Beni-Suef UniversityJournal of Basic and Applied Sciences vol 4 no 2 pp 109ndash1182015
[14] H Nouriani and R Ezzati ldquoQuadrature iterative method fornumerical solution of two-dimensional linear fuzzy Fredholmintegral equationsrdquoMathematical Sciences vol 11 no 1 pp 63ndash72 2017
[15] R Ezzati and S Ziari ldquoNumerical solution of two-dimensionalfuzzy Fredholm integral equations of the second kind usingfuzzy bivariate Bernstein polynomialsrdquo International Journal ofFuzzy Systems vol 15 no 1 pp 84ndash89 2013
[16] N Hassasi and R Ezzati ldquoNumerical solution of two-dimensional fuzzy Fredholm integral equations using collo-cation fuzzy wavelet like operatorrdquo International Journal ofIndustrial Mathematics vol 7 p 11 2015
[17] M A Vali M J Agheli and S Gohari Nezhad ldquoHomotopyanalysis method to solve two-dimensional fuzzy Fredholmintegral equationrdquo General Mathematics Notes vol 22 pp 31ndash43 2014
[18] A M Bica and S Ziari ldquoOpen fuzzy cubature rule withapplication tononlinear fuzzyVolterra integral equations in twodimensionsrdquo Fuzzy Sets and Systems In press Corrected Proof
[19] A Rivaz and F Yousefi ldquoKernel iterative method for solvingtwo-dimensional fuzzy Fredholm integral equations of thesecond kindrdquo Journal of Fuzzy Set Valued Analysis vol 2013 p9 2013
[20] A Rivaz and F Yousefi ldquoModified homotopy perturbationmethod for solving two-dimensional fuzzy Fredholm integralequationrdquo International Journal of AppliedMathematics vol 25no 4 pp 591ndash602 2012
[21] A Rivaz F Yousefi and H Salehinejad ldquoUsing block pulsefunctions for solving two-dimensional fuzzy Fredholm integralequations of the second kindrdquo International Journal of AppliedMathematics vol 25 no 4 pp 571ndash582 2012
[22] S M Sadatrasoul and R Ezzati ldquoNumerical solution of two-dimensional nonlinear Hammerstein fuzzy integral equationsbased on optimal fuzzy quadrature formulardquo Journal of Compu-tational and Applied Mathematics vol 292 pp 430ndash446 2016
[23] S Ziari ldquoIterative method for solving two-dimensional non-linear fuzzy integral equations using fuzzy bivariate block-pulse functions with error estimationrdquo Iranian Journal of FuzzySystems vol 15 no 1 pp 55ndash76 183 2018
[24] G A Anastassiou Fuzzy Mathematics Approximation eoryvol 251 Springer 2010
[25] A R BozorgmaneshMOtadi A A Safe Kordi et al ldquoBarkhor-dari Ahmadi Lagrange two-dimensional interpolationmethodfor modeling nanoparticle formation during RESS processrdquoInternational Journal of Industrial Mathematics vol 1 pp 175ndash181 2009
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6 Advances in Fuzzy Systems
Table 1 The absolute errors on the level sets with 119899 = 2 119896 = 5 for Example 13 by using the proposed method (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 123485119890 minus 501 666819119890 minus 7 1141 119890 minus 502 13501 119890 minus 6 105703119890 minus 503 204985119890 minus 6 982941119890 minus 604 276606119890 minus 6 918729119890 minus 605 349874119890 minus 6 864395119890 minus 606 424788119890 minus 6 819941119890 minus 607 501349119890 minus 6 785365119890 minus 608 579556119890 minus 6 760668119890 minus 609 65941 119890 minus 6 745849119890 minus 610 74091 119890 minus 6 74091 119890 minus 6
⊙ 119866119896minus1 (119909 119910) 119889119909 119889119910 (119865119877) int119889119888(119865119877)
sdot int119887119886K (119909 119910 119904 119905) ⊙ 119896minus1 (119909 119910) 119889119909 119889119910)
le 120598 + 120583int119889119888int119887119886
1003816100381610038161003816K (119909 119910 119904 119905)1003816100381610038161003816sdot 119863 (119866119896minus1 (119909 119910) 119896minus1 (119909 119910)) 119889119909 119889119910
(33)
We conclude that
119863lowast (119866119896 119896) le 120598 + 120583int119889119888int119887119886M119863lowast (119866119896minus1 119896minus1) 119889119909 119889119910
= 120598 + 119861119863lowast (119866119896minus1 119896minus1) (34)
and thus
119863lowast (119866119896 119896) le 120598 + 119861119863lowast (119866119896minus1 119896minus1)119863lowast (119866119896minus1 119896minus1) le 120598 + 119861119863lowast (119866119896minus2 119896minus2)
119863lowast (1198661 1) le 120598 + 119861119863lowast (1198660 0)
(35)
So
119863lowast (119866119896 119896) le 120598 + 119861 (120598 + 119861119863lowast (119866119896minus2 119896minus2))le 120598 + 119861120598 + 1198612 (120598 + 119861119863lowast (119866119896minus3 119896minus3))
le 120598 + 119861120598 + 1198612120598 + 1198613120598 + sdot sdot sdot + 119861119896119863lowast (1198660 0)le 120598 (1 + 119861 + 1198612 + 1198613 + sdot sdot sdot + 119861119896) le 1205981 minus 119861
(36)
Therefore
119863lowast (119896 V119896)le 11 minus 119861 (2119898119897 (1 + 2119862(119889 minus 119888) (119887 minus 119886)) + 120598) (37)
6 Numerical Examples
In this section we use the proposed method in two-dimensional linear fuzzy Fredholm integral equations forsolving two examples By using the proposed method for 119899 =2 119896 = 5 8 16 and 119903 isin 00 01 02 03 04 05 06 07 0809 10 in (119904 119905) = (05 05) we present the absolute errors inTables 1ndash4
Example 13 (see [11]) Consider the linear integral equation
119866 (119904 119905)= 119892 (119904 119905)
oplus (119865119877) int10(119865119877) int1
0K (119909 119910 119904 119905) ⊙ 119866 (119909 119910) 119889119909 119889119910
(38)
with
K (119909 119910 119904 119905) = 119909119910119904119905 (119904 119905 119903) = 119903 (13119903 + 83)(1 + 119904 + 119905 minus 712119904119905) 119892 (119904 119905 119903) = (21199032 minus 4119903 + 5) (1 + 119904 + 119905 minus 712119904119905)
(39)
The exact solution for 2DLFFIE (38) is
lowast (119904 119905 119903) = 119903 (13119903 + 83) (119904 + 119905 + 1) 119866lowast (119904 119905 119903) = (21199032 minus 4119903 + 5) (119904 + 119905 + 1)
(40)
Advances in Fuzzy Systems 7
Table 2 The absolute errors on the level sets with 119899 = 2 119896 = 16 for Example 13 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 177636119890 minus 1501 111022119890 minus 16 177636119890 minus 1502 0 177636119890 minus 1503 222045119890 minus 16 888178119890 minus 1604 444089119890 minus 16 888178119890 minus 1605 444089119890 minus 16 888178119890 minus 1606 444089119890 minus 16 007 888178119890 minus 16 008 0 009 0 010 0 0
Table 3 The absolute errors on the level sets with 119899 = 2 119896 = 5 for Example 14 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 960565119890 minus 1301 480276119890 minus 14 912548119890 minus 1302 960551119890 minus 14 864531119890 minus 1303 144079119890 minus 13 816458119890 minus 1304 19211 119890 minus 13 768441119890 minus 1305 240141119890 minus 13 720479119890 minus 1306 288158119890 minus 13 672407119890 minus 1307 336203119890 minus 13 624334119890 minus 1308 38422 119890 minus 13 576317119890 minus 1309 432265119890 minus 13 5283 119890 minus 1310 480282119890 minus 13 480282119890 minus 13
Table 4 The absolute errors on the level sets with 119899 = 2 119896 = 8 for Example 14 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 001 0 002 0 555112119890 minus 1703 0 004 0 005 0 555112119890 minus 1706 0 007 0 008 0 009 277556119890 minus 17 010 0 0
Example 14 (see [13]) Consider the following fuzzy Fred-holm integral equation (38) with
(119904 119905 119903) = 119903 (119904119905 + 1676 (1199042 + 1199052 minus 2)) 119892 (119904 119905 119903) = (2 minus 119903) (119904119905 + 1676 (1199042 + 1199052 minus 2)) (41)
and kernel
K (119909 119910 119904 119905) = 1169 (1199092 + 1199102 minus 2) (1199042 + 1199052 minus 2) 0 le 119904 119905 119909 119910 le 1 (42)
The exact solution is
lowast (119904 119905 119903) = 119903119904119905
8 Advances in Fuzzy Systems
119866lowast (119904 119905 119903) = (2 minus 119903) 119904119905(43)
7 Conclusion
The 2DLFFIE is solved by utilizing two-dimensional fuzzyLagrange interpolation and iterative method As it wasexpected the method used to approximate the integral inthis equation is a suitable one since convergence analysis andstability analysis have been proved and also absolute errorin examples is nearly zero As a result considering the factthat the proposed method does not lead to solve fuzzy linearsystem it can be utilized as an efficient method to solve thistype of equations As future researches we can use finite anddivided differences methods fuzzy spline interpolation andfuzzy quasi-interpolation for solving two-dimensional linearor nonlinear fuzzy Fredholm integral equations
Conflicts of Interest
The authors declare that there are no conflicts of interestrelated to this paper
References
[1] E Babolian H Sadeghi Goghary and S Abbasbandy ldquoNumer-ical solution of linear Fredholm fuzzy integral equations of thesecond kind by Adomian methodrdquo Applied Mathematics andComputation vol 161 no 3 pp 733ndash744 2005
[2] A M Bica ldquoError estimation in the approximation of thesolution of nonlinear fuzzy Fredholm integral equationsrdquo Infor-mation Sciences vol 178 no 5 pp 1279ndash1292 2008
[3] A M Bica and C Popescu ldquoApproximating the solution ofnonlinear Hammerstein fuzzy integral equationsrdquo Fuzzy Setsand Systems vol 245 pp 1ndash17 2014
[4] M A Fariborzi Araghi and G Kazemi Gelian ldquoSolving fuzzyFredholm linear integral equations using Sinc method anddouble exponential transformationrdquo So Computing vol 19 no4 pp 1063ndash1070 2015
[5] M A Faborzi Araghi and N Parandin ldquoNumerical solution offuzzy Fredholm integral equations by the Lagrange interpola-tion based on the extension principlerdquo So Computing vol 15no 12 pp 2449ndash2456 2011
[6] H Hosseini Fadravi R Buzhabadi and H Saberi Nik ldquoSolvinglinear Fredholm fuzzy integral equations of the second kind byartificial neural networksrdquo Alexandria Engineering Journal vol53 no 1 pp 249ndash257 2014
[7] Y Jafarzadeh ldquoNumerical solution for fuzzy Fredholm integralequations with upper-bound on error by splines interpolationrdquoFuzzy Information and Engineering vol 4 no 3 pp 339ndash3472012
[8] A Molabahrami A Shidfar and A Ghyasi ldquoAn analyticalmethod for solving linear Fredholm fuzzy integral equations ofthe second kindrdquo Computers amp Mathematics with Applicationsvol 61 no 9 pp 2754ndash2761 2011
[9] N Parandin and M A Fariborzi Araghi ldquoThe approximatesolution of linear fuzzy Fredholm integral equations of thesecond kind by using iterative interpolationrdquoWorld Academy ofScience Engineering and Technology vol 49 pp 978ndash984 2009
[10] A M Bica and S Ziari ldquoIterative numerical method for fuzzyVolterra linear integral equations in two dimensionsrdquo SoComputing vol 21 no 5 pp 1097ndash1108 2017
[11] SM Sadatrasoul andR Ezzati ldquoQuadrature Rules and IterativeMethod for Numerical Solution of Two-Dimensional FuzzyIntegral Equationsrdquo Abstract and Applied Analysis vol 2014Article ID 413570 18 pages 2014
[12] SM Sadatrasoul andR Ezzati ldquoIterativemethod for numericalsolution of two-dimensional nonlinear fuzzy integral equa-tionsrdquo Fuzzy Sets and Systems vol 280 pp 91ndash106 2015
[13] F Mirzaee M K Yari and E Hadadiyan ldquoNumerical solutionof two-dimensional fuzzy Fredholm integral equations of thesecond kind using triangular functionsrdquo Beni-Suef UniversityJournal of Basic and Applied Sciences vol 4 no 2 pp 109ndash1182015
[14] H Nouriani and R Ezzati ldquoQuadrature iterative method fornumerical solution of two-dimensional linear fuzzy Fredholmintegral equationsrdquoMathematical Sciences vol 11 no 1 pp 63ndash72 2017
[15] R Ezzati and S Ziari ldquoNumerical solution of two-dimensionalfuzzy Fredholm integral equations of the second kind usingfuzzy bivariate Bernstein polynomialsrdquo International Journal ofFuzzy Systems vol 15 no 1 pp 84ndash89 2013
[16] N Hassasi and R Ezzati ldquoNumerical solution of two-dimensional fuzzy Fredholm integral equations using collo-cation fuzzy wavelet like operatorrdquo International Journal ofIndustrial Mathematics vol 7 p 11 2015
[17] M A Vali M J Agheli and S Gohari Nezhad ldquoHomotopyanalysis method to solve two-dimensional fuzzy Fredholmintegral equationrdquo General Mathematics Notes vol 22 pp 31ndash43 2014
[18] A M Bica and S Ziari ldquoOpen fuzzy cubature rule withapplication tononlinear fuzzyVolterra integral equations in twodimensionsrdquo Fuzzy Sets and Systems In press Corrected Proof
[19] A Rivaz and F Yousefi ldquoKernel iterative method for solvingtwo-dimensional fuzzy Fredholm integral equations of thesecond kindrdquo Journal of Fuzzy Set Valued Analysis vol 2013 p9 2013
[20] A Rivaz and F Yousefi ldquoModified homotopy perturbationmethod for solving two-dimensional fuzzy Fredholm integralequationrdquo International Journal of AppliedMathematics vol 25no 4 pp 591ndash602 2012
[21] A Rivaz F Yousefi and H Salehinejad ldquoUsing block pulsefunctions for solving two-dimensional fuzzy Fredholm integralequations of the second kindrdquo International Journal of AppliedMathematics vol 25 no 4 pp 571ndash582 2012
[22] S M Sadatrasoul and R Ezzati ldquoNumerical solution of two-dimensional nonlinear Hammerstein fuzzy integral equationsbased on optimal fuzzy quadrature formulardquo Journal of Compu-tational and Applied Mathematics vol 292 pp 430ndash446 2016
[23] S Ziari ldquoIterative method for solving two-dimensional non-linear fuzzy integral equations using fuzzy bivariate block-pulse functions with error estimationrdquo Iranian Journal of FuzzySystems vol 15 no 1 pp 55ndash76 183 2018
[24] G A Anastassiou Fuzzy Mathematics Approximation eoryvol 251 Springer 2010
[25] A R BozorgmaneshMOtadi A A Safe Kordi et al ldquoBarkhor-dari Ahmadi Lagrange two-dimensional interpolationmethodfor modeling nanoparticle formation during RESS processrdquoInternational Journal of Industrial Mathematics vol 1 pp 175ndash181 2009
Computer Games Technology
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Advances in
FuzzySystems
Hindawiwwwhindawicom
Volume 2018
International Journal of
ReconfigurableComputing
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
thinspArtificial Intelligence
Hindawiwwwhindawicom Volumethinsp2018
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawiwwwhindawicom Volume 2018
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Computational Intelligence and Neuroscience
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018
Human-ComputerInteraction
Advances in
Hindawiwwwhindawicom Volume 2018
Scientic Programming
Submit your manuscripts atwwwhindawicom
Advances in Fuzzy Systems 7
Table 2 The absolute errors on the level sets with 119899 = 2 119896 = 16 for Example 13 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 177636119890 minus 1501 111022119890 minus 16 177636119890 minus 1502 0 177636119890 minus 1503 222045119890 minus 16 888178119890 minus 1604 444089119890 minus 16 888178119890 minus 1605 444089119890 minus 16 888178119890 minus 1606 444089119890 minus 16 007 888178119890 minus 16 008 0 009 0 010 0 0
Table 3 The absolute errors on the level sets with 119899 = 2 119896 = 5 for Example 14 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 960565119890 minus 1301 480276119890 minus 14 912548119890 minus 1302 960551119890 minus 14 864531119890 minus 1303 144079119890 minus 13 816458119890 minus 1304 19211 119890 minus 13 768441119890 minus 1305 240141119890 minus 13 720479119890 minus 1306 288158119890 minus 13 672407119890 minus 1307 336203119890 minus 13 624334119890 minus 1308 38422 119890 minus 13 576317119890 minus 1309 432265119890 minus 13 5283 119890 minus 1310 480282119890 minus 13 480282119890 minus 13
Table 4 The absolute errors on the level sets with 119899 = 2 119896 = 8 for Example 14 by using the proposed method in (119904 119905) = (05 05)119903 minus 119897119890V119890119897 119890119903 = 10038161003816100381610038161003816lowast(119904 119905 119903) minus 119896(119904 119905 119903)10038161003816100381610038161003816 119890119903 = 1003816100381610038161003816100381610038161003816119866
lowast(119904 119905 119903) minus 119896(119904 119905 119903)100381610038161003816100381610038161003816100381600 0 001 0 002 0 555112119890 minus 1703 0 004 0 005 0 555112119890 minus 1706 0 007 0 008 0 009 277556119890 minus 17 010 0 0
Example 14 (see [13]) Consider the following fuzzy Fred-holm integral equation (38) with
(119904 119905 119903) = 119903 (119904119905 + 1676 (1199042 + 1199052 minus 2)) 119892 (119904 119905 119903) = (2 minus 119903) (119904119905 + 1676 (1199042 + 1199052 minus 2)) (41)
and kernel
K (119909 119910 119904 119905) = 1169 (1199092 + 1199102 minus 2) (1199042 + 1199052 minus 2) 0 le 119904 119905 119909 119910 le 1 (42)
The exact solution is
lowast (119904 119905 119903) = 119903119904119905
8 Advances in Fuzzy Systems
119866lowast (119904 119905 119903) = (2 minus 119903) 119904119905(43)
7 Conclusion
The 2DLFFIE is solved by utilizing two-dimensional fuzzyLagrange interpolation and iterative method As it wasexpected the method used to approximate the integral inthis equation is a suitable one since convergence analysis andstability analysis have been proved and also absolute errorin examples is nearly zero As a result considering the factthat the proposed method does not lead to solve fuzzy linearsystem it can be utilized as an efficient method to solve thistype of equations As future researches we can use finite anddivided differences methods fuzzy spline interpolation andfuzzy quasi-interpolation for solving two-dimensional linearor nonlinear fuzzy Fredholm integral equations
Conflicts of Interest
The authors declare that there are no conflicts of interestrelated to this paper
References
[1] E Babolian H Sadeghi Goghary and S Abbasbandy ldquoNumer-ical solution of linear Fredholm fuzzy integral equations of thesecond kind by Adomian methodrdquo Applied Mathematics andComputation vol 161 no 3 pp 733ndash744 2005
[2] A M Bica ldquoError estimation in the approximation of thesolution of nonlinear fuzzy Fredholm integral equationsrdquo Infor-mation Sciences vol 178 no 5 pp 1279ndash1292 2008
[3] A M Bica and C Popescu ldquoApproximating the solution ofnonlinear Hammerstein fuzzy integral equationsrdquo Fuzzy Setsand Systems vol 245 pp 1ndash17 2014
[4] M A Fariborzi Araghi and G Kazemi Gelian ldquoSolving fuzzyFredholm linear integral equations using Sinc method anddouble exponential transformationrdquo So Computing vol 19 no4 pp 1063ndash1070 2015
[5] M A Faborzi Araghi and N Parandin ldquoNumerical solution offuzzy Fredholm integral equations by the Lagrange interpola-tion based on the extension principlerdquo So Computing vol 15no 12 pp 2449ndash2456 2011
[6] H Hosseini Fadravi R Buzhabadi and H Saberi Nik ldquoSolvinglinear Fredholm fuzzy integral equations of the second kind byartificial neural networksrdquo Alexandria Engineering Journal vol53 no 1 pp 249ndash257 2014
[7] Y Jafarzadeh ldquoNumerical solution for fuzzy Fredholm integralequations with upper-bound on error by splines interpolationrdquoFuzzy Information and Engineering vol 4 no 3 pp 339ndash3472012
[8] A Molabahrami A Shidfar and A Ghyasi ldquoAn analyticalmethod for solving linear Fredholm fuzzy integral equations ofthe second kindrdquo Computers amp Mathematics with Applicationsvol 61 no 9 pp 2754ndash2761 2011
[9] N Parandin and M A Fariborzi Araghi ldquoThe approximatesolution of linear fuzzy Fredholm integral equations of thesecond kind by using iterative interpolationrdquoWorld Academy ofScience Engineering and Technology vol 49 pp 978ndash984 2009
[10] A M Bica and S Ziari ldquoIterative numerical method for fuzzyVolterra linear integral equations in two dimensionsrdquo SoComputing vol 21 no 5 pp 1097ndash1108 2017
[11] SM Sadatrasoul andR Ezzati ldquoQuadrature Rules and IterativeMethod for Numerical Solution of Two-Dimensional FuzzyIntegral Equationsrdquo Abstract and Applied Analysis vol 2014Article ID 413570 18 pages 2014
[12] SM Sadatrasoul andR Ezzati ldquoIterativemethod for numericalsolution of two-dimensional nonlinear fuzzy integral equa-tionsrdquo Fuzzy Sets and Systems vol 280 pp 91ndash106 2015
[13] F Mirzaee M K Yari and E Hadadiyan ldquoNumerical solutionof two-dimensional fuzzy Fredholm integral equations of thesecond kind using triangular functionsrdquo Beni-Suef UniversityJournal of Basic and Applied Sciences vol 4 no 2 pp 109ndash1182015
[14] H Nouriani and R Ezzati ldquoQuadrature iterative method fornumerical solution of two-dimensional linear fuzzy Fredholmintegral equationsrdquoMathematical Sciences vol 11 no 1 pp 63ndash72 2017
[15] R Ezzati and S Ziari ldquoNumerical solution of two-dimensionalfuzzy Fredholm integral equations of the second kind usingfuzzy bivariate Bernstein polynomialsrdquo International Journal ofFuzzy Systems vol 15 no 1 pp 84ndash89 2013
[16] N Hassasi and R Ezzati ldquoNumerical solution of two-dimensional fuzzy Fredholm integral equations using collo-cation fuzzy wavelet like operatorrdquo International Journal ofIndustrial Mathematics vol 7 p 11 2015
[17] M A Vali M J Agheli and S Gohari Nezhad ldquoHomotopyanalysis method to solve two-dimensional fuzzy Fredholmintegral equationrdquo General Mathematics Notes vol 22 pp 31ndash43 2014
[18] A M Bica and S Ziari ldquoOpen fuzzy cubature rule withapplication tononlinear fuzzyVolterra integral equations in twodimensionsrdquo Fuzzy Sets and Systems In press Corrected Proof
[19] A Rivaz and F Yousefi ldquoKernel iterative method for solvingtwo-dimensional fuzzy Fredholm integral equations of thesecond kindrdquo Journal of Fuzzy Set Valued Analysis vol 2013 p9 2013
[20] A Rivaz and F Yousefi ldquoModified homotopy perturbationmethod for solving two-dimensional fuzzy Fredholm integralequationrdquo International Journal of AppliedMathematics vol 25no 4 pp 591ndash602 2012
[21] A Rivaz F Yousefi and H Salehinejad ldquoUsing block pulsefunctions for solving two-dimensional fuzzy Fredholm integralequations of the second kindrdquo International Journal of AppliedMathematics vol 25 no 4 pp 571ndash582 2012
[22] S M Sadatrasoul and R Ezzati ldquoNumerical solution of two-dimensional nonlinear Hammerstein fuzzy integral equationsbased on optimal fuzzy quadrature formulardquo Journal of Compu-tational and Applied Mathematics vol 292 pp 430ndash446 2016
[23] S Ziari ldquoIterative method for solving two-dimensional non-linear fuzzy integral equations using fuzzy bivariate block-pulse functions with error estimationrdquo Iranian Journal of FuzzySystems vol 15 no 1 pp 55ndash76 183 2018
[24] G A Anastassiou Fuzzy Mathematics Approximation eoryvol 251 Springer 2010
[25] A R BozorgmaneshMOtadi A A Safe Kordi et al ldquoBarkhor-dari Ahmadi Lagrange two-dimensional interpolationmethodfor modeling nanoparticle formation during RESS processrdquoInternational Journal of Industrial Mathematics vol 1 pp 175ndash181 2009
Computer Games Technology
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Advances in
FuzzySystems
Hindawiwwwhindawicom
Volume 2018
International Journal of
ReconfigurableComputing
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
thinspArtificial Intelligence
Hindawiwwwhindawicom Volumethinsp2018
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawiwwwhindawicom Volume 2018
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Computational Intelligence and Neuroscience
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018
Human-ComputerInteraction
Advances in
Hindawiwwwhindawicom Volume 2018
Scientic Programming
Submit your manuscripts atwwwhindawicom
8 Advances in Fuzzy Systems
119866lowast (119904 119905 119903) = (2 minus 119903) 119904119905(43)
7 Conclusion
The 2DLFFIE is solved by utilizing two-dimensional fuzzyLagrange interpolation and iterative method As it wasexpected the method used to approximate the integral inthis equation is a suitable one since convergence analysis andstability analysis have been proved and also absolute errorin examples is nearly zero As a result considering the factthat the proposed method does not lead to solve fuzzy linearsystem it can be utilized as an efficient method to solve thistype of equations As future researches we can use finite anddivided differences methods fuzzy spline interpolation andfuzzy quasi-interpolation for solving two-dimensional linearor nonlinear fuzzy Fredholm integral equations
Conflicts of Interest
The authors declare that there are no conflicts of interestrelated to this paper
References
[1] E Babolian H Sadeghi Goghary and S Abbasbandy ldquoNumer-ical solution of linear Fredholm fuzzy integral equations of thesecond kind by Adomian methodrdquo Applied Mathematics andComputation vol 161 no 3 pp 733ndash744 2005
[2] A M Bica ldquoError estimation in the approximation of thesolution of nonlinear fuzzy Fredholm integral equationsrdquo Infor-mation Sciences vol 178 no 5 pp 1279ndash1292 2008
[3] A M Bica and C Popescu ldquoApproximating the solution ofnonlinear Hammerstein fuzzy integral equationsrdquo Fuzzy Setsand Systems vol 245 pp 1ndash17 2014
[4] M A Fariborzi Araghi and G Kazemi Gelian ldquoSolving fuzzyFredholm linear integral equations using Sinc method anddouble exponential transformationrdquo So Computing vol 19 no4 pp 1063ndash1070 2015
[5] M A Faborzi Araghi and N Parandin ldquoNumerical solution offuzzy Fredholm integral equations by the Lagrange interpola-tion based on the extension principlerdquo So Computing vol 15no 12 pp 2449ndash2456 2011
[6] H Hosseini Fadravi R Buzhabadi and H Saberi Nik ldquoSolvinglinear Fredholm fuzzy integral equations of the second kind byartificial neural networksrdquo Alexandria Engineering Journal vol53 no 1 pp 249ndash257 2014
[7] Y Jafarzadeh ldquoNumerical solution for fuzzy Fredholm integralequations with upper-bound on error by splines interpolationrdquoFuzzy Information and Engineering vol 4 no 3 pp 339ndash3472012
[8] A Molabahrami A Shidfar and A Ghyasi ldquoAn analyticalmethod for solving linear Fredholm fuzzy integral equations ofthe second kindrdquo Computers amp Mathematics with Applicationsvol 61 no 9 pp 2754ndash2761 2011
[9] N Parandin and M A Fariborzi Araghi ldquoThe approximatesolution of linear fuzzy Fredholm integral equations of thesecond kind by using iterative interpolationrdquoWorld Academy ofScience Engineering and Technology vol 49 pp 978ndash984 2009
[10] A M Bica and S Ziari ldquoIterative numerical method for fuzzyVolterra linear integral equations in two dimensionsrdquo SoComputing vol 21 no 5 pp 1097ndash1108 2017
[11] SM Sadatrasoul andR Ezzati ldquoQuadrature Rules and IterativeMethod for Numerical Solution of Two-Dimensional FuzzyIntegral Equationsrdquo Abstract and Applied Analysis vol 2014Article ID 413570 18 pages 2014
[12] SM Sadatrasoul andR Ezzati ldquoIterativemethod for numericalsolution of two-dimensional nonlinear fuzzy integral equa-tionsrdquo Fuzzy Sets and Systems vol 280 pp 91ndash106 2015
[13] F Mirzaee M K Yari and E Hadadiyan ldquoNumerical solutionof two-dimensional fuzzy Fredholm integral equations of thesecond kind using triangular functionsrdquo Beni-Suef UniversityJournal of Basic and Applied Sciences vol 4 no 2 pp 109ndash1182015
[14] H Nouriani and R Ezzati ldquoQuadrature iterative method fornumerical solution of two-dimensional linear fuzzy Fredholmintegral equationsrdquoMathematical Sciences vol 11 no 1 pp 63ndash72 2017
[15] R Ezzati and S Ziari ldquoNumerical solution of two-dimensionalfuzzy Fredholm integral equations of the second kind usingfuzzy bivariate Bernstein polynomialsrdquo International Journal ofFuzzy Systems vol 15 no 1 pp 84ndash89 2013
[16] N Hassasi and R Ezzati ldquoNumerical solution of two-dimensional fuzzy Fredholm integral equations using collo-cation fuzzy wavelet like operatorrdquo International Journal ofIndustrial Mathematics vol 7 p 11 2015
[17] M A Vali M J Agheli and S Gohari Nezhad ldquoHomotopyanalysis method to solve two-dimensional fuzzy Fredholmintegral equationrdquo General Mathematics Notes vol 22 pp 31ndash43 2014
[18] A M Bica and S Ziari ldquoOpen fuzzy cubature rule withapplication tononlinear fuzzyVolterra integral equations in twodimensionsrdquo Fuzzy Sets and Systems In press Corrected Proof
[19] A Rivaz and F Yousefi ldquoKernel iterative method for solvingtwo-dimensional fuzzy Fredholm integral equations of thesecond kindrdquo Journal of Fuzzy Set Valued Analysis vol 2013 p9 2013
[20] A Rivaz and F Yousefi ldquoModified homotopy perturbationmethod for solving two-dimensional fuzzy Fredholm integralequationrdquo International Journal of AppliedMathematics vol 25no 4 pp 591ndash602 2012
[21] A Rivaz F Yousefi and H Salehinejad ldquoUsing block pulsefunctions for solving two-dimensional fuzzy Fredholm integralequations of the second kindrdquo International Journal of AppliedMathematics vol 25 no 4 pp 571ndash582 2012
[22] S M Sadatrasoul and R Ezzati ldquoNumerical solution of two-dimensional nonlinear Hammerstein fuzzy integral equationsbased on optimal fuzzy quadrature formulardquo Journal of Compu-tational and Applied Mathematics vol 292 pp 430ndash446 2016
[23] S Ziari ldquoIterative method for solving two-dimensional non-linear fuzzy integral equations using fuzzy bivariate block-pulse functions with error estimationrdquo Iranian Journal of FuzzySystems vol 15 no 1 pp 55ndash76 183 2018
[24] G A Anastassiou Fuzzy Mathematics Approximation eoryvol 251 Springer 2010
[25] A R BozorgmaneshMOtadi A A Safe Kordi et al ldquoBarkhor-dari Ahmadi Lagrange two-dimensional interpolationmethodfor modeling nanoparticle formation during RESS processrdquoInternational Journal of Industrial Mathematics vol 1 pp 175ndash181 2009
Computer Games Technology
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Advances in
FuzzySystems
Hindawiwwwhindawicom
Volume 2018
International Journal of
ReconfigurableComputing
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
thinspArtificial Intelligence
Hindawiwwwhindawicom Volumethinsp2018
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawiwwwhindawicom Volume 2018
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Computational Intelligence and Neuroscience
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018
Human-ComputerInteraction
Advances in
Hindawiwwwhindawicom Volume 2018
Scientic Programming
Submit your manuscripts atwwwhindawicom
Computer Games Technology
International Journal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Journal ofEngineeringVolume 2018
Advances in
FuzzySystems
Hindawiwwwhindawicom
Volume 2018
International Journal of
ReconfigurableComputing
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Applied Computational Intelligence and Soft Computing
thinspAdvancesthinspinthinsp
thinspArtificial Intelligence
Hindawiwwwhindawicom Volumethinsp2018
Hindawiwwwhindawicom Volume 2018
Civil EngineeringAdvances in
Hindawiwwwhindawicom Volume 2018
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawiwwwhindawicom Volume 2018
Hindawi
wwwhindawicom Volume 2018
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
RoboticsJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Computational Intelligence and Neuroscience
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018
Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom
The Scientific World Journal
Volume 2018
Hindawiwwwhindawicom Volume 2018
Human-ComputerInteraction
Advances in
Hindawiwwwhindawicom Volume 2018
Scientic Programming
Submit your manuscripts atwwwhindawicom
