Analytic Solution for Systems of Two-Dimensional Time...

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Research Article Analytic Solution for Systems of Two-Dimensional Time Conformable Fractional PDEs by Using CFRDTM Abir Chaouk and Maher Jneid Department of Mathematics and Computer Science, Faculty of Science, Beirut Arab University, Beirut, Lebanon Correspondence should be addressed to Maher Jneid; [email protected] Received 2 January 2019; Accepted 24 March 2019; Published 11 April 2019 Academic Editor: Birendra Nath Mandal Copyright © 2019 Abir Chaouk and Maher Jneid. isis 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 we use the conformable fractional reduced differential transform (CFRDTM) method to compute solutions for systems of nonlinear conformable fractional PDEs. e proposed method yields a numerical approximate solution in the form of an infinite series that converges to a closed form solution, which is in many cases the exact solution. We inspect its efficiency in solving systems of CFPDEs by working on four different nonlinear systems. e results show that CFRDTM gave similar solutions to exact solutions, confirming its proficiency as a competent technique for solving CFPDEs systems. It required very little computational work and hence consumed much less time compared to other numerical methods. 1. Introduction Fractional partial differential equations (FPDEs) are mathe- matical tools employed to depict a broad variety of phenom- ena that arise in distinct areas of science and engineering like biology, physics, fluid mechanics, chemistry, and so on [1–4]. Recently investigators have centered their attention on study- ing the solution of partial differential equations of fractional order. However, till now not a single definition of fractional order derivative has been found [5]. erefore, researchers raced to set definitions for fractional order derivatives and consequently many definitions, mostly complicated, were proposed such as the Riemann-Liouville, Caputo, Hadamard, and others [5]. However, in 2014 an intriguing perspective to the subject was presented by Khalil [6] who came up with the conformable fractional derivative, an extension of the classical simple limit definition of derivative we all know. is new definition unlocked a whole new chapter in the world of research of FPDEs, in which many recent works have been done. To fully grasp these problems and to be able to analyze those phenomena, it is a necessity to find solutions for those systems of FPDEs or CFPDEs. But then again, finding a numerical or analytical solution for those systems is a very tough task to achieve, and exact solutions for them are not easily obtained. To tackle this problem, researchers worked on finding adequate methods for those systems of FPDEs. One of the greatest methods found by researchers is the RDTM. What distinguishes this method is that it provides us with analytical approximations, which in many cases are exact solutions, through convergent series with terms that are relatively easy to compute. Not to mention its ability in solving a huge variety of problems, which makes it a very significant mathematical tool. For example, in 2006, Bildik et al. [7] solved different partial differential equations using the RDM and ADM and performed a comparison with approximate solution and analytic solutions. Guptav [8] obtained the approximate solution of Benney-Lin equation through the RDTM and the HPM. Acan and Baleanu [9] handled three linear and nonlinear equations by CRDTM and CADM. For more applications of the present method, see [9–12] and some references cited therein. In this paper we propose the conformable fractional reduced differential transform method (CFRDTM) as a reliable method to solve systems of nonlinear CFPDEs. is paper is arranged as follows. In the second section we demonstrate the properties and theorems of the CFRDTM, in addition to reviewing some of the properties of the conformable derivative which are needed in the following work. In Section 3, we solve 4 systems by using CFRDTM Hindawi International Journal of Mathematics and Mathematical Sciences Volume 2019, Article ID 7869516, 7 pages https://doi.org/10.1155/2019/7869516

Transcript of Analytic Solution for Systems of Two-Dimensional Time...

Page 1: Analytic Solution for Systems of Two-Dimensional Time …downloads.hindawi.com/journals/ijmms/2019/7869516.pdf · 2019-07-30 · ResearchArticle Analytic Solution for Systems of Two-Dimensional

Research ArticleAnalytic Solution for Systems of Two-Dimensional TimeConformable Fractional PDEs by Using CFRDTM

Abir Chaouk and Maher Jneid

Department of Mathematics and Computer Science Faculty of Science Beirut Arab University Beirut Lebanon

Correspondence should be addressed to Maher Jneid mjneidbauedulb

Received 2 January 2019 Accepted 24 March 2019 Published 11 April 2019

Academic Editor Birendra Nath Mandal

Copyright copy 2019 Abir Chaouk andMaher JneidThisis an open access article distributed under theCreativeCommonsAttributionLicensewhichpermits unrestricteduse distribution and reproduction in anymedium provided the original work is properly cited

In this studywe use the conformable fractional reduced differential transform (CFRDTM)method to compute solutions for systemsof nonlinear conformable fractional PDEsThe proposedmethod yields a numerical approximate solution in the form of an infiniteseries that converges to a closed form solution which is in many cases the exact solutionWe inspect its efficiency in solving systemsofCFPDEs byworking on four different nonlinear systemsThe results show that CFRDTMgave similar solutions to exact solutionsconfirming its proficiency as a competent technique for solving CFPDEs systems It required very little computational work andhence consumed much less time compared to other numerical methods

1 Introduction

Fractional partial differential equations (FPDEs) are mathe-matical tools employed to depict a broad variety of phenom-ena that arise in distinct areas of science and engineering likebiology physics fluid mechanics chemistry and so on [1ndash4]Recently investigators have centered their attention on study-ing the solution of partial differential equations of fractionalorder However till now not a single definition of fractionalorder derivative has been found [5] Therefore researchersraced to set definitions for fractional order derivatives andconsequently many definitions mostly complicated wereproposed such as the Riemann-Liouville Caputo Hadamardand others [5] However in 2014 an intriguing perspective tothe subject was presented by Khalil [6] who came up withthe conformable fractional derivative an extension of theclassical simple limit definition of derivative we all knowThis new definition unlocked a whole new chapter in theworld of research of FPDEs in whichmany recent works havebeen done To fully grasp these problems and to be able toanalyze those phenomena it is a necessity to find solutions forthose systems of FPDEs or CFPDEs But then again findinga numerical or analytical solution for those systems is a verytough task to achieve and exact solutions for them are noteasily obtained

To tackle this problem researchers worked on findingadequate methods for those systems of FPDEs One of thegreatest methods found by researchers is the RDTM Whatdistinguishes this method is that it provides us with analyticalapproximations which in many cases are exact solutionsthrough convergent series with terms that are relativelyeasy to compute Not to mention its ability in solving ahuge variety of problems which makes it a very significantmathematical tool For example in 2006 Bildik et al [7]solved different partial differential equations using the RDMand ADM and performed a comparison with approximatesolution and analytic solutions Guptav [8] obtained theapproximate solution of Benney-Lin equation through theRDTM and the HPM Acan and Baleanu [9] handled threelinear and nonlinear equations by CRDTM and CADMFor more applications of the present method see [9ndash12]and some references cited therein In this paper we proposethe conformable fractional reduced differential transformmethod (CFRDTM) as a reliable method to solve systems ofnonlinear CFPDEs

This paper is arranged as follows In the second sectionwedemonstrate the properties and theorems of the CFRDTMin addition to reviewing some of the properties of theconformable derivative which are needed in the followingwork In Section 3 we solve 4 systems by using CFRDTM

HindawiInternational Journal of Mathematics and Mathematical SciencesVolume 2019 Article ID 7869516 7 pageshttpsdoiorg10115520197869516

2 International Journal of Mathematics and Mathematical Sciences

to test its efficiency In Section 4 we conclude this article byanalyzing the results we have obtained briefly

2 CFRDTM

In this section we introduce the CFRDTM which we willbe using to solve the system of conformable FPDEs in thispaper Throughout this section let x isin R119899 and q isin (0 1]Formore details about the definitions and properties given inthis section the reader can refer to [13ndash16] and the referencestherin

Definition 1 Given a function u R119899 times (0infin) 997888rarr R theconformable fractional partial derivative of 119906(119909 119905) of orderq with respect to t is given as

120597119902119905 119906 (119909 119905) = lim120576997888rarr0

119906 (119909 119905 + 1205761199051minus119902) minus 119906 (119909 119905)120576

(1)

provided that this limit exists as a finite number

Definition 2 Let 119906(119909 119905) be an infinitely partially q-differen-tiable function near 0 with respect to 119905 of order 119902 then theCFRDT of 119906(119909 119905) is given as

119880119902119896 (119909) =

1119902119896119896 [(120597

119902119905 )119896 119906 (119909 119905)]

119905=0(2)

where (120597119902119905 )119896119906(119909 119905) resembles applying the conformable par-tial fractional derivative 119896-times and 119880119902

119896(119909) is the CFRDT

function

Definition 3 Let 119880119902119896(119909) be the CFRDT of 119906(119909 119905) Then the

inverse CFRDT of 119880119902119896(119909) is given as

119906 (119909 119905) =infin

sum119896=0

119880119902119896 (119909) 119905119902119896

=infin

sum119896=0

1119902119896119896 [(120597

119902119905 )119896 119906 (119909 119905)]

119905=0119905119902119896

(3)

Definition 4 TheCFRDT of 119906(119909 119905) of the initial conditions isdefined as

119880119902119896 (119909) =

1(119902119896) [(120597

119902119905 )119896 119906]

119905=0if 119896119902 isin Z+

0 if 119896119902 notin Z+(4)

for 119896 = 0 1 2 (119899119902 minus 1) where 119899 is the order of CFPDETheorem5 Let 119906(119909 119905) V(119909 119905) and119908(119909 119905)R119899times[0infin) 997888rarr R

be partially q-differentiable at a point 119905 gt 0 and ab isin R Thenthe following is obtained

(1) If 119906(119909 119905) = V(119909 119905)119908(119909 119905) then 119880119902119896(119909) = sum119896119904=0 =

119881119902119896(119909 119904)119882119902

119896minus119904

(2) If V(119909 119905) = 120597119902119905 119906(119909 119905) then 119881119902119896(119909) = 119902(119896 + 1)119880119902

119896+1(119909)

(3) If 119906(119909 119905) = 119886V(119909 119905) plusmn 119887119908(119909 119905) then 119880119902119896(119909) = 119886119881119902

119896(119909) plusmn

119887119882119902119896(119909)

In general for 119906(119909 119905) = V1(119909 119905)V2(119909 119905) sdot sdot sdot V119899(119909 119905) then wehave

119880119902119896 (119909)

=119896119899

sum119896119899minus1=0

sdot sdot sdot1198962

sum1198961=0

119881119902(1)1198961119881119902

(2)1198962minus1198961times sdot sdot sdot 119881119902(119899minus1)119896119899minus1119881

119902

(119899)119896119899minus119896119899minus1

(5)

(4) If 119906(119909 119905) = 119905119898ℎ(119909) then 119880119902119896(119909) = 120575(119896 minus 119898119902)ℎ(119909)where

120575 (119896) =

1 if 119896 = 00 if 119896 = 0

(6)

3 Applications

To demonstrate the excellent performance of this method weapply it on four different systems in this section

Example 6 Consider the nonlinear system of CFDEs

120597119902119905 119906 = 1199062V minus 2119906 + 14 (12059721199092119906 + 12059721199102119906)

120597119902119905 V = 119906 minus 1199062V + 14 (12059721199092V + 12059721199102V)

(7)

with initial conditions

119906 (119909 119910 0) = 119890minus119909minus119910V (119909 119910 0) = 119890119909+119910

(8)

By applying CFRDTM on (7) we obtain the two recurrencerelations

119902 (119896 + 1)119880119902119896+1 =119896

sum119894=0

119894

sum119895=0

119880119902119894minus119895119880119902119895119881119902119896minus119894 minus 2119880119902119896

+ 14 (12059721199092119880119902119896 + 12059721199102119880119902119896)

119902 (119896 + 1) 119881119902119896+1 = 119880119902119896minus119896

sum119894=0

119894

sum119895=0

119880119902119894minus119895119880119902119895119881119902119896minus119894

+ 14 (12059721199092119881119902119896 + 12059721199102119881119902119896 )

(9)

where the IC (8) can be transformed at 119905 = 0 to1198801199020 = 119890minus119909minus119910

1198811199020 = 119890119909+119910(10)

Now substitute 119896 = 0 into (9) one obtains

1198801199021 =minus12119902 119890minus119909minus119910

1198811199021 =12119902119890119909+119910

(11)

International Journal of Mathematics and Mathematical Sciences 3

For k=1 and k=2 we get

1198801199022 =181199022 119890minus119909minus119910 = 1

2221199022 119890minus119909minus119910

1198811199022 =181199022 119890119909+119910 = 1

2221199022 119890119909+119910

(12)

and

1198801199023 =minus1481199023 119890

minus119909minus119910 = minus12331199023 119890

minus119909minus119910

1198811199023 =1

481199023 119890119909+119910 = 1

2331199023 119890119909+119910

(13)

Carrying on the iterative calculations in a similar way forother values of k we obtain the general terms

119880119902119899 =(minus1)1198992119899119899119902119899 119890

minus119909minus119910

119881119902119899 =1

2119899119899119902119899 119890119909+119910

(14)

Applying the inverse transformation of CFRDTM the solu-tion of (7) is given as

119906 (119909 119910 119905) =infin

sum119896=0

119880119902119896 (119909 119910 119905) 119905119896119902 = 119890minus119909minus119910minus1199051199022119902

V (119909 119910 119905) =infin

sum119896=0

119881119902119896 (119909 119910 119905) 119905119896119902 = 119890119909+119910+1199051199022119902(15)

Example 7 Consider the following system of CFDEs

120597119902119905 119906 = minus120597119909119906120597119909V minus 120597119910119906120597119910V minus 119906120597119902119905 V = minus120597119909V120597119909119908 + 120597119910V120597119910119908 + V

120597119902119905119908 = minus120597119909119908120597119909119906 minus 120597119910119908120597119910119906 minus 119908(16)

with the initial conditions

119906 (119909 119910 0) = 119890119909+119910V (119909 119910 0) = 119890119909minus119910119908 (119909 119910 0) = 119890minus119909+119910

(17)

Applying theorems of CFRDTM on (16) we obtain thefollowing system of recurrence relations

(119896 + 1)119880119902119896+1

= minus119896

sum119903=0

120597119909119880119902119896120597119909119881119902119896 minus119896

sum119903=0

120597119910119880119902119896120597119910119881119902119896 minus 119880119902119896

(119896 + 1) 119881119902119896+1 = minus119896

sum119903=0

120597119909119881119902119896 120597119909119882119902

119896+119896

sum119903=0

120597119910119881119902119896 120597119910119882119902

119896+ 119881119902119896

(119896 + 1)119882119902119896+1 = minus119896

sum119903=0

120597119909119882119902119896 120597119909119880119902119896 minus119896

sum119903=0

120597119910119882119902119896 120597119910119880119902119896 minus119882119902119896

(18)

where the IC can be transformed at 119905 = 0 to1198801199020 = 119890119909+119910

1198811199020 = 119890119909minus119910

1198821199020 = 119890minus119909+119910(19)

Substituting 119896 = 0 into (16) we obtain

1198801199021 =minus1119902 119890119909+119910

1198811199020 =1119902119890119909minus119910

1198821199020 =minus1119902 119890minus119909+119910

(20)

Similarly for k=1 and k=2 we get

1198801199022 =121199022 119890119909+119910

1198811199022 =121199022 119890119909minus119910

1198821199022 =121199022 119890minus119909+119910

(21)

and

1198801199023 = minus 161199023 119890119909+119910

1198811199023 =161199023 119890119909minus119910

1198821199023 = minus 161199023 119890minus119909+119910

(22)

Continuing the iterative calculations in a similar way forother values of k we obtain the general terms

119880119902119899 =(minus1)119899119899119902119899 119890

119909+119910

119881119902119899 =1

119899119902119899 119890119909minus119910

119882119902119899 =(minus1)119899119899119902119899 119890

minus119909+119910

(23)

Applying the inverse transformation of CFRDTM the solu-tion of (16) is given as

119906 (119909 119910 119905) =infin

sum119896=0

119880119902119896(119909 119910 119905) 119905119896119902 = 119890119909+119910minus119905119902119902

V (119909 119910 119905) =infin

sum119896=0

119881119902119896(119909 119910 119905) 119905119896119902 = 119890119909minus119910+119905119902119902

119908 (119909 119910 119905) =infin

sum119896=0

119882119902119896(119909 119910 119905) 119905119896119902 = 119890minus119909+119910minus119905119902119902

(24)

4 International Journal of Mathematics and Mathematical Sciences

Example 8 Consider the following system of CFDEs

120597119902119905 119906 + 119906120597119909119906 + V120597119910119906 = 12059721199092119906 + 12059721199102119906

120597119902119905 V + 119906120597119909V + V120597119910V = 12059721199092V + 12059721199102V(25)

subject to the initial conditions

119906 (119909 119910 0) = 119909 + 119910V (119909 119910 0) = 119909 minus 119910

(26)

Applying theorems of CFRDTM on (25) we obtain thefollowing system of recurrence relations

119902 (119896 + 1)119880119902119896+1 +119896

sum119903=0

119880119902119896minus119903120597119909119880119902119903 +119896

sum119903=0

119881119902119896minus119903120597119910119880119902119903

= 12059721199092119880119902119896 + 12059721199102119880119902119896

119902 (119896 + 1)119881119902119896+1 +119896

sum119903=0

119880119902119896minus119903120597119909119881119902119903 +119896

sum119903=0

119881119902119896minus119903120597119910119881119902119903

= 12059721199102119881119902119896 + +12059721199102119881119902119896

(27)

where the IC can be transformed at 119905 = 0 to1198801199020 = 119909 + 1199101198811199020 = 119909 minus 119910

(28)

Substituting 119896 = 0 into (25) we obtain1198801199021 =

minus2119902 119909

1198811199021 =minus2119902 119910

(29)

Similarly for 119896 = 1 119896 = 2 119896 = 3 and 119896 = 4 we getrespectively

1198801199022 =21199022 (119909 + 119910)

1198811199022 =21199022 (119909 minus 119910)

(30)

1198801199023 = minus 41199023119909

1198811199023 = minus 41199023119910

(31)

1198801199024 =41199023 (119909 + 119910)

1198811199024 = minus 41199023 (119909 minus 119910)

(32)

and

1198801199025 = minus 81199025119909

1198811199025 = minus 81199025119910

(33)

Applying the inverse transformation of CFRDTM the solu-tion of (25) is given as

119906 (119909 119910 119905) =infin

sum119896=0

119880119902119896(119909 119910 119905) 119905119896119902

= 119909 + 119910 minus 2119902119909119905119902 + 2

1199022 (119909 + 119910) 1199052119902 minus 411990231199091199053119902

+ 41199023 (119909 + 119910) 1199054119902 minus 8

11990251199091199055119902 + sdot sdot sdot

V (119909 119910 119905) =infin

sum119896=0

119881119902119896(119909 119910 119905) 119905119896119902

= 119909 minus 119910 minus 2119902119910119905119902 + 2

1199022 (119909 minus 119910) 1199052119902 minus 411990231199101199053119902

+ 41199023 (119909 minus 119910) 1199054119902 minus 8

11990251199101199055119902 + sdot sdot sdot

(34)

And by performing some computations the solutions aregiven as

119906 (119909 119910 119905) = 119909 + 119910 minus 21199091199051199021199021 minus 211990521199021199022

V (119909 119910 119905) = 119909 minus 119910 minus 21199091199051199021199021 minus 211990521199021199022

(35)

Example 9 Consider the following system of CFDEs

120597119902119905 119906 = V120597119909119906 + 120597119902119905 V120597119910119906 + 1 minus 119909 + 119910 + 119905119902119902

120597119902119905 V = 119906120597119909V + 120597119902119905 119906120597119910V + 1 minus 119909 minus 119910 minus 119905119902119902

(36)

subject to the initial conditions

119906 (119909 119910 0) = 119909 + 119910 minus 1V (119909 119910 0) = 119909 minus 119910 + 1

(37)

Applying theorems of CFRDTM on (25) we obtain thefollowing system of recurrence relations

119902 (119896 + 1)119880119902119896+1 =119896

sum119903=0

119881119902119903 120597119909119880119902119896minus119903

+119896

sum119903=0

119902 (119896 + 1) 119881119902119896+1minus119903120597119910119880119902119903 + 120575 (119896)

minus 119909 + 119910 minus 1 + 120575 (119896 minus 1)

119902 (119896 + 1) 119881119902119896+1 =119896

sum119903=0

119880119902119903 120597119909119880119902119896minus119903

+119896

sum119903=0

119902 (119896 + 1)119880119902119896+1minus119903120597119910119881119902119903 + 120575 (119896)

minus 119909 minus 119910 minus 1 + 120575 (119896 minus 1)

(38)

International Journal of Mathematics and Mathematical Sciences 5

00

00

00

00

00

00

minus05

minus05minus05

minus05minus05

minus10

minus15

minus20

05

05

05

15

05

0510

u(xy)v(xy)

Figure 1 The approximate solution of Example 9 for 120572 = 099 and t = 01

00

00

00

00

minus05

minus05minus05

minus05

minus10

minus15

05

05

00

00

minus0505

0515

05

10

u(xy) v(xy)

Figure 2The exact solution of Example 9 for 120572 = 099 and t = 01

where the IC can be transformed at 119905 = 0 to

1198801199020 = 119909 + 119910 minus 11198811199020 = 119909 minus 119910 + 1

(39)

Substituting 119896 = 0 into (36) we obtain

1198801199021 =1119902

1198811199021 = minus1119902

(40)

Similarly for 119896 = 1 and 119896 = 2 we get1198801199022 = minus 119909

2119902

1198811199022 =1119902 minus 119910 minus 1

2119902 (41)

and

1198801199023 =minus2119902119909 minus 119909

61199022 + 1 minus 5119902121199023

1198811199023 =minus1199103119902 minus 119902 + 1

121199023 (42)

Therefore the approximate analytical solution of (36) is givenas

119906 = 119909 + 119910 minus 1 + 119905119902119902 minus 1199091199052119902

2119902 + minus2119902119909 minus 11990961199022 1199053119902

+ 1 minus 5119902121199023 119905

3119902 + sdot sdot sdot

V = 119909 minus 119910 + 1 minus 119905119902119902 + 1119902 minus 119910 minus 1

2119902 1199052119902 + minus1199103119902 1199053119902

minus 119902 + 1121199023 119905

3119902 + sdot sdot sdot

(43)

Since this solution does not equal the exact one we usemathematica program to show its accuracy In Figures 1 and3 we plot the approximate solution for Example 9 for 119902 = 099at 119905 = 01 and at 119905 = 02 respectively The graphs in Figures2 and 4 illustrate the exact solution for Example 9 for 119902 =099 at 119905 = 01 and at 119905 = 02 respectively We can clearlyobserve at the time 119905 = 01 and 119905 = 02 that the values of theapproximate solution byCFRDTMand the values of the exactsolution obtained by other methods are extremely close Inother words solution provided by CFRDTM is highly preciseand accurate Particularly CFRDTM is a successful techniquefor solving systems of conformable FPDEs

6 International Journal of Mathematics and Mathematical Sciences

00

minus05

00

00

minus0505

0515

05

10

00

00

00

minus05

minus05

minus05

minus10

minus15

05

05

u(xy) v(xy)

Figure 3 The approximate solution of Example 9 for 120572 = 099 and t = 02

00

00

00

minus05

minus05

minus05minus10minus15

05

05

00

00

minus05

minus05

05

05

u(xy) v(xy)

00

15

05

10

Figure 4 The exact solution of Example 9 for 120572 = 099 and t = 02

4 Conclusion

In this paper our main purpose was to inspect the com-petence of the CFRDTM as a valid technique to solvesystems of nonlinear CFPDEs We successfully applied itto four distinct systems of nonlinear conformable time andspace FPDEs CFRDTM produced a numerical approximatesolution having the form of an infinite series that convergedto a closed form solution which coincided with the exactsolution in the first 3 applications In the 4119905ℎ applicationhowever we obtained an approximate solution that turnedout to be extremely close to the exact solution as we haveinterpreted from the attached Figures 1ndash4 On this basis weconclude that the CFRDTM is a powerful tool and a facileapproach for obtaining solutions for systems of CFPDEs Itreduces the amount of required computational work whencompared to other techniques used for this purpose and givessolutions that are in outstanding agreement with the exactones This provides a good starting point for further researchvia CFRDTM

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations North-HollandMathematics Studies New York NY USA 2006

[2] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010

[3] M D Ortigueira Fractional Calculus for Scientists and Engi-neers Heidelberg Springer Germany 2011

[4] T J Freeborn ldquoA survey of fractional-order circuit modelsfor biology and biomedicinerdquo IEEE Journal on Emerging andSelected Topics in Circuits and Systems vol 3 no 3 pp 416ndash4242013

[5] E C de Oliveira and J A Tenreiro Machado ldquoA review ofdefinitions for fractional derivatives and integralrdquoMathematicalProblems in Engineering vol 2014 Article ID 238459 6 pages2014

[6] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

International Journal of Mathematics and Mathematical Sciences 7

[7] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationbydifferential transformmethodandAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006

[8] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers ampMathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011

[9] O Acan and D Baleanu ldquoA new numerical technique forsolving fractional partial differential equationsrdquo Miskolc Math-ematical Notes vol 19 no 1 pp 3ndash18 2018

[10] M Jneid and A El Chakik ldquoAnalytical solution for somesystems of nonlinear conformable fractional differential equa-tionsrdquo Far East Journal of Mathematical Sciences (FJMS) vol109 no 2 pp 243ndash259 2018

[11] O S Iyiola and G O Ojo ldquoOn the analytical solution ofFornberg-Whitham equation with the new fractional deriva-tiverdquo PramanamdashJournal of Physics vol 85 no 4 pp 567ndash5752015

[12] M Jneid and A Chaouk ldquoThe conformable reduced differ-ential transform method for solving Newell-Whitehead-Segelequation with non-integer orderrdquo Journal of Analysis andApplications 2019

[13] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 no 1 pp889ndash898 2015

[15] O Acan O Firat Y Keskin and G Oturanc ldquoSolution ofconformable fractional partial differential equations by reduceddifferential transform methodrdquo Selcuk Journal of Applied Math-ematics 2016

[16] H Thabet and S Kendre ldquoAnalytical solutions for conformablespace-time fractional partial differential equations via frac-tional differential transformrdquo Chaos Solitons amp Fractals vol109 pp 238ndash245 2018

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Page 2: Analytic Solution for Systems of Two-Dimensional Time …downloads.hindawi.com/journals/ijmms/2019/7869516.pdf · 2019-07-30 · ResearchArticle Analytic Solution for Systems of Two-Dimensional

2 International Journal of Mathematics and Mathematical Sciences

to test its efficiency In Section 4 we conclude this article byanalyzing the results we have obtained briefly

2 CFRDTM

In this section we introduce the CFRDTM which we willbe using to solve the system of conformable FPDEs in thispaper Throughout this section let x isin R119899 and q isin (0 1]Formore details about the definitions and properties given inthis section the reader can refer to [13ndash16] and the referencestherin

Definition 1 Given a function u R119899 times (0infin) 997888rarr R theconformable fractional partial derivative of 119906(119909 119905) of orderq with respect to t is given as

120597119902119905 119906 (119909 119905) = lim120576997888rarr0

119906 (119909 119905 + 1205761199051minus119902) minus 119906 (119909 119905)120576

(1)

provided that this limit exists as a finite number

Definition 2 Let 119906(119909 119905) be an infinitely partially q-differen-tiable function near 0 with respect to 119905 of order 119902 then theCFRDT of 119906(119909 119905) is given as

119880119902119896 (119909) =

1119902119896119896 [(120597

119902119905 )119896 119906 (119909 119905)]

119905=0(2)

where (120597119902119905 )119896119906(119909 119905) resembles applying the conformable par-tial fractional derivative 119896-times and 119880119902

119896(119909) is the CFRDT

function

Definition 3 Let 119880119902119896(119909) be the CFRDT of 119906(119909 119905) Then the

inverse CFRDT of 119880119902119896(119909) is given as

119906 (119909 119905) =infin

sum119896=0

119880119902119896 (119909) 119905119902119896

=infin

sum119896=0

1119902119896119896 [(120597

119902119905 )119896 119906 (119909 119905)]

119905=0119905119902119896

(3)

Definition 4 TheCFRDT of 119906(119909 119905) of the initial conditions isdefined as

119880119902119896 (119909) =

1(119902119896) [(120597

119902119905 )119896 119906]

119905=0if 119896119902 isin Z+

0 if 119896119902 notin Z+(4)

for 119896 = 0 1 2 (119899119902 minus 1) where 119899 is the order of CFPDETheorem5 Let 119906(119909 119905) V(119909 119905) and119908(119909 119905)R119899times[0infin) 997888rarr R

be partially q-differentiable at a point 119905 gt 0 and ab isin R Thenthe following is obtained

(1) If 119906(119909 119905) = V(119909 119905)119908(119909 119905) then 119880119902119896(119909) = sum119896119904=0 =

119881119902119896(119909 119904)119882119902

119896minus119904

(2) If V(119909 119905) = 120597119902119905 119906(119909 119905) then 119881119902119896(119909) = 119902(119896 + 1)119880119902

119896+1(119909)

(3) If 119906(119909 119905) = 119886V(119909 119905) plusmn 119887119908(119909 119905) then 119880119902119896(119909) = 119886119881119902

119896(119909) plusmn

119887119882119902119896(119909)

In general for 119906(119909 119905) = V1(119909 119905)V2(119909 119905) sdot sdot sdot V119899(119909 119905) then wehave

119880119902119896 (119909)

=119896119899

sum119896119899minus1=0

sdot sdot sdot1198962

sum1198961=0

119881119902(1)1198961119881119902

(2)1198962minus1198961times sdot sdot sdot 119881119902(119899minus1)119896119899minus1119881

119902

(119899)119896119899minus119896119899minus1

(5)

(4) If 119906(119909 119905) = 119905119898ℎ(119909) then 119880119902119896(119909) = 120575(119896 minus 119898119902)ℎ(119909)where

120575 (119896) =

1 if 119896 = 00 if 119896 = 0

(6)

3 Applications

To demonstrate the excellent performance of this method weapply it on four different systems in this section

Example 6 Consider the nonlinear system of CFDEs

120597119902119905 119906 = 1199062V minus 2119906 + 14 (12059721199092119906 + 12059721199102119906)

120597119902119905 V = 119906 minus 1199062V + 14 (12059721199092V + 12059721199102V)

(7)

with initial conditions

119906 (119909 119910 0) = 119890minus119909minus119910V (119909 119910 0) = 119890119909+119910

(8)

By applying CFRDTM on (7) we obtain the two recurrencerelations

119902 (119896 + 1)119880119902119896+1 =119896

sum119894=0

119894

sum119895=0

119880119902119894minus119895119880119902119895119881119902119896minus119894 minus 2119880119902119896

+ 14 (12059721199092119880119902119896 + 12059721199102119880119902119896)

119902 (119896 + 1) 119881119902119896+1 = 119880119902119896minus119896

sum119894=0

119894

sum119895=0

119880119902119894minus119895119880119902119895119881119902119896minus119894

+ 14 (12059721199092119881119902119896 + 12059721199102119881119902119896 )

(9)

where the IC (8) can be transformed at 119905 = 0 to1198801199020 = 119890minus119909minus119910

1198811199020 = 119890119909+119910(10)

Now substitute 119896 = 0 into (9) one obtains

1198801199021 =minus12119902 119890minus119909minus119910

1198811199021 =12119902119890119909+119910

(11)

International Journal of Mathematics and Mathematical Sciences 3

For k=1 and k=2 we get

1198801199022 =181199022 119890minus119909minus119910 = 1

2221199022 119890minus119909minus119910

1198811199022 =181199022 119890119909+119910 = 1

2221199022 119890119909+119910

(12)

and

1198801199023 =minus1481199023 119890

minus119909minus119910 = minus12331199023 119890

minus119909minus119910

1198811199023 =1

481199023 119890119909+119910 = 1

2331199023 119890119909+119910

(13)

Carrying on the iterative calculations in a similar way forother values of k we obtain the general terms

119880119902119899 =(minus1)1198992119899119899119902119899 119890

minus119909minus119910

119881119902119899 =1

2119899119899119902119899 119890119909+119910

(14)

Applying the inverse transformation of CFRDTM the solu-tion of (7) is given as

119906 (119909 119910 119905) =infin

sum119896=0

119880119902119896 (119909 119910 119905) 119905119896119902 = 119890minus119909minus119910minus1199051199022119902

V (119909 119910 119905) =infin

sum119896=0

119881119902119896 (119909 119910 119905) 119905119896119902 = 119890119909+119910+1199051199022119902(15)

Example 7 Consider the following system of CFDEs

120597119902119905 119906 = minus120597119909119906120597119909V minus 120597119910119906120597119910V minus 119906120597119902119905 V = minus120597119909V120597119909119908 + 120597119910V120597119910119908 + V

120597119902119905119908 = minus120597119909119908120597119909119906 minus 120597119910119908120597119910119906 minus 119908(16)

with the initial conditions

119906 (119909 119910 0) = 119890119909+119910V (119909 119910 0) = 119890119909minus119910119908 (119909 119910 0) = 119890minus119909+119910

(17)

Applying theorems of CFRDTM on (16) we obtain thefollowing system of recurrence relations

(119896 + 1)119880119902119896+1

= minus119896

sum119903=0

120597119909119880119902119896120597119909119881119902119896 minus119896

sum119903=0

120597119910119880119902119896120597119910119881119902119896 minus 119880119902119896

(119896 + 1) 119881119902119896+1 = minus119896

sum119903=0

120597119909119881119902119896 120597119909119882119902

119896+119896

sum119903=0

120597119910119881119902119896 120597119910119882119902

119896+ 119881119902119896

(119896 + 1)119882119902119896+1 = minus119896

sum119903=0

120597119909119882119902119896 120597119909119880119902119896 minus119896

sum119903=0

120597119910119882119902119896 120597119910119880119902119896 minus119882119902119896

(18)

where the IC can be transformed at 119905 = 0 to1198801199020 = 119890119909+119910

1198811199020 = 119890119909minus119910

1198821199020 = 119890minus119909+119910(19)

Substituting 119896 = 0 into (16) we obtain

1198801199021 =minus1119902 119890119909+119910

1198811199020 =1119902119890119909minus119910

1198821199020 =minus1119902 119890minus119909+119910

(20)

Similarly for k=1 and k=2 we get

1198801199022 =121199022 119890119909+119910

1198811199022 =121199022 119890119909minus119910

1198821199022 =121199022 119890minus119909+119910

(21)

and

1198801199023 = minus 161199023 119890119909+119910

1198811199023 =161199023 119890119909minus119910

1198821199023 = minus 161199023 119890minus119909+119910

(22)

Continuing the iterative calculations in a similar way forother values of k we obtain the general terms

119880119902119899 =(minus1)119899119899119902119899 119890

119909+119910

119881119902119899 =1

119899119902119899 119890119909minus119910

119882119902119899 =(minus1)119899119899119902119899 119890

minus119909+119910

(23)

Applying the inverse transformation of CFRDTM the solu-tion of (16) is given as

119906 (119909 119910 119905) =infin

sum119896=0

119880119902119896(119909 119910 119905) 119905119896119902 = 119890119909+119910minus119905119902119902

V (119909 119910 119905) =infin

sum119896=0

119881119902119896(119909 119910 119905) 119905119896119902 = 119890119909minus119910+119905119902119902

119908 (119909 119910 119905) =infin

sum119896=0

119882119902119896(119909 119910 119905) 119905119896119902 = 119890minus119909+119910minus119905119902119902

(24)

4 International Journal of Mathematics and Mathematical Sciences

Example 8 Consider the following system of CFDEs

120597119902119905 119906 + 119906120597119909119906 + V120597119910119906 = 12059721199092119906 + 12059721199102119906

120597119902119905 V + 119906120597119909V + V120597119910V = 12059721199092V + 12059721199102V(25)

subject to the initial conditions

119906 (119909 119910 0) = 119909 + 119910V (119909 119910 0) = 119909 minus 119910

(26)

Applying theorems of CFRDTM on (25) we obtain thefollowing system of recurrence relations

119902 (119896 + 1)119880119902119896+1 +119896

sum119903=0

119880119902119896minus119903120597119909119880119902119903 +119896

sum119903=0

119881119902119896minus119903120597119910119880119902119903

= 12059721199092119880119902119896 + 12059721199102119880119902119896

119902 (119896 + 1)119881119902119896+1 +119896

sum119903=0

119880119902119896minus119903120597119909119881119902119903 +119896

sum119903=0

119881119902119896minus119903120597119910119881119902119903

= 12059721199102119881119902119896 + +12059721199102119881119902119896

(27)

where the IC can be transformed at 119905 = 0 to1198801199020 = 119909 + 1199101198811199020 = 119909 minus 119910

(28)

Substituting 119896 = 0 into (25) we obtain1198801199021 =

minus2119902 119909

1198811199021 =minus2119902 119910

(29)

Similarly for 119896 = 1 119896 = 2 119896 = 3 and 119896 = 4 we getrespectively

1198801199022 =21199022 (119909 + 119910)

1198811199022 =21199022 (119909 minus 119910)

(30)

1198801199023 = minus 41199023119909

1198811199023 = minus 41199023119910

(31)

1198801199024 =41199023 (119909 + 119910)

1198811199024 = minus 41199023 (119909 minus 119910)

(32)

and

1198801199025 = minus 81199025119909

1198811199025 = minus 81199025119910

(33)

Applying the inverse transformation of CFRDTM the solu-tion of (25) is given as

119906 (119909 119910 119905) =infin

sum119896=0

119880119902119896(119909 119910 119905) 119905119896119902

= 119909 + 119910 minus 2119902119909119905119902 + 2

1199022 (119909 + 119910) 1199052119902 minus 411990231199091199053119902

+ 41199023 (119909 + 119910) 1199054119902 minus 8

11990251199091199055119902 + sdot sdot sdot

V (119909 119910 119905) =infin

sum119896=0

119881119902119896(119909 119910 119905) 119905119896119902

= 119909 minus 119910 minus 2119902119910119905119902 + 2

1199022 (119909 minus 119910) 1199052119902 minus 411990231199101199053119902

+ 41199023 (119909 minus 119910) 1199054119902 minus 8

11990251199101199055119902 + sdot sdot sdot

(34)

And by performing some computations the solutions aregiven as

119906 (119909 119910 119905) = 119909 + 119910 minus 21199091199051199021199021 minus 211990521199021199022

V (119909 119910 119905) = 119909 minus 119910 minus 21199091199051199021199021 minus 211990521199021199022

(35)

Example 9 Consider the following system of CFDEs

120597119902119905 119906 = V120597119909119906 + 120597119902119905 V120597119910119906 + 1 minus 119909 + 119910 + 119905119902119902

120597119902119905 V = 119906120597119909V + 120597119902119905 119906120597119910V + 1 minus 119909 minus 119910 minus 119905119902119902

(36)

subject to the initial conditions

119906 (119909 119910 0) = 119909 + 119910 minus 1V (119909 119910 0) = 119909 minus 119910 + 1

(37)

Applying theorems of CFRDTM on (25) we obtain thefollowing system of recurrence relations

119902 (119896 + 1)119880119902119896+1 =119896

sum119903=0

119881119902119903 120597119909119880119902119896minus119903

+119896

sum119903=0

119902 (119896 + 1) 119881119902119896+1minus119903120597119910119880119902119903 + 120575 (119896)

minus 119909 + 119910 minus 1 + 120575 (119896 minus 1)

119902 (119896 + 1) 119881119902119896+1 =119896

sum119903=0

119880119902119903 120597119909119880119902119896minus119903

+119896

sum119903=0

119902 (119896 + 1)119880119902119896+1minus119903120597119910119881119902119903 + 120575 (119896)

minus 119909 minus 119910 minus 1 + 120575 (119896 minus 1)

(38)

International Journal of Mathematics and Mathematical Sciences 5

00

00

00

00

00

00

minus05

minus05minus05

minus05minus05

minus10

minus15

minus20

05

05

05

15

05

0510

u(xy)v(xy)

Figure 1 The approximate solution of Example 9 for 120572 = 099 and t = 01

00

00

00

00

minus05

minus05minus05

minus05

minus10

minus15

05

05

00

00

minus0505

0515

05

10

u(xy) v(xy)

Figure 2The exact solution of Example 9 for 120572 = 099 and t = 01

where the IC can be transformed at 119905 = 0 to

1198801199020 = 119909 + 119910 minus 11198811199020 = 119909 minus 119910 + 1

(39)

Substituting 119896 = 0 into (36) we obtain

1198801199021 =1119902

1198811199021 = minus1119902

(40)

Similarly for 119896 = 1 and 119896 = 2 we get1198801199022 = minus 119909

2119902

1198811199022 =1119902 minus 119910 minus 1

2119902 (41)

and

1198801199023 =minus2119902119909 minus 119909

61199022 + 1 minus 5119902121199023

1198811199023 =minus1199103119902 minus 119902 + 1

121199023 (42)

Therefore the approximate analytical solution of (36) is givenas

119906 = 119909 + 119910 minus 1 + 119905119902119902 minus 1199091199052119902

2119902 + minus2119902119909 minus 11990961199022 1199053119902

+ 1 minus 5119902121199023 119905

3119902 + sdot sdot sdot

V = 119909 minus 119910 + 1 minus 119905119902119902 + 1119902 minus 119910 minus 1

2119902 1199052119902 + minus1199103119902 1199053119902

minus 119902 + 1121199023 119905

3119902 + sdot sdot sdot

(43)

Since this solution does not equal the exact one we usemathematica program to show its accuracy In Figures 1 and3 we plot the approximate solution for Example 9 for 119902 = 099at 119905 = 01 and at 119905 = 02 respectively The graphs in Figures2 and 4 illustrate the exact solution for Example 9 for 119902 =099 at 119905 = 01 and at 119905 = 02 respectively We can clearlyobserve at the time 119905 = 01 and 119905 = 02 that the values of theapproximate solution byCFRDTMand the values of the exactsolution obtained by other methods are extremely close Inother words solution provided by CFRDTM is highly preciseand accurate Particularly CFRDTM is a successful techniquefor solving systems of conformable FPDEs

6 International Journal of Mathematics and Mathematical Sciences

00

minus05

00

00

minus0505

0515

05

10

00

00

00

minus05

minus05

minus05

minus10

minus15

05

05

u(xy) v(xy)

Figure 3 The approximate solution of Example 9 for 120572 = 099 and t = 02

00

00

00

minus05

minus05

minus05minus10minus15

05

05

00

00

minus05

minus05

05

05

u(xy) v(xy)

00

15

05

10

Figure 4 The exact solution of Example 9 for 120572 = 099 and t = 02

4 Conclusion

In this paper our main purpose was to inspect the com-petence of the CFRDTM as a valid technique to solvesystems of nonlinear CFPDEs We successfully applied itto four distinct systems of nonlinear conformable time andspace FPDEs CFRDTM produced a numerical approximatesolution having the form of an infinite series that convergedto a closed form solution which coincided with the exactsolution in the first 3 applications In the 4119905ℎ applicationhowever we obtained an approximate solution that turnedout to be extremely close to the exact solution as we haveinterpreted from the attached Figures 1ndash4 On this basis weconclude that the CFRDTM is a powerful tool and a facileapproach for obtaining solutions for systems of CFPDEs Itreduces the amount of required computational work whencompared to other techniques used for this purpose and givessolutions that are in outstanding agreement with the exactones This provides a good starting point for further researchvia CFRDTM

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations North-HollandMathematics Studies New York NY USA 2006

[2] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010

[3] M D Ortigueira Fractional Calculus for Scientists and Engi-neers Heidelberg Springer Germany 2011

[4] T J Freeborn ldquoA survey of fractional-order circuit modelsfor biology and biomedicinerdquo IEEE Journal on Emerging andSelected Topics in Circuits and Systems vol 3 no 3 pp 416ndash4242013

[5] E C de Oliveira and J A Tenreiro Machado ldquoA review ofdefinitions for fractional derivatives and integralrdquoMathematicalProblems in Engineering vol 2014 Article ID 238459 6 pages2014

[6] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

International Journal of Mathematics and Mathematical Sciences 7

[7] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationbydifferential transformmethodandAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006

[8] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers ampMathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011

[9] O Acan and D Baleanu ldquoA new numerical technique forsolving fractional partial differential equationsrdquo Miskolc Math-ematical Notes vol 19 no 1 pp 3ndash18 2018

[10] M Jneid and A El Chakik ldquoAnalytical solution for somesystems of nonlinear conformable fractional differential equa-tionsrdquo Far East Journal of Mathematical Sciences (FJMS) vol109 no 2 pp 243ndash259 2018

[11] O S Iyiola and G O Ojo ldquoOn the analytical solution ofFornberg-Whitham equation with the new fractional deriva-tiverdquo PramanamdashJournal of Physics vol 85 no 4 pp 567ndash5752015

[12] M Jneid and A Chaouk ldquoThe conformable reduced differ-ential transform method for solving Newell-Whitehead-Segelequation with non-integer orderrdquo Journal of Analysis andApplications 2019

[13] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 no 1 pp889ndash898 2015

[15] O Acan O Firat Y Keskin and G Oturanc ldquoSolution ofconformable fractional partial differential equations by reduceddifferential transform methodrdquo Selcuk Journal of Applied Math-ematics 2016

[16] H Thabet and S Kendre ldquoAnalytical solutions for conformablespace-time fractional partial differential equations via frac-tional differential transformrdquo Chaos Solitons amp Fractals vol109 pp 238ndash245 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 3: Analytic Solution for Systems of Two-Dimensional Time …downloads.hindawi.com/journals/ijmms/2019/7869516.pdf · 2019-07-30 · ResearchArticle Analytic Solution for Systems of Two-Dimensional

International Journal of Mathematics and Mathematical Sciences 3

For k=1 and k=2 we get

1198801199022 =181199022 119890minus119909minus119910 = 1

2221199022 119890minus119909minus119910

1198811199022 =181199022 119890119909+119910 = 1

2221199022 119890119909+119910

(12)

and

1198801199023 =minus1481199023 119890

minus119909minus119910 = minus12331199023 119890

minus119909minus119910

1198811199023 =1

481199023 119890119909+119910 = 1

2331199023 119890119909+119910

(13)

Carrying on the iterative calculations in a similar way forother values of k we obtain the general terms

119880119902119899 =(minus1)1198992119899119899119902119899 119890

minus119909minus119910

119881119902119899 =1

2119899119899119902119899 119890119909+119910

(14)

Applying the inverse transformation of CFRDTM the solu-tion of (7) is given as

119906 (119909 119910 119905) =infin

sum119896=0

119880119902119896 (119909 119910 119905) 119905119896119902 = 119890minus119909minus119910minus1199051199022119902

V (119909 119910 119905) =infin

sum119896=0

119881119902119896 (119909 119910 119905) 119905119896119902 = 119890119909+119910+1199051199022119902(15)

Example 7 Consider the following system of CFDEs

120597119902119905 119906 = minus120597119909119906120597119909V minus 120597119910119906120597119910V minus 119906120597119902119905 V = minus120597119909V120597119909119908 + 120597119910V120597119910119908 + V

120597119902119905119908 = minus120597119909119908120597119909119906 minus 120597119910119908120597119910119906 minus 119908(16)

with the initial conditions

119906 (119909 119910 0) = 119890119909+119910V (119909 119910 0) = 119890119909minus119910119908 (119909 119910 0) = 119890minus119909+119910

(17)

Applying theorems of CFRDTM on (16) we obtain thefollowing system of recurrence relations

(119896 + 1)119880119902119896+1

= minus119896

sum119903=0

120597119909119880119902119896120597119909119881119902119896 minus119896

sum119903=0

120597119910119880119902119896120597119910119881119902119896 minus 119880119902119896

(119896 + 1) 119881119902119896+1 = minus119896

sum119903=0

120597119909119881119902119896 120597119909119882119902

119896+119896

sum119903=0

120597119910119881119902119896 120597119910119882119902

119896+ 119881119902119896

(119896 + 1)119882119902119896+1 = minus119896

sum119903=0

120597119909119882119902119896 120597119909119880119902119896 minus119896

sum119903=0

120597119910119882119902119896 120597119910119880119902119896 minus119882119902119896

(18)

where the IC can be transformed at 119905 = 0 to1198801199020 = 119890119909+119910

1198811199020 = 119890119909minus119910

1198821199020 = 119890minus119909+119910(19)

Substituting 119896 = 0 into (16) we obtain

1198801199021 =minus1119902 119890119909+119910

1198811199020 =1119902119890119909minus119910

1198821199020 =minus1119902 119890minus119909+119910

(20)

Similarly for k=1 and k=2 we get

1198801199022 =121199022 119890119909+119910

1198811199022 =121199022 119890119909minus119910

1198821199022 =121199022 119890minus119909+119910

(21)

and

1198801199023 = minus 161199023 119890119909+119910

1198811199023 =161199023 119890119909minus119910

1198821199023 = minus 161199023 119890minus119909+119910

(22)

Continuing the iterative calculations in a similar way forother values of k we obtain the general terms

119880119902119899 =(minus1)119899119899119902119899 119890

119909+119910

119881119902119899 =1

119899119902119899 119890119909minus119910

119882119902119899 =(minus1)119899119899119902119899 119890

minus119909+119910

(23)

Applying the inverse transformation of CFRDTM the solu-tion of (16) is given as

119906 (119909 119910 119905) =infin

sum119896=0

119880119902119896(119909 119910 119905) 119905119896119902 = 119890119909+119910minus119905119902119902

V (119909 119910 119905) =infin

sum119896=0

119881119902119896(119909 119910 119905) 119905119896119902 = 119890119909minus119910+119905119902119902

119908 (119909 119910 119905) =infin

sum119896=0

119882119902119896(119909 119910 119905) 119905119896119902 = 119890minus119909+119910minus119905119902119902

(24)

4 International Journal of Mathematics and Mathematical Sciences

Example 8 Consider the following system of CFDEs

120597119902119905 119906 + 119906120597119909119906 + V120597119910119906 = 12059721199092119906 + 12059721199102119906

120597119902119905 V + 119906120597119909V + V120597119910V = 12059721199092V + 12059721199102V(25)

subject to the initial conditions

119906 (119909 119910 0) = 119909 + 119910V (119909 119910 0) = 119909 minus 119910

(26)

Applying theorems of CFRDTM on (25) we obtain thefollowing system of recurrence relations

119902 (119896 + 1)119880119902119896+1 +119896

sum119903=0

119880119902119896minus119903120597119909119880119902119903 +119896

sum119903=0

119881119902119896minus119903120597119910119880119902119903

= 12059721199092119880119902119896 + 12059721199102119880119902119896

119902 (119896 + 1)119881119902119896+1 +119896

sum119903=0

119880119902119896minus119903120597119909119881119902119903 +119896

sum119903=0

119881119902119896minus119903120597119910119881119902119903

= 12059721199102119881119902119896 + +12059721199102119881119902119896

(27)

where the IC can be transformed at 119905 = 0 to1198801199020 = 119909 + 1199101198811199020 = 119909 minus 119910

(28)

Substituting 119896 = 0 into (25) we obtain1198801199021 =

minus2119902 119909

1198811199021 =minus2119902 119910

(29)

Similarly for 119896 = 1 119896 = 2 119896 = 3 and 119896 = 4 we getrespectively

1198801199022 =21199022 (119909 + 119910)

1198811199022 =21199022 (119909 minus 119910)

(30)

1198801199023 = minus 41199023119909

1198811199023 = minus 41199023119910

(31)

1198801199024 =41199023 (119909 + 119910)

1198811199024 = minus 41199023 (119909 minus 119910)

(32)

and

1198801199025 = minus 81199025119909

1198811199025 = minus 81199025119910

(33)

Applying the inverse transformation of CFRDTM the solu-tion of (25) is given as

119906 (119909 119910 119905) =infin

sum119896=0

119880119902119896(119909 119910 119905) 119905119896119902

= 119909 + 119910 minus 2119902119909119905119902 + 2

1199022 (119909 + 119910) 1199052119902 minus 411990231199091199053119902

+ 41199023 (119909 + 119910) 1199054119902 minus 8

11990251199091199055119902 + sdot sdot sdot

V (119909 119910 119905) =infin

sum119896=0

119881119902119896(119909 119910 119905) 119905119896119902

= 119909 minus 119910 minus 2119902119910119905119902 + 2

1199022 (119909 minus 119910) 1199052119902 minus 411990231199101199053119902

+ 41199023 (119909 minus 119910) 1199054119902 minus 8

11990251199101199055119902 + sdot sdot sdot

(34)

And by performing some computations the solutions aregiven as

119906 (119909 119910 119905) = 119909 + 119910 minus 21199091199051199021199021 minus 211990521199021199022

V (119909 119910 119905) = 119909 minus 119910 minus 21199091199051199021199021 minus 211990521199021199022

(35)

Example 9 Consider the following system of CFDEs

120597119902119905 119906 = V120597119909119906 + 120597119902119905 V120597119910119906 + 1 minus 119909 + 119910 + 119905119902119902

120597119902119905 V = 119906120597119909V + 120597119902119905 119906120597119910V + 1 minus 119909 minus 119910 minus 119905119902119902

(36)

subject to the initial conditions

119906 (119909 119910 0) = 119909 + 119910 minus 1V (119909 119910 0) = 119909 minus 119910 + 1

(37)

Applying theorems of CFRDTM on (25) we obtain thefollowing system of recurrence relations

119902 (119896 + 1)119880119902119896+1 =119896

sum119903=0

119881119902119903 120597119909119880119902119896minus119903

+119896

sum119903=0

119902 (119896 + 1) 119881119902119896+1minus119903120597119910119880119902119903 + 120575 (119896)

minus 119909 + 119910 minus 1 + 120575 (119896 minus 1)

119902 (119896 + 1) 119881119902119896+1 =119896

sum119903=0

119880119902119903 120597119909119880119902119896minus119903

+119896

sum119903=0

119902 (119896 + 1)119880119902119896+1minus119903120597119910119881119902119903 + 120575 (119896)

minus 119909 minus 119910 minus 1 + 120575 (119896 minus 1)

(38)

International Journal of Mathematics and Mathematical Sciences 5

00

00

00

00

00

00

minus05

minus05minus05

minus05minus05

minus10

minus15

minus20

05

05

05

15

05

0510

u(xy)v(xy)

Figure 1 The approximate solution of Example 9 for 120572 = 099 and t = 01

00

00

00

00

minus05

minus05minus05

minus05

minus10

minus15

05

05

00

00

minus0505

0515

05

10

u(xy) v(xy)

Figure 2The exact solution of Example 9 for 120572 = 099 and t = 01

where the IC can be transformed at 119905 = 0 to

1198801199020 = 119909 + 119910 minus 11198811199020 = 119909 minus 119910 + 1

(39)

Substituting 119896 = 0 into (36) we obtain

1198801199021 =1119902

1198811199021 = minus1119902

(40)

Similarly for 119896 = 1 and 119896 = 2 we get1198801199022 = minus 119909

2119902

1198811199022 =1119902 minus 119910 minus 1

2119902 (41)

and

1198801199023 =minus2119902119909 minus 119909

61199022 + 1 minus 5119902121199023

1198811199023 =minus1199103119902 minus 119902 + 1

121199023 (42)

Therefore the approximate analytical solution of (36) is givenas

119906 = 119909 + 119910 minus 1 + 119905119902119902 minus 1199091199052119902

2119902 + minus2119902119909 minus 11990961199022 1199053119902

+ 1 minus 5119902121199023 119905

3119902 + sdot sdot sdot

V = 119909 minus 119910 + 1 minus 119905119902119902 + 1119902 minus 119910 minus 1

2119902 1199052119902 + minus1199103119902 1199053119902

minus 119902 + 1121199023 119905

3119902 + sdot sdot sdot

(43)

Since this solution does not equal the exact one we usemathematica program to show its accuracy In Figures 1 and3 we plot the approximate solution for Example 9 for 119902 = 099at 119905 = 01 and at 119905 = 02 respectively The graphs in Figures2 and 4 illustrate the exact solution for Example 9 for 119902 =099 at 119905 = 01 and at 119905 = 02 respectively We can clearlyobserve at the time 119905 = 01 and 119905 = 02 that the values of theapproximate solution byCFRDTMand the values of the exactsolution obtained by other methods are extremely close Inother words solution provided by CFRDTM is highly preciseand accurate Particularly CFRDTM is a successful techniquefor solving systems of conformable FPDEs

6 International Journal of Mathematics and Mathematical Sciences

00

minus05

00

00

minus0505

0515

05

10

00

00

00

minus05

minus05

minus05

minus10

minus15

05

05

u(xy) v(xy)

Figure 3 The approximate solution of Example 9 for 120572 = 099 and t = 02

00

00

00

minus05

minus05

minus05minus10minus15

05

05

00

00

minus05

minus05

05

05

u(xy) v(xy)

00

15

05

10

Figure 4 The exact solution of Example 9 for 120572 = 099 and t = 02

4 Conclusion

In this paper our main purpose was to inspect the com-petence of the CFRDTM as a valid technique to solvesystems of nonlinear CFPDEs We successfully applied itto four distinct systems of nonlinear conformable time andspace FPDEs CFRDTM produced a numerical approximatesolution having the form of an infinite series that convergedto a closed form solution which coincided with the exactsolution in the first 3 applications In the 4119905ℎ applicationhowever we obtained an approximate solution that turnedout to be extremely close to the exact solution as we haveinterpreted from the attached Figures 1ndash4 On this basis weconclude that the CFRDTM is a powerful tool and a facileapproach for obtaining solutions for systems of CFPDEs Itreduces the amount of required computational work whencompared to other techniques used for this purpose and givessolutions that are in outstanding agreement with the exactones This provides a good starting point for further researchvia CFRDTM

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations North-HollandMathematics Studies New York NY USA 2006

[2] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010

[3] M D Ortigueira Fractional Calculus for Scientists and Engi-neers Heidelberg Springer Germany 2011

[4] T J Freeborn ldquoA survey of fractional-order circuit modelsfor biology and biomedicinerdquo IEEE Journal on Emerging andSelected Topics in Circuits and Systems vol 3 no 3 pp 416ndash4242013

[5] E C de Oliveira and J A Tenreiro Machado ldquoA review ofdefinitions for fractional derivatives and integralrdquoMathematicalProblems in Engineering vol 2014 Article ID 238459 6 pages2014

[6] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

International Journal of Mathematics and Mathematical Sciences 7

[7] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationbydifferential transformmethodandAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006

[8] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers ampMathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011

[9] O Acan and D Baleanu ldquoA new numerical technique forsolving fractional partial differential equationsrdquo Miskolc Math-ematical Notes vol 19 no 1 pp 3ndash18 2018

[10] M Jneid and A El Chakik ldquoAnalytical solution for somesystems of nonlinear conformable fractional differential equa-tionsrdquo Far East Journal of Mathematical Sciences (FJMS) vol109 no 2 pp 243ndash259 2018

[11] O S Iyiola and G O Ojo ldquoOn the analytical solution ofFornberg-Whitham equation with the new fractional deriva-tiverdquo PramanamdashJournal of Physics vol 85 no 4 pp 567ndash5752015

[12] M Jneid and A Chaouk ldquoThe conformable reduced differ-ential transform method for solving Newell-Whitehead-Segelequation with non-integer orderrdquo Journal of Analysis andApplications 2019

[13] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 no 1 pp889ndash898 2015

[15] O Acan O Firat Y Keskin and G Oturanc ldquoSolution ofconformable fractional partial differential equations by reduceddifferential transform methodrdquo Selcuk Journal of Applied Math-ematics 2016

[16] H Thabet and S Kendre ldquoAnalytical solutions for conformablespace-time fractional partial differential equations via frac-tional differential transformrdquo Chaos Solitons amp Fractals vol109 pp 238ndash245 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 4: Analytic Solution for Systems of Two-Dimensional Time …downloads.hindawi.com/journals/ijmms/2019/7869516.pdf · 2019-07-30 · ResearchArticle Analytic Solution for Systems of Two-Dimensional

4 International Journal of Mathematics and Mathematical Sciences

Example 8 Consider the following system of CFDEs

120597119902119905 119906 + 119906120597119909119906 + V120597119910119906 = 12059721199092119906 + 12059721199102119906

120597119902119905 V + 119906120597119909V + V120597119910V = 12059721199092V + 12059721199102V(25)

subject to the initial conditions

119906 (119909 119910 0) = 119909 + 119910V (119909 119910 0) = 119909 minus 119910

(26)

Applying theorems of CFRDTM on (25) we obtain thefollowing system of recurrence relations

119902 (119896 + 1)119880119902119896+1 +119896

sum119903=0

119880119902119896minus119903120597119909119880119902119903 +119896

sum119903=0

119881119902119896minus119903120597119910119880119902119903

= 12059721199092119880119902119896 + 12059721199102119880119902119896

119902 (119896 + 1)119881119902119896+1 +119896

sum119903=0

119880119902119896minus119903120597119909119881119902119903 +119896

sum119903=0

119881119902119896minus119903120597119910119881119902119903

= 12059721199102119881119902119896 + +12059721199102119881119902119896

(27)

where the IC can be transformed at 119905 = 0 to1198801199020 = 119909 + 1199101198811199020 = 119909 minus 119910

(28)

Substituting 119896 = 0 into (25) we obtain1198801199021 =

minus2119902 119909

1198811199021 =minus2119902 119910

(29)

Similarly for 119896 = 1 119896 = 2 119896 = 3 and 119896 = 4 we getrespectively

1198801199022 =21199022 (119909 + 119910)

1198811199022 =21199022 (119909 minus 119910)

(30)

1198801199023 = minus 41199023119909

1198811199023 = minus 41199023119910

(31)

1198801199024 =41199023 (119909 + 119910)

1198811199024 = minus 41199023 (119909 minus 119910)

(32)

and

1198801199025 = minus 81199025119909

1198811199025 = minus 81199025119910

(33)

Applying the inverse transformation of CFRDTM the solu-tion of (25) is given as

119906 (119909 119910 119905) =infin

sum119896=0

119880119902119896(119909 119910 119905) 119905119896119902

= 119909 + 119910 minus 2119902119909119905119902 + 2

1199022 (119909 + 119910) 1199052119902 minus 411990231199091199053119902

+ 41199023 (119909 + 119910) 1199054119902 minus 8

11990251199091199055119902 + sdot sdot sdot

V (119909 119910 119905) =infin

sum119896=0

119881119902119896(119909 119910 119905) 119905119896119902

= 119909 minus 119910 minus 2119902119910119905119902 + 2

1199022 (119909 minus 119910) 1199052119902 minus 411990231199101199053119902

+ 41199023 (119909 minus 119910) 1199054119902 minus 8

11990251199101199055119902 + sdot sdot sdot

(34)

And by performing some computations the solutions aregiven as

119906 (119909 119910 119905) = 119909 + 119910 minus 21199091199051199021199021 minus 211990521199021199022

V (119909 119910 119905) = 119909 minus 119910 minus 21199091199051199021199021 minus 211990521199021199022

(35)

Example 9 Consider the following system of CFDEs

120597119902119905 119906 = V120597119909119906 + 120597119902119905 V120597119910119906 + 1 minus 119909 + 119910 + 119905119902119902

120597119902119905 V = 119906120597119909V + 120597119902119905 119906120597119910V + 1 minus 119909 minus 119910 minus 119905119902119902

(36)

subject to the initial conditions

119906 (119909 119910 0) = 119909 + 119910 minus 1V (119909 119910 0) = 119909 minus 119910 + 1

(37)

Applying theorems of CFRDTM on (25) we obtain thefollowing system of recurrence relations

119902 (119896 + 1)119880119902119896+1 =119896

sum119903=0

119881119902119903 120597119909119880119902119896minus119903

+119896

sum119903=0

119902 (119896 + 1) 119881119902119896+1minus119903120597119910119880119902119903 + 120575 (119896)

minus 119909 + 119910 minus 1 + 120575 (119896 minus 1)

119902 (119896 + 1) 119881119902119896+1 =119896

sum119903=0

119880119902119903 120597119909119880119902119896minus119903

+119896

sum119903=0

119902 (119896 + 1)119880119902119896+1minus119903120597119910119881119902119903 + 120575 (119896)

minus 119909 minus 119910 minus 1 + 120575 (119896 minus 1)

(38)

International Journal of Mathematics and Mathematical Sciences 5

00

00

00

00

00

00

minus05

minus05minus05

minus05minus05

minus10

minus15

minus20

05

05

05

15

05

0510

u(xy)v(xy)

Figure 1 The approximate solution of Example 9 for 120572 = 099 and t = 01

00

00

00

00

minus05

minus05minus05

minus05

minus10

minus15

05

05

00

00

minus0505

0515

05

10

u(xy) v(xy)

Figure 2The exact solution of Example 9 for 120572 = 099 and t = 01

where the IC can be transformed at 119905 = 0 to

1198801199020 = 119909 + 119910 minus 11198811199020 = 119909 minus 119910 + 1

(39)

Substituting 119896 = 0 into (36) we obtain

1198801199021 =1119902

1198811199021 = minus1119902

(40)

Similarly for 119896 = 1 and 119896 = 2 we get1198801199022 = minus 119909

2119902

1198811199022 =1119902 minus 119910 minus 1

2119902 (41)

and

1198801199023 =minus2119902119909 minus 119909

61199022 + 1 minus 5119902121199023

1198811199023 =minus1199103119902 minus 119902 + 1

121199023 (42)

Therefore the approximate analytical solution of (36) is givenas

119906 = 119909 + 119910 minus 1 + 119905119902119902 minus 1199091199052119902

2119902 + minus2119902119909 minus 11990961199022 1199053119902

+ 1 minus 5119902121199023 119905

3119902 + sdot sdot sdot

V = 119909 minus 119910 + 1 minus 119905119902119902 + 1119902 minus 119910 minus 1

2119902 1199052119902 + minus1199103119902 1199053119902

minus 119902 + 1121199023 119905

3119902 + sdot sdot sdot

(43)

Since this solution does not equal the exact one we usemathematica program to show its accuracy In Figures 1 and3 we plot the approximate solution for Example 9 for 119902 = 099at 119905 = 01 and at 119905 = 02 respectively The graphs in Figures2 and 4 illustrate the exact solution for Example 9 for 119902 =099 at 119905 = 01 and at 119905 = 02 respectively We can clearlyobserve at the time 119905 = 01 and 119905 = 02 that the values of theapproximate solution byCFRDTMand the values of the exactsolution obtained by other methods are extremely close Inother words solution provided by CFRDTM is highly preciseand accurate Particularly CFRDTM is a successful techniquefor solving systems of conformable FPDEs

6 International Journal of Mathematics and Mathematical Sciences

00

minus05

00

00

minus0505

0515

05

10

00

00

00

minus05

minus05

minus05

minus10

minus15

05

05

u(xy) v(xy)

Figure 3 The approximate solution of Example 9 for 120572 = 099 and t = 02

00

00

00

minus05

minus05

minus05minus10minus15

05

05

00

00

minus05

minus05

05

05

u(xy) v(xy)

00

15

05

10

Figure 4 The exact solution of Example 9 for 120572 = 099 and t = 02

4 Conclusion

In this paper our main purpose was to inspect the com-petence of the CFRDTM as a valid technique to solvesystems of nonlinear CFPDEs We successfully applied itto four distinct systems of nonlinear conformable time andspace FPDEs CFRDTM produced a numerical approximatesolution having the form of an infinite series that convergedto a closed form solution which coincided with the exactsolution in the first 3 applications In the 4119905ℎ applicationhowever we obtained an approximate solution that turnedout to be extremely close to the exact solution as we haveinterpreted from the attached Figures 1ndash4 On this basis weconclude that the CFRDTM is a powerful tool and a facileapproach for obtaining solutions for systems of CFPDEs Itreduces the amount of required computational work whencompared to other techniques used for this purpose and givessolutions that are in outstanding agreement with the exactones This provides a good starting point for further researchvia CFRDTM

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations North-HollandMathematics Studies New York NY USA 2006

[2] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010

[3] M D Ortigueira Fractional Calculus for Scientists and Engi-neers Heidelberg Springer Germany 2011

[4] T J Freeborn ldquoA survey of fractional-order circuit modelsfor biology and biomedicinerdquo IEEE Journal on Emerging andSelected Topics in Circuits and Systems vol 3 no 3 pp 416ndash4242013

[5] E C de Oliveira and J A Tenreiro Machado ldquoA review ofdefinitions for fractional derivatives and integralrdquoMathematicalProblems in Engineering vol 2014 Article ID 238459 6 pages2014

[6] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

International Journal of Mathematics and Mathematical Sciences 7

[7] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationbydifferential transformmethodandAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006

[8] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers ampMathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011

[9] O Acan and D Baleanu ldquoA new numerical technique forsolving fractional partial differential equationsrdquo Miskolc Math-ematical Notes vol 19 no 1 pp 3ndash18 2018

[10] M Jneid and A El Chakik ldquoAnalytical solution for somesystems of nonlinear conformable fractional differential equa-tionsrdquo Far East Journal of Mathematical Sciences (FJMS) vol109 no 2 pp 243ndash259 2018

[11] O S Iyiola and G O Ojo ldquoOn the analytical solution ofFornberg-Whitham equation with the new fractional deriva-tiverdquo PramanamdashJournal of Physics vol 85 no 4 pp 567ndash5752015

[12] M Jneid and A Chaouk ldquoThe conformable reduced differ-ential transform method for solving Newell-Whitehead-Segelequation with non-integer orderrdquo Journal of Analysis andApplications 2019

[13] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 no 1 pp889ndash898 2015

[15] O Acan O Firat Y Keskin and G Oturanc ldquoSolution ofconformable fractional partial differential equations by reduceddifferential transform methodrdquo Selcuk Journal of Applied Math-ematics 2016

[16] H Thabet and S Kendre ldquoAnalytical solutions for conformablespace-time fractional partial differential equations via frac-tional differential transformrdquo Chaos Solitons amp Fractals vol109 pp 238ndash245 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 5: Analytic Solution for Systems of Two-Dimensional Time …downloads.hindawi.com/journals/ijmms/2019/7869516.pdf · 2019-07-30 · ResearchArticle Analytic Solution for Systems of Two-Dimensional

International Journal of Mathematics and Mathematical Sciences 5

00

00

00

00

00

00

minus05

minus05minus05

minus05minus05

minus10

minus15

minus20

05

05

05

15

05

0510

u(xy)v(xy)

Figure 1 The approximate solution of Example 9 for 120572 = 099 and t = 01

00

00

00

00

minus05

minus05minus05

minus05

minus10

minus15

05

05

00

00

minus0505

0515

05

10

u(xy) v(xy)

Figure 2The exact solution of Example 9 for 120572 = 099 and t = 01

where the IC can be transformed at 119905 = 0 to

1198801199020 = 119909 + 119910 minus 11198811199020 = 119909 minus 119910 + 1

(39)

Substituting 119896 = 0 into (36) we obtain

1198801199021 =1119902

1198811199021 = minus1119902

(40)

Similarly for 119896 = 1 and 119896 = 2 we get1198801199022 = minus 119909

2119902

1198811199022 =1119902 minus 119910 minus 1

2119902 (41)

and

1198801199023 =minus2119902119909 minus 119909

61199022 + 1 minus 5119902121199023

1198811199023 =minus1199103119902 minus 119902 + 1

121199023 (42)

Therefore the approximate analytical solution of (36) is givenas

119906 = 119909 + 119910 minus 1 + 119905119902119902 minus 1199091199052119902

2119902 + minus2119902119909 minus 11990961199022 1199053119902

+ 1 minus 5119902121199023 119905

3119902 + sdot sdot sdot

V = 119909 minus 119910 + 1 minus 119905119902119902 + 1119902 minus 119910 minus 1

2119902 1199052119902 + minus1199103119902 1199053119902

minus 119902 + 1121199023 119905

3119902 + sdot sdot sdot

(43)

Since this solution does not equal the exact one we usemathematica program to show its accuracy In Figures 1 and3 we plot the approximate solution for Example 9 for 119902 = 099at 119905 = 01 and at 119905 = 02 respectively The graphs in Figures2 and 4 illustrate the exact solution for Example 9 for 119902 =099 at 119905 = 01 and at 119905 = 02 respectively We can clearlyobserve at the time 119905 = 01 and 119905 = 02 that the values of theapproximate solution byCFRDTMand the values of the exactsolution obtained by other methods are extremely close Inother words solution provided by CFRDTM is highly preciseand accurate Particularly CFRDTM is a successful techniquefor solving systems of conformable FPDEs

6 International Journal of Mathematics and Mathematical Sciences

00

minus05

00

00

minus0505

0515

05

10

00

00

00

minus05

minus05

minus05

minus10

minus15

05

05

u(xy) v(xy)

Figure 3 The approximate solution of Example 9 for 120572 = 099 and t = 02

00

00

00

minus05

minus05

minus05minus10minus15

05

05

00

00

minus05

minus05

05

05

u(xy) v(xy)

00

15

05

10

Figure 4 The exact solution of Example 9 for 120572 = 099 and t = 02

4 Conclusion

In this paper our main purpose was to inspect the com-petence of the CFRDTM as a valid technique to solvesystems of nonlinear CFPDEs We successfully applied itto four distinct systems of nonlinear conformable time andspace FPDEs CFRDTM produced a numerical approximatesolution having the form of an infinite series that convergedto a closed form solution which coincided with the exactsolution in the first 3 applications In the 4119905ℎ applicationhowever we obtained an approximate solution that turnedout to be extremely close to the exact solution as we haveinterpreted from the attached Figures 1ndash4 On this basis weconclude that the CFRDTM is a powerful tool and a facileapproach for obtaining solutions for systems of CFPDEs Itreduces the amount of required computational work whencompared to other techniques used for this purpose and givessolutions that are in outstanding agreement with the exactones This provides a good starting point for further researchvia CFRDTM

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations North-HollandMathematics Studies New York NY USA 2006

[2] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010

[3] M D Ortigueira Fractional Calculus for Scientists and Engi-neers Heidelberg Springer Germany 2011

[4] T J Freeborn ldquoA survey of fractional-order circuit modelsfor biology and biomedicinerdquo IEEE Journal on Emerging andSelected Topics in Circuits and Systems vol 3 no 3 pp 416ndash4242013

[5] E C de Oliveira and J A Tenreiro Machado ldquoA review ofdefinitions for fractional derivatives and integralrdquoMathematicalProblems in Engineering vol 2014 Article ID 238459 6 pages2014

[6] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

International Journal of Mathematics and Mathematical Sciences 7

[7] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationbydifferential transformmethodandAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006

[8] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers ampMathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011

[9] O Acan and D Baleanu ldquoA new numerical technique forsolving fractional partial differential equationsrdquo Miskolc Math-ematical Notes vol 19 no 1 pp 3ndash18 2018

[10] M Jneid and A El Chakik ldquoAnalytical solution for somesystems of nonlinear conformable fractional differential equa-tionsrdquo Far East Journal of Mathematical Sciences (FJMS) vol109 no 2 pp 243ndash259 2018

[11] O S Iyiola and G O Ojo ldquoOn the analytical solution ofFornberg-Whitham equation with the new fractional deriva-tiverdquo PramanamdashJournal of Physics vol 85 no 4 pp 567ndash5752015

[12] M Jneid and A Chaouk ldquoThe conformable reduced differ-ential transform method for solving Newell-Whitehead-Segelequation with non-integer orderrdquo Journal of Analysis andApplications 2019

[13] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 no 1 pp889ndash898 2015

[15] O Acan O Firat Y Keskin and G Oturanc ldquoSolution ofconformable fractional partial differential equations by reduceddifferential transform methodrdquo Selcuk Journal of Applied Math-ematics 2016

[16] H Thabet and S Kendre ldquoAnalytical solutions for conformablespace-time fractional partial differential equations via frac-tional differential transformrdquo Chaos Solitons amp Fractals vol109 pp 238ndash245 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 6: Analytic Solution for Systems of Two-Dimensional Time …downloads.hindawi.com/journals/ijmms/2019/7869516.pdf · 2019-07-30 · ResearchArticle Analytic Solution for Systems of Two-Dimensional

6 International Journal of Mathematics and Mathematical Sciences

00

minus05

00

00

minus0505

0515

05

10

00

00

00

minus05

minus05

minus05

minus10

minus15

05

05

u(xy) v(xy)

Figure 3 The approximate solution of Example 9 for 120572 = 099 and t = 02

00

00

00

minus05

minus05

minus05minus10minus15

05

05

00

00

minus05

minus05

05

05

u(xy) v(xy)

00

15

05

10

Figure 4 The exact solution of Example 9 for 120572 = 099 and t = 02

4 Conclusion

In this paper our main purpose was to inspect the com-petence of the CFRDTM as a valid technique to solvesystems of nonlinear CFPDEs We successfully applied itto four distinct systems of nonlinear conformable time andspace FPDEs CFRDTM produced a numerical approximatesolution having the form of an infinite series that convergedto a closed form solution which coincided with the exactsolution in the first 3 applications In the 4119905ℎ applicationhowever we obtained an approximate solution that turnedout to be extremely close to the exact solution as we haveinterpreted from the attached Figures 1ndash4 On this basis weconclude that the CFRDTM is a powerful tool and a facileapproach for obtaining solutions for systems of CFPDEs Itreduces the amount of required computational work whencompared to other techniques used for this purpose and givessolutions that are in outstanding agreement with the exactones This provides a good starting point for further researchvia CFRDTM

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations North-HollandMathematics Studies New York NY USA 2006

[2] A S Elwakil ldquoFractional-order circuits and systems an emerg-ing interdisciplinary research areardquo IEEE Circuits and SystemsMagazine vol 10 no 4 pp 40ndash50 2010

[3] M D Ortigueira Fractional Calculus for Scientists and Engi-neers Heidelberg Springer Germany 2011

[4] T J Freeborn ldquoA survey of fractional-order circuit modelsfor biology and biomedicinerdquo IEEE Journal on Emerging andSelected Topics in Circuits and Systems vol 3 no 3 pp 416ndash4242013

[5] E C de Oliveira and J A Tenreiro Machado ldquoA review ofdefinitions for fractional derivatives and integralrdquoMathematicalProblems in Engineering vol 2014 Article ID 238459 6 pages2014

[6] R Khalil M Al Horani A Yousef and M Sababheh ldquoA newdefinition of fractional derivativerdquo Journal of Computationaland Applied Mathematics vol 264 pp 65ndash70 2014

International Journal of Mathematics and Mathematical Sciences 7

[7] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationbydifferential transformmethodandAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006

[8] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers ampMathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011

[9] O Acan and D Baleanu ldquoA new numerical technique forsolving fractional partial differential equationsrdquo Miskolc Math-ematical Notes vol 19 no 1 pp 3ndash18 2018

[10] M Jneid and A El Chakik ldquoAnalytical solution for somesystems of nonlinear conformable fractional differential equa-tionsrdquo Far East Journal of Mathematical Sciences (FJMS) vol109 no 2 pp 243ndash259 2018

[11] O S Iyiola and G O Ojo ldquoOn the analytical solution ofFornberg-Whitham equation with the new fractional deriva-tiverdquo PramanamdashJournal of Physics vol 85 no 4 pp 567ndash5752015

[12] M Jneid and A Chaouk ldquoThe conformable reduced differ-ential transform method for solving Newell-Whitehead-Segelequation with non-integer orderrdquo Journal of Analysis andApplications 2019

[13] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 no 1 pp889ndash898 2015

[15] O Acan O Firat Y Keskin and G Oturanc ldquoSolution ofconformable fractional partial differential equations by reduceddifferential transform methodrdquo Selcuk Journal of Applied Math-ematics 2016

[16] H Thabet and S Kendre ldquoAnalytical solutions for conformablespace-time fractional partial differential equations via frac-tional differential transformrdquo Chaos Solitons amp Fractals vol109 pp 238ndash245 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 7: Analytic Solution for Systems of Two-Dimensional Time …downloads.hindawi.com/journals/ijmms/2019/7869516.pdf · 2019-07-30 · ResearchArticle Analytic Solution for Systems of Two-Dimensional

International Journal of Mathematics and Mathematical Sciences 7

[7] N Bildik A Konuralp F Orak Bek and S KucukarslanldquoSolution of different type of the partial differential equationbydifferential transformmethodandAdomianrsquos decompositionmethodrdquo Applied Mathematics and Computation vol 172 no 1pp 551ndash567 2006

[8] P K Gupta ldquoApproximate analytical solutions of fractionalBenney-Lin equation by reduced differential transformmethodand the homotopy perturbation methodrdquo Computers ampMathe-matics with Applications vol 61 no 9 pp 2829ndash2842 2011

[9] O Acan and D Baleanu ldquoA new numerical technique forsolving fractional partial differential equationsrdquo Miskolc Math-ematical Notes vol 19 no 1 pp 3ndash18 2018

[10] M Jneid and A El Chakik ldquoAnalytical solution for somesystems of nonlinear conformable fractional differential equa-tionsrdquo Far East Journal of Mathematical Sciences (FJMS) vol109 no 2 pp 243ndash259 2018

[11] O S Iyiola and G O Ojo ldquoOn the analytical solution ofFornberg-Whitham equation with the new fractional deriva-tiverdquo PramanamdashJournal of Physics vol 85 no 4 pp 567ndash5752015

[12] M Jneid and A Chaouk ldquoThe conformable reduced differ-ential transform method for solving Newell-Whitehead-Segelequation with non-integer orderrdquo Journal of Analysis andApplications 2019

[13] T Abdeljawad ldquoOn conformable fractional calculusrdquo Journal ofComputational and Applied Mathematics vol 279 pp 57ndash662015

[14] A Atangana D Baleanu and A Alsaedi ldquoNew properties ofconformable derivativerdquo Open Mathematics vol 13 no 1 pp889ndash898 2015

[15] O Acan O Firat Y Keskin and G Oturanc ldquoSolution ofconformable fractional partial differential equations by reduceddifferential transform methodrdquo Selcuk Journal of Applied Math-ematics 2016

[16] H Thabet and S Kendre ldquoAnalytical solutions for conformablespace-time fractional partial differential equations via frac-tional differential transformrdquo Chaos Solitons amp Fractals vol109 pp 238ndash245 2018

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 8: Analytic Solution for Systems of Two-Dimensional Time …downloads.hindawi.com/journals/ijmms/2019/7869516.pdf · 2019-07-30 · ResearchArticle Analytic Solution for Systems of Two-Dimensional

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


ANSIG – An Analytic Signature for Permutation-Invariant Two Dimensional Shape Representation

ANSIG – An Analytic Signature for Permutation-Invariant Two Dimensional Shape Representation

Research Article Analytic Approximate Solution for Falkner-Skan …downloads.hindawi.com/journals/tswj/2014/617453.pdf · 2019-07-31 · Research Article Analytic Approximate Solution

Research Article Analytic Approximate Solution for Falkner-Skan …downloads.hindawi.com/journals/tswj/2014/617453.pdf · 2019-07-31 · Research Article Analytic Approximate Solution

Approximate solution methods for one-dimensional ...

Approximate solution methods for one-dimensional ...

Preliminary Computational Technique Numerical Methods · Analytic solution– linear equation, simple geometry, simple initial and boundary conditions Analytic solution techniques–

Preliminary Computational Technique Numerical Methods · Analytic solution– linear equation, simple geometry, simple initial and boundary conditions Analytic solution techniques–

NUMERICAL SOLUTION OF THE ONE-DIMENSIONAL

NUMERICAL SOLUTION OF THE ONE-DIMENSIONAL

ANALYTIC NUMERIC SOLUTION OF CORONAVIRUS (COVID 19 ...

ANALYTIC NUMERIC SOLUTION OF CORONAVIRUS (COVID 19 ...

An improved analytic solution for analysis of particle

An improved analytic solution for analysis of particle

Analytic Solution of Liquid Flow in Rectangular in PDMS MC

Analytic Solution of Liquid Flow in Rectangular in PDMS MC

Analytic solution for nonlinear shock acceleration in the ...

Analytic solution for nonlinear shock acceleration in the ...

An analytic problem whose solution follows from a simple algebraic ...

An analytic problem whose solution follows from a simple algebraic ...

analytic solution of stochastic activity networks with exponential and

analytic solution of stochastic activity networks with exponential and

Finite Analytic Method for One-Dimensional Nonlinear Consolidation … · 2019. 7. 30. · ResearchArticle Finite Analytic Method for One-Dimensional Nonlinear Consolidation under

Finite Analytic Method for One-Dimensional Nonlinear Consolidation … · 2019. 7. 30. · ResearchArticle Finite Analytic Method for One-Dimensional Nonlinear Consolidation under

Analytic Solution of Partial Differential Equations

Analytic Solution of Partial Differential Equations

Analytic Solution of Fractional-Order Heat- and Wave …znaturforsch.com/s66a/s66a0581.pdfAnalytic Solution of Fractional-Order Heat- and Wave-Like Equations Using Generalized n-dimensional

Analytic Solution of Fractional-Order Heat- and Wave …znaturforsch.com/s66a/s66a0581.pdfAnalytic Solution of Fractional-Order Heat- and Wave-Like Equations Using Generalized n-dimensional

Analytic adjoint solutions for the quasi-one-dimensional ... · present work investigates the analytic properties of adjoint solutions for the quasi-one-dimensional Euler equations.

Analytic adjoint solutions for the quasi-one-dimensional ... · present work investigates the analytic properties of adjoint solutions for the quasi-one-dimensional Euler equations.

Lagrange: A Three-dimensional Analytic Lens Design Method ...

Lagrange: A Three-dimensional Analytic Lens Design Method ...

Analytic for data-driven decision-making in complex high-dimensional ...rx917b36b/... · Analytic for Data-Driven Decision-Making in Complex High-Dimensional Time-to-Event Data A

Analytic for data-driven decision-making in complex high-dimensional ...rx917b36b/... · Analytic for Data-Driven Decision-Making in Complex High-Dimensional Time-to-Event Data A

Analytic Banking Services: A solution to facilitate financial and risk management ... · 2016-06-08 · Analytic Banking Services: A solution to facilitate financial and risk management,

Analytic Banking Services: A solution to facilitate financial and risk management ... · 2016-06-08 · Analytic Banking Services: A solution to facilitate financial and risk management,

Two-Dimensional Green’s Function Poisson Solution ...

Two-Dimensional Green’s Function Poisson Solution ...

Analytic solution of nonlinear batch reaction kinetics ...

Analytic solution of nonlinear batch reaction kinetics ...