Research Article Solving Systems of Volterra Integral and...
6
Research Article Solving Systems of Volterra Integral and Integrodifferential Equations with Proportional Delays by Differential Transformation Method Fuayip YüzbaGJ and Nurbol Ismailov Department of Mathematics, Faculty of Science, Akdeniz University, 07058 Antalya, Turkey Correspondence should be addressed to S ¸uayip Y¨ uzbas ¸ı; [email protected] Received 1 May 2014; Revised 7 July 2014; Accepted 15 July 2014; Published 12 August 2014 Academic Editor: Stefan Siegmund Copyright © 2014 S ¸. Y¨ uzbas ¸ı and N. Ismailov. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this paper, the differential transformation method is applied to the system of Volterra integral and integrodifferential equations with proportional delays. e method is useful for both linear and nonlinear equations. By using this method, the solutions are obtained in series forms. If the solutions of the problem can be expanded to Taylor series, then the method gives opportunity to determine the coefficients of Taylor series. Hence, the exact solution can be obtained in Taylor series form. In illustrative examples, the method is applied to a few types of systems. 1. Introduction Integral and integrodifferential equations have found applica- tions in engineering, physics, chemistry, and insurance math- ematics [1–3]. In particular, functional-differential equations with proportional delays have described some models such as motion of particle in liquid and polymer crystallization which can be found in [4]. ere are a lot of methods of approach for solutions of systems of integral and integrodifferential equations. For example, the linear and nonlinear systems of integrodif- ferential equations have been solved by Haar functions [5]; Maleknejad and Tavassoli Kajani [6] used the hybrid Legendre functions, the Chebyshev polynomial method [7], the Bessel collocation method [8, 9], the Taylor collocation method [10], the homotopy perturbation method [11, 12], the variational iteration method [13], the differential transforma- tion method [14], and the Taylor series method [15]. Biazar et al. [16] have obtained the solutions of systems of Volterra integral equations of the first kind by the Adomian method. In addition, the homotopy perturbation method has been used for systems of Abel’s integral equations [17]. On the other hand, the special systems of integral equations have been solved by the differential transformation method [18]. Katani and Shahmorad [19] have presented Romberg quadrature for the systems of Urysohn type Volterra integral equations. e nonlinear systems of Volterra integrodifferential equations with delay arguments have been studied by Yalc ¸ınbas ¸ and Erdem [20]. In this paper, we consider the system of Volterra integral and integrodifferential equations with proportional delays: (, () ( ) , ∫ 0 (, , ( )) ) = 0, (1) where , are given functions, , , ∈ (0,1], = 1, 2,...,, and = 0, 1, . . . , . 2. Differential Transformation Method In 1987, the differential transformation method is introduced by Zhou [21] in the study of electric circuits. e method based on Taylor series and yields of differential transforma- tion are difference equations which solutions give the exact values of derivatives of origin function at the given point. e method has been used for a wide class of problems [22–25]. e main advantage of differential transformation from Laplace and Fourier transformations is that it can be Hindawi Publishing Corporation Journal of Mathematics Volume 2014, Article ID 725648, 5 pages http://dx.doi.org/10.1155/2014/725648
Transcript of Research Article Solving Systems of Volterra Integral and...
Research ArticleSolving Systems of Volterra Integral andIntegrodifferential Equations with ProportionalDelays by Differential Transformation Method
Fuayip YuumlzbaGJ and Nurbol Ismailov
Department of Mathematics Faculty of Science Akdeniz University 07058 Antalya Turkey
Correspondence should be addressed to Suayip Yuzbası syuzbasiakdenizedutr
Received 1 May 2014 Revised 7 July 2014 Accepted 15 July 2014 Published 12 August 2014
Academic Editor Stefan Siegmund
Copyright copy 2014 S Yuzbası and N Ismailov This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
In this paper the differential transformation method is applied to the system of Volterra integral and integrodifferential equationswith proportional delays The method is useful for both linear and nonlinear equations By using this method the solutions areobtained in series forms If the solutions of the problem can be expanded to Taylor series then the method gives opportunity todetermine the coefficients of Taylor series Hence the exact solution can be obtained in Taylor series form In illustrative examplesthe method is applied to a few types of systems
1 Introduction
Integral and integrodifferential equations have found applica-tions in engineering physics chemistry and insurancemath-ematics [1ndash3] In particular functional-differential equationswith proportional delays have described somemodels such asmotion of particle in liquid and polymer crystallizationwhichcan be found in [4]
There are a lot of methods of approach for solutionsof systems of integral and integrodifferential equations Forexample the linear and nonlinear systems of integrodif-ferential equations have been solved by Haar functions[5] Maleknejad and Tavassoli Kajani [6] used the hybridLegendre functions the Chebyshev polynomial method [7]the Bessel collocation method [8 9] the Taylor collocationmethod [10] the homotopy perturbation method [11 12] thevariational iterationmethod [13] the differential transforma-tion method [14] and the Taylor series method [15] Biazaret al [16] have obtained the solutions of systems of Volterraintegral equations of the first kind by the Adomian methodIn addition the homotopy perturbation method has beenused for systems ofAbelrsquos integral equations [17]On the otherhand the special systems of integral equations have beensolved by the differential transformation method [18] Katani
and Shahmorad [19] have presented Romberg quadrature forthe systems of Urysohn type Volterra integral equations Thenonlinear systems of Volterra integrodifferential equationswith delay arguments have been studied by Yalcınbas andErdem [20]
In this paper we consider the system of Volterra integraland integrodifferential equations with proportional delays
119865119896 (119909 119910(119897)
119896(119901119896119909) int
119903119909
0
119866119896 (119909 119905 119910119896 (119902119896119905)) 119889119905) = 0 (1)
where 119865119896 119866119896 are given functions 119901119896 119902119896 119903 isin (0 1] 119896 = 1
2 119899 and 119897 = 0 1 119898
2 Differential Transformation Method
In 1987 the differential transformation method is introducedby Zhou [21] in the study of electric circuits The methodbased on Taylor series and yields of differential transforma-tion are difference equations which solutions give the exactvalues of derivatives of origin function at the given pointThe method has been used for a wide class of problems[22ndash25] The main advantage of differential transformationfrom Laplace and Fourier transformations is that it can be
Hindawi Publishing CorporationJournal of MathematicsVolume 2014 Article ID 725648 5 pageshttpdxdoiorg1011552014725648
2 Journal of Mathematics
applied easily to linear equations with constant and variablecoefficients and some nonlinear equations
The differential transformation of the 119896th derivative offunction 119910(119909) is defined by
119884 (119896) =1
119896[119889119896119910(119909)
119889119909119896]
119909=1199090
(2)
and the inverse transformation is defined as follows
119910 (119909) =
infin
sum
119896=0
119884 (119896) (119909 minus 1199090)119896 (3)
The following theorems can be obtained from definitions(2) and (3)
Theorem 1 Assume that 119865(119896) 119866(119896)119867(119896) are the differentialtransformations at the 1199090 = 0 of the functions 119891(119909) 119892(119909)ℎ(119909) respectively then one has the following
If 119891(119909) = 119889119899119892(119909)119889119909
119899 then 119865(119896) = ((119896 + 119899)119896)119866(119896 +
119899)If 119891(119909) = 119892(119909)ℎ(119909) then 119865(119896) = sum
119896
119897=0119866(119897)119867(119896 minus 119897)
If 119891(119909) = 119909119899 then 119865(119896) = 120575(119896 minus 119899) 120575 is the Kronecker
delta symbolIf 119891(119909) = 119890
120582119909 then 119865(119896) = 120582119896119896
If 119891(119909) = sin(120596119909 + 120582) then 119865(119896) = (120596119896119896) sin(1198961205872 +
120582)If 119891(119909) = cos(120596119909+120582) then 119865(119896) = (120596
119896119896) cos(1198961205872+
120582)If 119891(119909) = 119892(119909) int
119909
0ℎ(119905)119889119905 then 119865(119896) = sum
119896minus1
119897=0(1(119896 minus
119897))119866(119897)119867(119896 minus 119897 minus 1)
Theorem 2 Assume that119882(119896) 119884(119896) and 119884119894(119896) 119894 = 1 119899are the differential transformations at the 1199090 = 0 of thefunctions 119908(119909) 119910(119909) and 119910119894(119909) respectively and 119902 119902119894 119903 isin
(0 1] 119894 = 1 2 Then one has the following
If 119908(119909) = 119910(119902119909) then119882(119896) = 119902119896119884(119896)
If 119908(119909) = 1199101(1199021119909)1199102(1199022119909) then 119882(119896) =
sum119896
119897=0119902119897
1119902119896minus119897
21198841(119897)1198842(119896 minus 119897)
If 119908(119909) = 119889119898119910(119902119909)119889119909
119898 then 119882(119896) = ((119896 +
119898)119896)119902119896+119898
119884(119896 + 119898)If 119908(119909) = int
119903119909
0119910(119902119905)119889119905 then 119882(119896) = (119903
119896119896)119902119896minus1
119884(119896 minus
1)If 119908(119909) = int
119909
01199101(1199021119905)1199102(1199022119905)119889119905 then 119882(119896) =
(1119896)sum119896minus1
119897=0119902119897
1119902119896minus119897minus1
21198841(119897)1198842(119896 minus 119897 minus 1)
If 119908(119909) = 119910(119902119909) int119909
01199101(1199021119905)1199102(1199022119905)119889119905 then
119882(119896) =
119896minus1
sum
119897=0
119896minus119897minus1
sum
119904=0
1
119896 minus 119897119902119897119902119904
1119902119896minus119897minus119904minus1
2119884 (119897)
times 1198841 (119904) 1198842 (119896 minus 119897 minus 119904 minus 1)
(4)
where 119896 isin 119873
The proofs of Theorems 1 and 2 are given in [22 25]
3 Illustrate Examples
Example 1 Let us consider the following linear systemof Volterra integrodifferential equations with proportionaldelays and separable kernels
1199101 (119909) minus 1199101015840
1(119909
3) + 9119910
1015840
2(119909
3)
= 4 cos119909 + 8int119909
0
cos (119909 minus 119905) 1199102 (119905
3) 119889119905
+1
9int
119909
0
1199101 (119905
3) 119889119905
1199102 (119909) + 1199101015840
1(119909) minus 4119910
1015840
2(119909)
= minus2 cos 1199092minus 2 + 3int
119909
0
sin (119909 minus 119905) 1199101 (119905
2) 119889119905
+ int
119909
0
1199102 (119905
2) 119889119905
(5)
with the initial conditions 1199101(0) = 1 and 1199102(0) = 0
The differential transformation of the last system is
1198841 (119896) minus119896 + 1
3119896+11198841 (119896 + 1) + 9
119896 + 1
3119896+11198842 (119896 + 1)
=4
119896cos 120587119896
2+8
119896minus1
sum
119897=0
119896minus119897minus1
sum
119904=0
1198842 (119896 minus 119897 minus 119904 minus 1)
3119896minus119897minus119904minus1 (119896 minus 119897) 119897119904cos(120587
2[119897 minus 119904])
+1198841 (119896 minus 1)
3119896+1119896
1198842 (119896) + (119896 + 1) 1198841 (119896 + 1) minus 4 (119896 + 1) 1198842 (119896 + 1)
= minus1
2119896minus1119896cos 120587119896
2minus 2120575 (119896)
+ 3
119896minus1
sum
119897=0
119896minus119897minus1
sum
119904=0
1198842 (119896 minus 119897 minus 119904 minus 1)
2119896minus119897minus119904minus1 (119896 minus 119897) 119897119904sin(120587
2[119897 minus 119904])
+1198842 (119896 minus 1)
2119896minus1119896
(6)
Substituting 119909 = 0 in Example 1 we obtain values of firstderivatives of unknown functions that is 1198841(1) = 0 and1198842(1) = 1
For 119896 = 1 in (6) we have the following system
1198841 (1) minus2
91198841 (2) + 21198842 (2) = 81198842 (0) +
1
91198841 (0)
1198842 (1) + 21198841 (2) minus 81198842 (2) = 1198842 (0)
(7)
Solving the last system we get 1198841(2) = minus12 and 1198842(2) =0
Journal of Mathematics 3
Substituting 119896 = 2 in (6) we obtain the following systemof equations
1
91198841 (3) minus 1198842 (3) minus
2
3= minus
1
2
minus31198841 (3) + 121198842 (3) + 2 = 0
(8)
Solving the last system with two unknown we have1198841(3) = 0 and 1198842(3) = minus16
Continue this process and use inverse transformation weget 1199101(119909) = cos119909 and 1199102(119909) = sin119909 which are the exactsolutions of Example 1
Example 2 Consider the following system of nonlinearVolterra integrodifferential equations with proportionaldelays
1199101 (119909) + 1199101015840
2(119909) = 119909 minus
1
41199092+ int
119909
0
1199101 (119905
2) 119889119905
1199102 (119909) + 1199101015840
1(119909) = 119890
119909minus 3 (119909 minus 3) 119890
1199093+ int
119909
0
1199101 (119905) 1199102 (119905
3) 119889119905
(9)
with the initial conditions 1199101(0) = 0 and 1199102(0) = 1
Analogously for119909 = 0 in (9) we get11991010158401(0) = 1 and1199101015840
2(0) =
1Applying the differential transformation to (9) we have
the following system of difference equations
1198841 (119896) + (119896 + 1) 1198842 (119896 + 1)
=1
119896+ 120575 (119896 minus 1) minus
1
4120575 (119896 minus 2) +
1
1198962119896 minus 11198841 (119896 minus 1)
1198842 (119896) + (119896 + 1) 1198841 (119896 + 1)
=1
119896minus 3
119896
sum
119897=0
120575 (119897 minus 1)1
3119896minus119897 (119896 minus 119897)+
1
3119896minus2119896
+1
119896
119896minus1
sum
119897=0
1198841 (119897) 1198842 (119896 minus 119897 minus 1)
3119896minus119897minus1
(10)
For 119896 = 1 in (10) we get the following system
1198841 (1) + 21198842 (2) = 2 + 1198841 (0)
1198842 (1) + 21198841 (2) = 1
(11)
Solving the last system we obtain 1198841(2) = 0 and 1198842(2) =12 = 12
Substituting 119896 = 2 in (10) and solving correspondingsystem we have 1198841(3) = 0 and 1198842(3) = 16 = 13 and for119896 ge 3 1198841(119896) = 0 and 1198842(119896) = 1119896 Then using (3) we gain1199101(119909) = 119909 and 1199102(119909) = 119890
119909 which are the exact solutions ofsystem (9)
Example 3 Consider the following system of nonlinearVolterra integral equations
1199101 (119909) =1199093
324+891199092
90minus int
1199093
0
[1199101 (119905
2) minus 1199102 (
119905
5)] 119889119905
1199102 (119909) = 119909 +81199093
27+ int
119909
0
[1199101 (119905
3) minus 1199102
2(119905)] 119889119905
(12)
with the initial condition 1199101(0) = 1199102(0) = 0
Now applying the differential transformation to (12) weget the following system of difference equations
1198841 (119896) =120575 (119896 minus 3)
324+89120575 (119896 minus 2)
90minus1198841 (119896 minus 1)
31198962119896minus1119896+1198842 (119896 minus 1)
31198965119896minus1119896
1198842 (119896) = 120575 (119896 minus 1) +8120575 (119896 minus 3)
27+1198841 (119896 minus 1)
3119896minus1119896
minus1
119896
119896minus1
sum
119897=0
1198842 (119897) 1198842 (119896 minus 119897 minus 1)
(13)
From initial conditions we have 1198841(0) = 1198842(0) = 0Using (13) we get the following values
1198841 (1) = 0 1198841 (2) = 1
1198842 (1) = 1 1198842 (2) = 0
(14)
and for 119896 ge 3 1198841(119896) = 1198842(119896) = 0Using (3) we have 1199101(119909) = 119909 and 1199102(119909) = 119909
2 which areexact solutions of system (12)
Example 4 In last we consider the linear systemwith variablecoefficients of Volterra integral equations with proportionaldelays
11990921199101 (
119909
4) + 2 sin1199091199102 (
119909
3)
= sin 4119909
3+ sin 2119909
3+ 2int
1199092
0
1199091199101 (119905) 119889119905
sin 119909
31199101 (119909) + 1199091199102 (
119909
2) = 119909 cos 119909
2+ int
1199093
0
1199091199102 (119905) 119889119905
(15)
with exact solutions 1199101(119909) = 119909 and 1199102(119909) = cos119909
4 Journal of Mathematics
Applying DTM we have
119896
sum
119897=0
120575 (119897 minus 2) 1198841 (119896 minus 119897)1
4119896minus119897+ 2
119896
sum
119897=0
1
1198973119896minus119897sin 119897120587
21198842 (119896 minus 119897)
=4119896
3119896119896sin 119896120587
2+
2119896
3119896119896sin 119896120587
2
+ 2
119896
sum
119897=0
120575 (119897 minus 1) 1198841 (119896 minus 119897 minus 1)1
(119896 minus 119897) 2119896minus119897
119896
sum
119897=0
1
1198973119897sin 119897120587
21198841 (119896 minus 119897) +
119896
sum
119897=0
120575 (119897 minus 1) 1198842 (119896 minus 119897)1
2119896minus119897
=
119896
sum
119897=0
120575 (119897 minus 1)1
(119896 minus 119897)2119896minus119897
cos(119896 minus 1198972
120587)
+
119896minus1
sum
119897=0
120575 (119897 minus 1) 1198842 (119896 minus 119897 minus 1)1
(119896 minus 119897) 3119896minus119897
(16)
Solving the last systemwe have 1199101(119909) = 119909 and 1199102(119909) = 1minus
(11990922)+(119909
44)minus sdot sdot sdot which are Taylor series of exact solutions
of Example 4
4 Conclusions
In this study the differential transformationmethod has beenpresented for solving system of integral and integrodifferen-tial equations with proportional delaysThemajor benefits ofmethod from integral transformations are that the methodcan be applied for linear equations with variable coefficientsand nonlinear equations and the method gives the exactsolutions in series forms
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the reviewers for their con-structive comments and suggestions to improve the paper
References
[1] R P Agarwal D OrsquoRegan and P J Y Wong ldquoEigenvalues ofa system of Fredholm integral equationsrdquo Mathematical andComputer Modelling vol 39 no 9-10 pp 1113ndash1150 2004
[2] RF Churchhouse Handbook of Applicable Mathematics JohnWiley amp Sons New York NY USA 1981
[3] R P Kanwal Linear Integral Equations Birkhaauser BostonMass USA 1997
[4] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional Differential Equations KluwerAcademic Publishers Dordrecht The Netherlands 1999
[5] K Maleknejad F Mirzaee and S Abbasbandy ldquoSolving linearintegro-differential equations systemby using rationalizedHaarfunctions methodrdquo Applied Mathematics and Computation vol155 no 2 pp 317ndash328 2004
[6] K Maleknejad andM Tavassoli Kajani ldquoSolving linear integro-differential equation system by Galerkin methods with hydridfunctionsrdquo Applied Mathematics and Computation vol 159 no3 pp 603ndash612 2004
[7] A Dascioglu and M Sezer ldquoChebyshev polynomial solutionsof systems of higher-order linear Fredholm-Volterra integro-differential equationsrdquo Journal of the Franklin Institute vol 342pp 688ndash701 2005
[8] S Yuzbası N Sahin and MSezer ldquoNumerical solutions ofsystems of linear Fredholm integro-differential equations withBessel polynomial basesrdquoComputersampMathematics withAppli-cations vol 61 no 10 pp 3079ndash3096 2011
[9] N Sahin S Yuzbası and M Gulsu ldquoA collocation approachfor solving systems of linear Volterra integral equations withvariable coefficientsrdquo Computers and Mathematics with Appli-cations vol 62 no 2 pp 755ndash769 2011
[10] M Gulsu andM Sezer ldquoTaylor collocationmethod for solutionof systems of high-order linear Fredholm-Volterra integro-differential equationsrdquo International Journal of ComputerMath-ematics vol 83 no 4 pp 429ndash448 2006
[11] E Yusufoglu ldquoAn efficient algorithm for solving integro-differential equations systemrdquo Applied Mathematics and Com-putation vol 192 no 1 pp 51ndash55 2007
[12] E Yusufoglu ldquoA homotopy perturbation algorithm to solve asystem of Fredholm-Volterra type integral equationsrdquo Mathe-matical andComputerModelling vol 47 no 11-12 pp 1099ndash11072008
[13] J Saberi-Nadjafi and M Tamamgar ldquoThe variational iterationmethod a highly promising method for solving the system ofintegro-differential equationsrdquo Computers amp Mathematics withApplications vol 56 no 2 pp 346ndash351 2008
[14] A Arikoglu and I Ozkol ldquoSolutions of integral and integro-differential equation systems by using differential transformmethodrdquo Computers amp Mathematics with Applications vol 56no 9 pp 2411ndash2417 2008
[15] H H Sorkun and S Yalcinbas ldquoApproximate solutions of linearVolterra integral equation systems with variable coefficientsrdquoApplied Mathematical Modelling vol 34 no 11 pp 3451ndash34642010
[16] J Biazar E Babolian and R M Islam ldquoSolution of a systemof Volterra integral equations of the first kind by Adomianmethodrdquo Applied Mathematics and Computation vol 139 no2-3 pp 249ndash258 2003
[17] S Kumar O P Singh and S Dixit ldquoHomotopy perturbationmethod for solving system of generalized Abelrsquos integral equa-tionsrdquo Applications and Applied Mathematics vol 6 no 11 pp2009ndash2024 2011
[18] J Biazar M Eslami and M R Islam ldquoDifferential transformmethod for special systems of integral equationsrdquo Journal ofKing Saud UniversitymdashScience vol 24 no 3 pp 211ndash214 2012
[19] R Katani and S Shahmorad ldquoA block by block method withRomberg quadrature for the system of Urysohn type Volterraintegral equationsrdquo Computational and Applied Mathematicsvol 31 no 1 pp 191ndash203 2012
[20] S Yalcınbas and K Erdem ldquoA new approximation method forthe systems of nonlinear fredholm integral equationsrdquo AppliedMathematics and Physics vol 2 no 2 pp 40ndash48 2014
Journal of Mathematics 5
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhong University Press Wuhan China1986 (Chinese)
[22] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[23] F Ayaz ldquoApplications of differential transform method todifferential-algebraic equationsrdquo Applied Mathematics andComputation vol 152 no 3 pp 649ndash657 2004
[24] Y Khan Z Svoboda and Z Smarda ldquoSolving certain classes ofLane-Emden type equations using the differential transforma-tionmethodrdquoAdvances inDifference Equations vol 2012 article174 2012
[25] Z Smarda J Diblık and Y Khan ldquoExtension of the differentialtransformation method to nonlinear differential and integro-differential equations with proportional delaysrdquo Advances inDifference Equations vol 2013 article 69 2013
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Mathematics
applied easily to linear equations with constant and variablecoefficients and some nonlinear equations
The differential transformation of the 119896th derivative offunction 119910(119909) is defined by
119884 (119896) =1
119896[119889119896119910(119909)
119889119909119896]
119909=1199090
(2)
and the inverse transformation is defined as follows
119910 (119909) =
infin
sum
119896=0
119884 (119896) (119909 minus 1199090)119896 (3)
The following theorems can be obtained from definitions(2) and (3)
Theorem 1 Assume that 119865(119896) 119866(119896)119867(119896) are the differentialtransformations at the 1199090 = 0 of the functions 119891(119909) 119892(119909)ℎ(119909) respectively then one has the following
If 119891(119909) = 119889119899119892(119909)119889119909
119899 then 119865(119896) = ((119896 + 119899)119896)119866(119896 +
119899)If 119891(119909) = 119892(119909)ℎ(119909) then 119865(119896) = sum
119896
119897=0119866(119897)119867(119896 minus 119897)
If 119891(119909) = 119909119899 then 119865(119896) = 120575(119896 minus 119899) 120575 is the Kronecker
delta symbolIf 119891(119909) = 119890
120582119909 then 119865(119896) = 120582119896119896
If 119891(119909) = sin(120596119909 + 120582) then 119865(119896) = (120596119896119896) sin(1198961205872 +
120582)If 119891(119909) = cos(120596119909+120582) then 119865(119896) = (120596
119896119896) cos(1198961205872+
120582)If 119891(119909) = 119892(119909) int
119909
0ℎ(119905)119889119905 then 119865(119896) = sum
119896minus1
119897=0(1(119896 minus
119897))119866(119897)119867(119896 minus 119897 minus 1)
Theorem 2 Assume that119882(119896) 119884(119896) and 119884119894(119896) 119894 = 1 119899are the differential transformations at the 1199090 = 0 of thefunctions 119908(119909) 119910(119909) and 119910119894(119909) respectively and 119902 119902119894 119903 isin
(0 1] 119894 = 1 2 Then one has the following
If 119908(119909) = 119910(119902119909) then119882(119896) = 119902119896119884(119896)
If 119908(119909) = 1199101(1199021119909)1199102(1199022119909) then 119882(119896) =
sum119896
119897=0119902119897
1119902119896minus119897
21198841(119897)1198842(119896 minus 119897)
If 119908(119909) = 119889119898119910(119902119909)119889119909
119898 then 119882(119896) = ((119896 +
119898)119896)119902119896+119898
119884(119896 + 119898)If 119908(119909) = int
119903119909
0119910(119902119905)119889119905 then 119882(119896) = (119903
119896119896)119902119896minus1
119884(119896 minus
1)If 119908(119909) = int
119909
01199101(1199021119905)1199102(1199022119905)119889119905 then 119882(119896) =
(1119896)sum119896minus1
119897=0119902119897
1119902119896minus119897minus1
21198841(119897)1198842(119896 minus 119897 minus 1)
If 119908(119909) = 119910(119902119909) int119909
01199101(1199021119905)1199102(1199022119905)119889119905 then
119882(119896) =
119896minus1
sum
119897=0
119896minus119897minus1
sum
119904=0
1
119896 minus 119897119902119897119902119904
1119902119896minus119897minus119904minus1
2119884 (119897)
times 1198841 (119904) 1198842 (119896 minus 119897 minus 119904 minus 1)
(4)
where 119896 isin 119873
The proofs of Theorems 1 and 2 are given in [22 25]
3 Illustrate Examples
Example 1 Let us consider the following linear systemof Volterra integrodifferential equations with proportionaldelays and separable kernels
1199101 (119909) minus 1199101015840
1(119909
3) + 9119910
1015840
2(119909
3)
= 4 cos119909 + 8int119909
0
cos (119909 minus 119905) 1199102 (119905
3) 119889119905
+1
9int
119909
0
1199101 (119905
3) 119889119905
1199102 (119909) + 1199101015840
1(119909) minus 4119910
1015840
2(119909)
= minus2 cos 1199092minus 2 + 3int
119909
0
sin (119909 minus 119905) 1199101 (119905
2) 119889119905
+ int
119909
0
1199102 (119905
2) 119889119905
(5)
with the initial conditions 1199101(0) = 1 and 1199102(0) = 0
The differential transformation of the last system is
1198841 (119896) minus119896 + 1
3119896+11198841 (119896 + 1) + 9
119896 + 1
3119896+11198842 (119896 + 1)
=4
119896cos 120587119896
2+8
119896minus1
sum
119897=0
119896minus119897minus1
sum
119904=0
1198842 (119896 minus 119897 minus 119904 minus 1)
3119896minus119897minus119904minus1 (119896 minus 119897) 119897119904cos(120587
2[119897 minus 119904])
+1198841 (119896 minus 1)
3119896+1119896
1198842 (119896) + (119896 + 1) 1198841 (119896 + 1) minus 4 (119896 + 1) 1198842 (119896 + 1)
= minus1
2119896minus1119896cos 120587119896
2minus 2120575 (119896)
+ 3
119896minus1
sum
119897=0
119896minus119897minus1
sum
119904=0
1198842 (119896 minus 119897 minus 119904 minus 1)
2119896minus119897minus119904minus1 (119896 minus 119897) 119897119904sin(120587
2[119897 minus 119904])
+1198842 (119896 minus 1)
2119896minus1119896
(6)
Substituting 119909 = 0 in Example 1 we obtain values of firstderivatives of unknown functions that is 1198841(1) = 0 and1198842(1) = 1
For 119896 = 1 in (6) we have the following system
1198841 (1) minus2
91198841 (2) + 21198842 (2) = 81198842 (0) +
1
91198841 (0)
1198842 (1) + 21198841 (2) minus 81198842 (2) = 1198842 (0)
(7)
Solving the last system we get 1198841(2) = minus12 and 1198842(2) =0
Journal of Mathematics 3
Substituting 119896 = 2 in (6) we obtain the following systemof equations
1
91198841 (3) minus 1198842 (3) minus
2
3= minus
1
2
minus31198841 (3) + 121198842 (3) + 2 = 0
(8)
Solving the last system with two unknown we have1198841(3) = 0 and 1198842(3) = minus16
Continue this process and use inverse transformation weget 1199101(119909) = cos119909 and 1199102(119909) = sin119909 which are the exactsolutions of Example 1
Example 2 Consider the following system of nonlinearVolterra integrodifferential equations with proportionaldelays
1199101 (119909) + 1199101015840
2(119909) = 119909 minus
1
41199092+ int
119909
0
1199101 (119905
2) 119889119905
1199102 (119909) + 1199101015840
1(119909) = 119890
119909minus 3 (119909 minus 3) 119890
1199093+ int
119909
0
1199101 (119905) 1199102 (119905
3) 119889119905
(9)
with the initial conditions 1199101(0) = 0 and 1199102(0) = 1
Analogously for119909 = 0 in (9) we get11991010158401(0) = 1 and1199101015840
2(0) =
1Applying the differential transformation to (9) we have
the following system of difference equations
1198841 (119896) + (119896 + 1) 1198842 (119896 + 1)
=1
119896+ 120575 (119896 minus 1) minus
1
4120575 (119896 minus 2) +
1
1198962119896 minus 11198841 (119896 minus 1)
1198842 (119896) + (119896 + 1) 1198841 (119896 + 1)
=1
119896minus 3
119896
sum
119897=0
120575 (119897 minus 1)1
3119896minus119897 (119896 minus 119897)+
1
3119896minus2119896
+1
119896
119896minus1
sum
119897=0
1198841 (119897) 1198842 (119896 minus 119897 minus 1)
3119896minus119897minus1
(10)
For 119896 = 1 in (10) we get the following system
1198841 (1) + 21198842 (2) = 2 + 1198841 (0)
1198842 (1) + 21198841 (2) = 1
(11)
Solving the last system we obtain 1198841(2) = 0 and 1198842(2) =12 = 12
Substituting 119896 = 2 in (10) and solving correspondingsystem we have 1198841(3) = 0 and 1198842(3) = 16 = 13 and for119896 ge 3 1198841(119896) = 0 and 1198842(119896) = 1119896 Then using (3) we gain1199101(119909) = 119909 and 1199102(119909) = 119890
119909 which are the exact solutions ofsystem (9)
Example 3 Consider the following system of nonlinearVolterra integral equations
1199101 (119909) =1199093
324+891199092
90minus int
1199093
0
[1199101 (119905
2) minus 1199102 (
119905
5)] 119889119905
1199102 (119909) = 119909 +81199093
27+ int
119909
0
[1199101 (119905
3) minus 1199102
2(119905)] 119889119905
(12)
with the initial condition 1199101(0) = 1199102(0) = 0
Now applying the differential transformation to (12) weget the following system of difference equations
1198841 (119896) =120575 (119896 minus 3)
324+89120575 (119896 minus 2)
90minus1198841 (119896 minus 1)
31198962119896minus1119896+1198842 (119896 minus 1)
31198965119896minus1119896
1198842 (119896) = 120575 (119896 minus 1) +8120575 (119896 minus 3)
27+1198841 (119896 minus 1)
3119896minus1119896
minus1
119896
119896minus1
sum
119897=0
1198842 (119897) 1198842 (119896 minus 119897 minus 1)
(13)
From initial conditions we have 1198841(0) = 1198842(0) = 0Using (13) we get the following values
1198841 (1) = 0 1198841 (2) = 1
1198842 (1) = 1 1198842 (2) = 0
(14)
and for 119896 ge 3 1198841(119896) = 1198842(119896) = 0Using (3) we have 1199101(119909) = 119909 and 1199102(119909) = 119909
2 which areexact solutions of system (12)
Example 4 In last we consider the linear systemwith variablecoefficients of Volterra integral equations with proportionaldelays
11990921199101 (
119909
4) + 2 sin1199091199102 (
119909
3)
= sin 4119909
3+ sin 2119909
3+ 2int
1199092
0
1199091199101 (119905) 119889119905
sin 119909
31199101 (119909) + 1199091199102 (
119909
2) = 119909 cos 119909
2+ int
1199093
0
1199091199102 (119905) 119889119905
(15)
with exact solutions 1199101(119909) = 119909 and 1199102(119909) = cos119909
4 Journal of Mathematics
Applying DTM we have
119896
sum
119897=0
120575 (119897 minus 2) 1198841 (119896 minus 119897)1
4119896minus119897+ 2
119896
sum
119897=0
1
1198973119896minus119897sin 119897120587
21198842 (119896 minus 119897)
=4119896
3119896119896sin 119896120587
2+
2119896
3119896119896sin 119896120587
2
+ 2
119896
sum
119897=0
120575 (119897 minus 1) 1198841 (119896 minus 119897 minus 1)1
(119896 minus 119897) 2119896minus119897
119896
sum
119897=0
1
1198973119897sin 119897120587
21198841 (119896 minus 119897) +
119896
sum
119897=0
120575 (119897 minus 1) 1198842 (119896 minus 119897)1
2119896minus119897
=
119896
sum
119897=0
120575 (119897 minus 1)1
(119896 minus 119897)2119896minus119897
cos(119896 minus 1198972
120587)
+
119896minus1
sum
119897=0
120575 (119897 minus 1) 1198842 (119896 minus 119897 minus 1)1
(119896 minus 119897) 3119896minus119897
(16)
Solving the last systemwe have 1199101(119909) = 119909 and 1199102(119909) = 1minus
(11990922)+(119909
44)minus sdot sdot sdot which are Taylor series of exact solutions
of Example 4
4 Conclusions
In this study the differential transformationmethod has beenpresented for solving system of integral and integrodifferen-tial equations with proportional delaysThemajor benefits ofmethod from integral transformations are that the methodcan be applied for linear equations with variable coefficientsand nonlinear equations and the method gives the exactsolutions in series forms
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the reviewers for their con-structive comments and suggestions to improve the paper
References
[1] R P Agarwal D OrsquoRegan and P J Y Wong ldquoEigenvalues ofa system of Fredholm integral equationsrdquo Mathematical andComputer Modelling vol 39 no 9-10 pp 1113ndash1150 2004
[2] RF Churchhouse Handbook of Applicable Mathematics JohnWiley amp Sons New York NY USA 1981
[3] R P Kanwal Linear Integral Equations Birkhaauser BostonMass USA 1997
[4] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional Differential Equations KluwerAcademic Publishers Dordrecht The Netherlands 1999
[5] K Maleknejad F Mirzaee and S Abbasbandy ldquoSolving linearintegro-differential equations systemby using rationalizedHaarfunctions methodrdquo Applied Mathematics and Computation vol155 no 2 pp 317ndash328 2004
[6] K Maleknejad andM Tavassoli Kajani ldquoSolving linear integro-differential equation system by Galerkin methods with hydridfunctionsrdquo Applied Mathematics and Computation vol 159 no3 pp 603ndash612 2004
[7] A Dascioglu and M Sezer ldquoChebyshev polynomial solutionsof systems of higher-order linear Fredholm-Volterra integro-differential equationsrdquo Journal of the Franklin Institute vol 342pp 688ndash701 2005
[8] S Yuzbası N Sahin and MSezer ldquoNumerical solutions ofsystems of linear Fredholm integro-differential equations withBessel polynomial basesrdquoComputersampMathematics withAppli-cations vol 61 no 10 pp 3079ndash3096 2011
[9] N Sahin S Yuzbası and M Gulsu ldquoA collocation approachfor solving systems of linear Volterra integral equations withvariable coefficientsrdquo Computers and Mathematics with Appli-cations vol 62 no 2 pp 755ndash769 2011
[10] M Gulsu andM Sezer ldquoTaylor collocationmethod for solutionof systems of high-order linear Fredholm-Volterra integro-differential equationsrdquo International Journal of ComputerMath-ematics vol 83 no 4 pp 429ndash448 2006
[11] E Yusufoglu ldquoAn efficient algorithm for solving integro-differential equations systemrdquo Applied Mathematics and Com-putation vol 192 no 1 pp 51ndash55 2007
[12] E Yusufoglu ldquoA homotopy perturbation algorithm to solve asystem of Fredholm-Volterra type integral equationsrdquo Mathe-matical andComputerModelling vol 47 no 11-12 pp 1099ndash11072008
[13] J Saberi-Nadjafi and M Tamamgar ldquoThe variational iterationmethod a highly promising method for solving the system ofintegro-differential equationsrdquo Computers amp Mathematics withApplications vol 56 no 2 pp 346ndash351 2008
[14] A Arikoglu and I Ozkol ldquoSolutions of integral and integro-differential equation systems by using differential transformmethodrdquo Computers amp Mathematics with Applications vol 56no 9 pp 2411ndash2417 2008
[15] H H Sorkun and S Yalcinbas ldquoApproximate solutions of linearVolterra integral equation systems with variable coefficientsrdquoApplied Mathematical Modelling vol 34 no 11 pp 3451ndash34642010
[16] J Biazar E Babolian and R M Islam ldquoSolution of a systemof Volterra integral equations of the first kind by Adomianmethodrdquo Applied Mathematics and Computation vol 139 no2-3 pp 249ndash258 2003
[17] S Kumar O P Singh and S Dixit ldquoHomotopy perturbationmethod for solving system of generalized Abelrsquos integral equa-tionsrdquo Applications and Applied Mathematics vol 6 no 11 pp2009ndash2024 2011
[18] J Biazar M Eslami and M R Islam ldquoDifferential transformmethod for special systems of integral equationsrdquo Journal ofKing Saud UniversitymdashScience vol 24 no 3 pp 211ndash214 2012
[19] R Katani and S Shahmorad ldquoA block by block method withRomberg quadrature for the system of Urysohn type Volterraintegral equationsrdquo Computational and Applied Mathematicsvol 31 no 1 pp 191ndash203 2012
[20] S Yalcınbas and K Erdem ldquoA new approximation method forthe systems of nonlinear fredholm integral equationsrdquo AppliedMathematics and Physics vol 2 no 2 pp 40ndash48 2014
Journal of Mathematics 5
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhong University Press Wuhan China1986 (Chinese)
[22] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[23] F Ayaz ldquoApplications of differential transform method todifferential-algebraic equationsrdquo Applied Mathematics andComputation vol 152 no 3 pp 649ndash657 2004
[24] Y Khan Z Svoboda and Z Smarda ldquoSolving certain classes ofLane-Emden type equations using the differential transforma-tionmethodrdquoAdvances inDifference Equations vol 2012 article174 2012
[25] Z Smarda J Diblık and Y Khan ldquoExtension of the differentialtransformation method to nonlinear differential and integro-differential equations with proportional delaysrdquo Advances inDifference Equations vol 2013 article 69 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Mathematics 3
Substituting 119896 = 2 in (6) we obtain the following systemof equations
1
91198841 (3) minus 1198842 (3) minus
2
3= minus
1
2
minus31198841 (3) + 121198842 (3) + 2 = 0
(8)
Solving the last system with two unknown we have1198841(3) = 0 and 1198842(3) = minus16
Continue this process and use inverse transformation weget 1199101(119909) = cos119909 and 1199102(119909) = sin119909 which are the exactsolutions of Example 1
Example 2 Consider the following system of nonlinearVolterra integrodifferential equations with proportionaldelays
1199101 (119909) + 1199101015840
2(119909) = 119909 minus
1
41199092+ int
119909
0
1199101 (119905
2) 119889119905
1199102 (119909) + 1199101015840
1(119909) = 119890
119909minus 3 (119909 minus 3) 119890
1199093+ int
119909
0
1199101 (119905) 1199102 (119905
3) 119889119905
(9)
with the initial conditions 1199101(0) = 0 and 1199102(0) = 1
Analogously for119909 = 0 in (9) we get11991010158401(0) = 1 and1199101015840
2(0) =
1Applying the differential transformation to (9) we have
the following system of difference equations
1198841 (119896) + (119896 + 1) 1198842 (119896 + 1)
=1
119896+ 120575 (119896 minus 1) minus
1
4120575 (119896 minus 2) +
1
1198962119896 minus 11198841 (119896 minus 1)
1198842 (119896) + (119896 + 1) 1198841 (119896 + 1)
=1
119896minus 3
119896
sum
119897=0
120575 (119897 minus 1)1
3119896minus119897 (119896 minus 119897)+
1
3119896minus2119896
+1
119896
119896minus1
sum
119897=0
1198841 (119897) 1198842 (119896 minus 119897 minus 1)
3119896minus119897minus1
(10)
For 119896 = 1 in (10) we get the following system
1198841 (1) + 21198842 (2) = 2 + 1198841 (0)
1198842 (1) + 21198841 (2) = 1
(11)
Solving the last system we obtain 1198841(2) = 0 and 1198842(2) =12 = 12
Substituting 119896 = 2 in (10) and solving correspondingsystem we have 1198841(3) = 0 and 1198842(3) = 16 = 13 and for119896 ge 3 1198841(119896) = 0 and 1198842(119896) = 1119896 Then using (3) we gain1199101(119909) = 119909 and 1199102(119909) = 119890
119909 which are the exact solutions ofsystem (9)
Example 3 Consider the following system of nonlinearVolterra integral equations
1199101 (119909) =1199093
324+891199092
90minus int
1199093
0
[1199101 (119905
2) minus 1199102 (
119905
5)] 119889119905
1199102 (119909) = 119909 +81199093
27+ int
119909
0
[1199101 (119905
3) minus 1199102
2(119905)] 119889119905
(12)
with the initial condition 1199101(0) = 1199102(0) = 0
Now applying the differential transformation to (12) weget the following system of difference equations
1198841 (119896) =120575 (119896 minus 3)
324+89120575 (119896 minus 2)
90minus1198841 (119896 minus 1)
31198962119896minus1119896+1198842 (119896 minus 1)
31198965119896minus1119896
1198842 (119896) = 120575 (119896 minus 1) +8120575 (119896 minus 3)
27+1198841 (119896 minus 1)
3119896minus1119896
minus1
119896
119896minus1
sum
119897=0
1198842 (119897) 1198842 (119896 minus 119897 minus 1)
(13)
From initial conditions we have 1198841(0) = 1198842(0) = 0Using (13) we get the following values
1198841 (1) = 0 1198841 (2) = 1
1198842 (1) = 1 1198842 (2) = 0
(14)
and for 119896 ge 3 1198841(119896) = 1198842(119896) = 0Using (3) we have 1199101(119909) = 119909 and 1199102(119909) = 119909
2 which areexact solutions of system (12)
Example 4 In last we consider the linear systemwith variablecoefficients of Volterra integral equations with proportionaldelays
11990921199101 (
119909
4) + 2 sin1199091199102 (
119909
3)
= sin 4119909
3+ sin 2119909
3+ 2int
1199092
0
1199091199101 (119905) 119889119905
sin 119909
31199101 (119909) + 1199091199102 (
119909
2) = 119909 cos 119909
2+ int
1199093
0
1199091199102 (119905) 119889119905
(15)
with exact solutions 1199101(119909) = 119909 and 1199102(119909) = cos119909
4 Journal of Mathematics
Applying DTM we have
119896
sum
119897=0
120575 (119897 minus 2) 1198841 (119896 minus 119897)1
4119896minus119897+ 2
119896
sum
119897=0
1
1198973119896minus119897sin 119897120587
21198842 (119896 minus 119897)
=4119896
3119896119896sin 119896120587
2+
2119896
3119896119896sin 119896120587
2
+ 2
119896
sum
119897=0
120575 (119897 minus 1) 1198841 (119896 minus 119897 minus 1)1
(119896 minus 119897) 2119896minus119897
119896
sum
119897=0
1
1198973119897sin 119897120587
21198841 (119896 minus 119897) +
119896
sum
119897=0
120575 (119897 minus 1) 1198842 (119896 minus 119897)1
2119896minus119897
=
119896
sum
119897=0
120575 (119897 minus 1)1
(119896 minus 119897)2119896minus119897
cos(119896 minus 1198972
120587)
+
119896minus1
sum
119897=0
120575 (119897 minus 1) 1198842 (119896 minus 119897 minus 1)1
(119896 minus 119897) 3119896minus119897
(16)
Solving the last systemwe have 1199101(119909) = 119909 and 1199102(119909) = 1minus
(11990922)+(119909
44)minus sdot sdot sdot which are Taylor series of exact solutions
of Example 4
4 Conclusions
In this study the differential transformationmethod has beenpresented for solving system of integral and integrodifferen-tial equations with proportional delaysThemajor benefits ofmethod from integral transformations are that the methodcan be applied for linear equations with variable coefficientsand nonlinear equations and the method gives the exactsolutions in series forms
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the reviewers for their con-structive comments and suggestions to improve the paper
References
[1] R P Agarwal D OrsquoRegan and P J Y Wong ldquoEigenvalues ofa system of Fredholm integral equationsrdquo Mathematical andComputer Modelling vol 39 no 9-10 pp 1113ndash1150 2004
[2] RF Churchhouse Handbook of Applicable Mathematics JohnWiley amp Sons New York NY USA 1981
[3] R P Kanwal Linear Integral Equations Birkhaauser BostonMass USA 1997
[4] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional Differential Equations KluwerAcademic Publishers Dordrecht The Netherlands 1999
[5] K Maleknejad F Mirzaee and S Abbasbandy ldquoSolving linearintegro-differential equations systemby using rationalizedHaarfunctions methodrdquo Applied Mathematics and Computation vol155 no 2 pp 317ndash328 2004
[6] K Maleknejad andM Tavassoli Kajani ldquoSolving linear integro-differential equation system by Galerkin methods with hydridfunctionsrdquo Applied Mathematics and Computation vol 159 no3 pp 603ndash612 2004
[7] A Dascioglu and M Sezer ldquoChebyshev polynomial solutionsof systems of higher-order linear Fredholm-Volterra integro-differential equationsrdquo Journal of the Franklin Institute vol 342pp 688ndash701 2005
[8] S Yuzbası N Sahin and MSezer ldquoNumerical solutions ofsystems of linear Fredholm integro-differential equations withBessel polynomial basesrdquoComputersampMathematics withAppli-cations vol 61 no 10 pp 3079ndash3096 2011
[9] N Sahin S Yuzbası and M Gulsu ldquoA collocation approachfor solving systems of linear Volterra integral equations withvariable coefficientsrdquo Computers and Mathematics with Appli-cations vol 62 no 2 pp 755ndash769 2011
[10] M Gulsu andM Sezer ldquoTaylor collocationmethod for solutionof systems of high-order linear Fredholm-Volterra integro-differential equationsrdquo International Journal of ComputerMath-ematics vol 83 no 4 pp 429ndash448 2006
[11] E Yusufoglu ldquoAn efficient algorithm for solving integro-differential equations systemrdquo Applied Mathematics and Com-putation vol 192 no 1 pp 51ndash55 2007
[12] E Yusufoglu ldquoA homotopy perturbation algorithm to solve asystem of Fredholm-Volterra type integral equationsrdquo Mathe-matical andComputerModelling vol 47 no 11-12 pp 1099ndash11072008
[13] J Saberi-Nadjafi and M Tamamgar ldquoThe variational iterationmethod a highly promising method for solving the system ofintegro-differential equationsrdquo Computers amp Mathematics withApplications vol 56 no 2 pp 346ndash351 2008
[14] A Arikoglu and I Ozkol ldquoSolutions of integral and integro-differential equation systems by using differential transformmethodrdquo Computers amp Mathematics with Applications vol 56no 9 pp 2411ndash2417 2008
[15] H H Sorkun and S Yalcinbas ldquoApproximate solutions of linearVolterra integral equation systems with variable coefficientsrdquoApplied Mathematical Modelling vol 34 no 11 pp 3451ndash34642010
[16] J Biazar E Babolian and R M Islam ldquoSolution of a systemof Volterra integral equations of the first kind by Adomianmethodrdquo Applied Mathematics and Computation vol 139 no2-3 pp 249ndash258 2003
[17] S Kumar O P Singh and S Dixit ldquoHomotopy perturbationmethod for solving system of generalized Abelrsquos integral equa-tionsrdquo Applications and Applied Mathematics vol 6 no 11 pp2009ndash2024 2011
[18] J Biazar M Eslami and M R Islam ldquoDifferential transformmethod for special systems of integral equationsrdquo Journal ofKing Saud UniversitymdashScience vol 24 no 3 pp 211ndash214 2012
[19] R Katani and S Shahmorad ldquoA block by block method withRomberg quadrature for the system of Urysohn type Volterraintegral equationsrdquo Computational and Applied Mathematicsvol 31 no 1 pp 191ndash203 2012
[20] S Yalcınbas and K Erdem ldquoA new approximation method forthe systems of nonlinear fredholm integral equationsrdquo AppliedMathematics and Physics vol 2 no 2 pp 40ndash48 2014
Journal of Mathematics 5
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhong University Press Wuhan China1986 (Chinese)
[22] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[23] F Ayaz ldquoApplications of differential transform method todifferential-algebraic equationsrdquo Applied Mathematics andComputation vol 152 no 3 pp 649ndash657 2004
[24] Y Khan Z Svoboda and Z Smarda ldquoSolving certain classes ofLane-Emden type equations using the differential transforma-tionmethodrdquoAdvances inDifference Equations vol 2012 article174 2012
[25] Z Smarda J Diblık and Y Khan ldquoExtension of the differentialtransformation method to nonlinear differential and integro-differential equations with proportional delaysrdquo Advances inDifference Equations vol 2013 article 69 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Mathematics
Applying DTM we have
119896
sum
119897=0
120575 (119897 minus 2) 1198841 (119896 minus 119897)1
4119896minus119897+ 2
119896
sum
119897=0
1
1198973119896minus119897sin 119897120587
21198842 (119896 minus 119897)
=4119896
3119896119896sin 119896120587
2+
2119896
3119896119896sin 119896120587
2
+ 2
119896
sum
119897=0
120575 (119897 minus 1) 1198841 (119896 minus 119897 minus 1)1
(119896 minus 119897) 2119896minus119897
119896
sum
119897=0
1
1198973119897sin 119897120587
21198841 (119896 minus 119897) +
119896
sum
119897=0
120575 (119897 minus 1) 1198842 (119896 minus 119897)1
2119896minus119897
=
119896
sum
119897=0
120575 (119897 minus 1)1
(119896 minus 119897)2119896minus119897
cos(119896 minus 1198972
120587)
+
119896minus1
sum
119897=0
120575 (119897 minus 1) 1198842 (119896 minus 119897 minus 1)1
(119896 minus 119897) 3119896minus119897
(16)
Solving the last systemwe have 1199101(119909) = 119909 and 1199102(119909) = 1minus
(11990922)+(119909
44)minus sdot sdot sdot which are Taylor series of exact solutions
of Example 4
4 Conclusions
In this study the differential transformationmethod has beenpresented for solving system of integral and integrodifferen-tial equations with proportional delaysThemajor benefits ofmethod from integral transformations are that the methodcan be applied for linear equations with variable coefficientsand nonlinear equations and the method gives the exactsolutions in series forms
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The authors would like to thank the reviewers for their con-structive comments and suggestions to improve the paper
References
[1] R P Agarwal D OrsquoRegan and P J Y Wong ldquoEigenvalues ofa system of Fredholm integral equationsrdquo Mathematical andComputer Modelling vol 39 no 9-10 pp 1113ndash1150 2004
[2] RF Churchhouse Handbook of Applicable Mathematics JohnWiley amp Sons New York NY USA 1981
[3] R P Kanwal Linear Integral Equations Birkhaauser BostonMass USA 1997
[4] V Kolmanovskii and A Myshkis Introduction to the Theoryand Applications of Functional Differential Equations KluwerAcademic Publishers Dordrecht The Netherlands 1999
[5] K Maleknejad F Mirzaee and S Abbasbandy ldquoSolving linearintegro-differential equations systemby using rationalizedHaarfunctions methodrdquo Applied Mathematics and Computation vol155 no 2 pp 317ndash328 2004
[6] K Maleknejad andM Tavassoli Kajani ldquoSolving linear integro-differential equation system by Galerkin methods with hydridfunctionsrdquo Applied Mathematics and Computation vol 159 no3 pp 603ndash612 2004
[7] A Dascioglu and M Sezer ldquoChebyshev polynomial solutionsof systems of higher-order linear Fredholm-Volterra integro-differential equationsrdquo Journal of the Franklin Institute vol 342pp 688ndash701 2005
[8] S Yuzbası N Sahin and MSezer ldquoNumerical solutions ofsystems of linear Fredholm integro-differential equations withBessel polynomial basesrdquoComputersampMathematics withAppli-cations vol 61 no 10 pp 3079ndash3096 2011
[9] N Sahin S Yuzbası and M Gulsu ldquoA collocation approachfor solving systems of linear Volterra integral equations withvariable coefficientsrdquo Computers and Mathematics with Appli-cations vol 62 no 2 pp 755ndash769 2011
[10] M Gulsu andM Sezer ldquoTaylor collocationmethod for solutionof systems of high-order linear Fredholm-Volterra integro-differential equationsrdquo International Journal of ComputerMath-ematics vol 83 no 4 pp 429ndash448 2006
[11] E Yusufoglu ldquoAn efficient algorithm for solving integro-differential equations systemrdquo Applied Mathematics and Com-putation vol 192 no 1 pp 51ndash55 2007
[12] E Yusufoglu ldquoA homotopy perturbation algorithm to solve asystem of Fredholm-Volterra type integral equationsrdquo Mathe-matical andComputerModelling vol 47 no 11-12 pp 1099ndash11072008
[13] J Saberi-Nadjafi and M Tamamgar ldquoThe variational iterationmethod a highly promising method for solving the system ofintegro-differential equationsrdquo Computers amp Mathematics withApplications vol 56 no 2 pp 346ndash351 2008
[14] A Arikoglu and I Ozkol ldquoSolutions of integral and integro-differential equation systems by using differential transformmethodrdquo Computers amp Mathematics with Applications vol 56no 9 pp 2411ndash2417 2008
[15] H H Sorkun and S Yalcinbas ldquoApproximate solutions of linearVolterra integral equation systems with variable coefficientsrdquoApplied Mathematical Modelling vol 34 no 11 pp 3451ndash34642010
[16] J Biazar E Babolian and R M Islam ldquoSolution of a systemof Volterra integral equations of the first kind by Adomianmethodrdquo Applied Mathematics and Computation vol 139 no2-3 pp 249ndash258 2003
[17] S Kumar O P Singh and S Dixit ldquoHomotopy perturbationmethod for solving system of generalized Abelrsquos integral equa-tionsrdquo Applications and Applied Mathematics vol 6 no 11 pp2009ndash2024 2011
[18] J Biazar M Eslami and M R Islam ldquoDifferential transformmethod for special systems of integral equationsrdquo Journal ofKing Saud UniversitymdashScience vol 24 no 3 pp 211ndash214 2012
[19] R Katani and S Shahmorad ldquoA block by block method withRomberg quadrature for the system of Urysohn type Volterraintegral equationsrdquo Computational and Applied Mathematicsvol 31 no 1 pp 191ndash203 2012
[20] S Yalcınbas and K Erdem ldquoA new approximation method forthe systems of nonlinear fredholm integral equationsrdquo AppliedMathematics and Physics vol 2 no 2 pp 40ndash48 2014
Journal of Mathematics 5
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhong University Press Wuhan China1986 (Chinese)
[22] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[23] F Ayaz ldquoApplications of differential transform method todifferential-algebraic equationsrdquo Applied Mathematics andComputation vol 152 no 3 pp 649ndash657 2004
[24] Y Khan Z Svoboda and Z Smarda ldquoSolving certain classes ofLane-Emden type equations using the differential transforma-tionmethodrdquoAdvances inDifference Equations vol 2012 article174 2012
[25] Z Smarda J Diblık and Y Khan ldquoExtension of the differentialtransformation method to nonlinear differential and integro-differential equations with proportional delaysrdquo Advances inDifference Equations vol 2013 article 69 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Mathematics 5
[21] J K Zhou Differential Transformation and Its Applications forElectrical Circuits Huazhong University Press Wuhan China1986 (Chinese)
[22] A Arikoglu and I Ozkol ldquoSolution of boundary value problemsfor integro-differential equations by using differential transformmethodrdquoAppliedMathematics and Computation vol 168 no 2pp 1145ndash1158 2005
[23] F Ayaz ldquoApplications of differential transform method todifferential-algebraic equationsrdquo Applied Mathematics andComputation vol 152 no 3 pp 649ndash657 2004
[24] Y Khan Z Svoboda and Z Smarda ldquoSolving certain classes ofLane-Emden type equations using the differential transforma-tionmethodrdquoAdvances inDifference Equations vol 2012 article174 2012
[25] Z Smarda J Diblık and Y Khan ldquoExtension of the differentialtransformation method to nonlinear differential and integro-differential equations with proportional delaysrdquo Advances inDifference Equations vol 2013 article 69 2013
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
