The Adomian Decomposition Method and the Differential Transform Method for Numerical Solution of Multi-Pantograph Delay Differential Equations

Abstract

In this paper, the Adomian Decomposition Method (ADM) and the Differential Transform Method (DTM) are applied to solve the multi-pantograph delay equations. The sufficient conditions are given to assure the convergence of these methods. Several examples are presented to demonstrate the efficiency and reliability of the ADM and the DTM; numerical results are discussed, compared with exact solution. The results of the ADM and the DTM show its better performance than others. These methods give the desired accurate results only in a few terms and in a series form of the solution. The approach is simple and effective. These methods are used to solve many linear and nonlinear problems and reduce the size of computational work.

Share and Cite:

Cakir, M. and Arslan, D. (2015) The Adomian Decomposition Method and the Differential Transform Method for Numerical Solution of Multi-Pantograph Delay Differential Equations. Applied Mathematics, 6, 1332-1343. doi: 10.4236/am.2015.68126.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] Ockendon, J.R. and Taylor, A.B. (1971) The Dynamics of a Current Collection System for an Electric Locomotive. Proceedings of the Royal Society of London A, 322, 447-468. http://dx.doi.org/10.1098/rspa.1971.0078 [2] Li, D. and Liu, M. (2005) Runge-Kutta Methods for the Multi-pantograph Delay Equation. Applied Mathematics and Computation, 163, 383-395. http://dx.doi.org/10.1016/j.amc.2004.02.013 [3] Evans, D.J. and Raslan, K.R. (2005) The Adomian Decomposition Method for Solving Delay Differential Equation. International Journal of Computer Mathematics, 82, 49-54. http://dx.doi.org/10.1080/00207160412331286815 [4] Keskin, Y., at al. (2007) Approximate Solutions of Generalized Pantograph Equations by the Differential Transform Method. International Journal of Nonlinear Sciences and Numerical, 8, 159-164. http://dx.doi.org/10.1515/IJNSNS.20078.2.159 [5] Sezer, M. and Dascioglu, A.A. (2007) A Taylor Method for Numerical Solution of Generalized Pantograph Equations with Linear Functional Argument. Journal of Computational and Applied Mathematics, 200, 217-225. http://dx.doi.org/10.1016/j.cam.2005.12.015 [6] Yu, Z. (2008) Variational ?teration Method for Solving the Multi-Pantograph Delay Equation. Physics Letters A, 372, 6475-6479. http://dx.doi.org/10.1016/j.physleta.2008.09.013 [7] Sezer, M., Yalcinbas, S. and Sahin, N. (2008) Approximate Solution of Multi-Pantograph Equation with Variable Coefficients. Journal of Computational and Applied Mathematics, 214, 406-416. http://dx.doi.org/10.1016/j.cam.2007.03.024 [8] Geng, F.Z. and Qian, S.P. (2014) Solving Singularly Perturbed Multi-Pantograph Delay Equations Based on the Reprociding Kernel Medhod. Abstract and Applied Analysis, 2014, 6 p. http://dx.doi.org/10.1155/2014/794716 [9] Cherruault, Y., Adomian, G., Abbaoui, K. and Rach, R. (1995) Further Remarks on Convergence of Decomposition Method. International Journal of Bio-Medical Computing, 38, 89-93. http://dx.doi.org/10.1016/0020-7101(94)01042-Y [10] Ismail, H.N., Raslan, K.R. and Salem, G.S. (2004) Solitary Wave Solutions for the General KdV Equation by Adomian Decomposition Method. Applied Mathematics and Computation, 154, 17-29. http://dx.doi.org/10.1016/S0096-3003(03)00686-6 [11] El-Safty, A., Salim, M.S. and El-Khatib, M.A. (2003) Convergent of the Spline Functions for Delay Dynamic System. International Journal of Computer Mathematics, 80, 509-518. http://dx.doi.org/10.1080/0020716021000014204 [12] Rostam, K., Saeed, B. and Rahman, M. (2011) Differential Transform Method for Solving System of Delay Differential Equation. Australian Journal of Basic and Applied Sciences, 5, 201-206. [13] Adomian, G. (1994) Solving Frontier Problems of Physics: The Decomposition Method. Kluwer Academic Publishers, Boston. http://dx.doi.org/10.1007/978-94-015-8289-6 [14] Adomian, G. and Rach, R. (1993) Analytic Solution of Nonlinear Boundary-Value Problems in Several Dimensions by Decomposition. Journal of Mathematical Analysis and Applications, 174, 118-137. http://dx.doi.org/10.1006/jmaa.1993.1105 [15] Zhou, J.K. (1986) Differential Transform and Its Application for Electrical Circuits. Huazhong University Press, Wuhan. [16] Feng, X. (2013) An Analytic Study on the Multi-Pantograph Delay Equations with Variable Coefficients. Bulletin Mathematique de la society des Sciences mathématiques de Roumanie Tome, 56, 205-215. http://ssmr.ro/bulletin/pdf/56-2/articol_7.pdf [17] Ayaz, F. (2004) Applications of Differential Transform Method to Differential-Algebraic Equations. Applied Mathematics and Computation, 152, 649-657. http://dx.doi.org/10.1016/S0096-3003(03)00581-2 [18] Kurnaz, A. and Oturanc, G. (2005) The Differential Transform Approximation for the System Ordinary Differential Equations. International Journal of Computer Mathematics, 82, 709-719. http://dx.doi.org/10.1080/00207160512331329050 [19] Cherruault, Y. (1989) Convergence of Adomian’s Method. Kybernetes, 18, 31-38. http://dx.doi.org/10.1108/eb005812 [20] Hosseini, M.M. and Nasabzadeh, H. (2006) On the Convergence of Adomian Decomposition Method. Applied Mathematics and Computation, 182, 536-543. http://dx.doi.org/10.1016/j.amc.2006.04.015 [21] Liu, M.Z. and Li, D. (2004) Properties of Analytic Solution and Numerical Solution of Multi-Pantograph Equation. Applied Mathematics and Computation, 155, 853-871. http://dx.doi.org/10.1016/j.amc.2003.07.017