Applied Mathematics, 2015, 6, 1332-1343 Published Online July 2015 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2015.68126 How to cite this paper: 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. http://dx.doi.org/10.4236/am.2015.68126 The Adomian Decomposition Method and the Differential Transform Method for Numerical Solution of Multi-Pantograph Delay Differential Equations Musa Cakir, Derya Arslan Department of Mathematics, Faculty of Science, University of Yuzuncu Yil, Van, Turkey Email: cakirmusa@hotmail.com , ayredlanu@gmail.com Received 15 May 2015; accepted 24 July 2015; published 27 July 2015 Copyright © 2015 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ 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 so- lution. The approach is simple and effective. These methods are used to solve many linear and nonlinear problems and reduce the size of computational work. Keywords Multi-Pantograph Delay Differential Equations, Adomian Decomposition Method (ADM), Differential Transform Method (DTM), Convergence of Adomian Decomposition Method 1. Introduction Pantograph is a device located on the electriclocomotive. The first time, electric locomotive was made in Amer- ica in 1851. It was commissioned in 1895. Mathematical model of pantograph was first developed by Taylor and Ockendon (1971) [1]. Pantograph equations belong to a special class of functional-differential equations with proportional delays and arise in many applications such as, astrophysics, nonlinear dynamical systems, probabi- lity theory on algebraic structures, electro dynamics, quantum mechanics and cell growth, number theory, mix- ing problems, population models, etc.