Research Article Modified Differential Transform Method for Solving the Model of Pollution for a System of Lakes Brahim Benhammouda, 1 Hector Vazquez-Leal, 2 and Luis Hernandez-Martinez 3 1 Higher Colleges of Technology, Abu Dhabi Men’s College, P.O. Box 25035, Abu Dhabi, UAE 2 Electronic Instrumentation and Atmospheric Sciences School, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltr´ an S/N, 91000 Xalapa, VER, Mexico 3 National Institute for Astrophysics, Optics, and Electronics, Luis Enrique Erro No. 1, Santa Maria 72840 Tonantzintla, PUE, Mexico Correspondence should be addressed to Hector Vazquez-Leal; hvazquez@uv.mx Received 19 May 2014; Revised 19 August 2014; Accepted 1 September 2014; Published 15 September 2014 Academic Editor: Carlo Piccardi Copyright © 2014 Brahim Benhammouda et al. Tis 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. Tis work presents the application of the diferential transform method (DTM) to the model of pollution for a system of three lakes interconnected by channels. Tree input models (periodic, exponentially decaying, and linear) are solved to show that DTM can provide analytical solutions of pollution model in convergent series form. In addition, we present the posttreatment of the power series solutions with the Laplace-Pad´ e resummation method as a useful strategy to extend the domain of convergence of the approximate solutions. Te Fehlberg fourth-ffh order Runge-Kutta method with degree four interpolant (RKF45) numerical solution of the lakes system problem is used as a reference to compare with the analytical approximations showing the high accuracy of the results. Te main advantage of the proposed technique is that it is based on a few straightforward steps and does not generate secular terms or depend of a perturbation parameter. 1. Introduction Semianalytical methods like diferential transform method (DTM) [1–4], reduced diferential transform method (RDTM) [5–7], homotopy perturbation method (HPM) [8–16], homotopy analysis method (HAM) [17], variational iteration method (VIM) [18], and generalized homotopy method [19], multivariate Pad´ e series [20], among others, are powerful tools to approximate linear and nonlinear problems in physics and engineering. Analytical solutions aid researchers to study the efect of diferent variables or parameters on the function under study easily [21]. Among the above-mentioned methods, the DTM is high- lighted by its simplicity and versatility to solve nonlinear diferential equations. Tis method does not rely on a perturbation parameter or a trial function as other popular approximative methods. In [1], the DTM was introduced to the engineering feld as a tool to fnd approximate solutions of electrical circuits. DTM produces approximations based on an iterative procedure derived from the Taylor series expansion. Tis method is very efective and powerful for solving various kinds of diferential equations as nonlinear biochemical reaction model [2], two point boundary-value problems [22], diferential-algebraic equations [23], the KdV and mKdV equations [24], the Schrodinger equations [25], fractional diferential equations [26], and the Riccati diferential equation [27], among others. Terefore, in this paper, we present the application of a hybrid technique combining DTM, Laplace transform, and Pad´ e approximant [28] to fnd approximate analytical solutions for a pollution model [29–34]. Te aim of the model is to describe the pollution of a system of three lakes [35– 39] as depicted in Figure 1. Each lake is considered to be as large compartment and the interconnecting channels as pipes between the compartments with given fow directions. Initially, a pollutant is introduced into the frst lake at a given rate which may be constant or may vary with time. Terefore, we are interested in knowing the level of pollution in each lake at any time. We assume the pollutant in each lake to be uniformly distributed throughout the lake by some mixing Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2014, Article ID 645726, 12 pages http://dx.doi.org/10.1155/2014/645726