Research Article | Open Access
Modified Differential Transform Method for Solving the Model of Pollution for a System of Lakes
This work presents the application of the differential transform method (DTM) to the model of pollution for a system of three lakes interconnected by channels. Three 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-Padé resummation method as a useful strategy to extend the domain of convergence of the approximate solutions. The Fehlberg fourth-fifth 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. The 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.
Semianalytical methods like differential transform method (DTM) [1–4], reduced differential transform method (RDTM) [5–7], homotopy perturbation method (HPM) [8–16], homotopy analysis method (HAM) , variational iteration method (VIM) , and generalized homotopy method , multivariate Padé series , among others, are powerful tools to approximate linear and nonlinear problems in physics and engineering. Analytical solutions aid researchers to study the effect of different variables or parameters on the function under study easily . Among the above-mentioned methods, the DTM is highlighted by its simplicity and versatility to solve nonlinear differential equations. This method does not rely on a perturbation parameter or a trial function as other popular approximative methods. In , the DTM was introduced to the engineering field as a tool to find approximate solutions of electrical circuits. DTM produces approximations based on an iterative procedure derived from the Taylor series expansion. This method is very effective and powerful for solving various kinds of differential equations as nonlinear biochemical reaction model , two point boundary-value problems , differential-algebraic equations , the KdV and mKdV equations , the Schrodinger equations , fractional differential equations , and the Riccati differential equation , among others.
Therefore, in this paper, we present the application of a hybrid technique combining DTM, Laplace transform, and Padé approximant  to find approximate analytical solutions for a pollution model [29–34]. The 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 flow directions. Initially, a pollutant is introduced into the first lake at a given rate which may be constant or may vary with time. Therefore, 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 process, and the volume of water in any of the three lakes remains constant. Also we assume that the type of pollution is persistent and not degrading to other forms.
We will consider three input models for the pollution source: periodic, exponentially decaying, and linear. To the best of the authors’ knowledge, the periodic and exponential decaying input models are not reported in literature as potential behaviours for the pollution source. Solutions to this problem are first obtained in convergent series form using the DTM. To improve the solution obtained from DTM’s truncated series, we apply Laplace transform to it, and then convert the transformed series into a meromorphic function by forming its Padé approximant. Finally, we take the inverse Laplace transform of the Padé approximant to obtain the approximate analytical solution. This hybrid method (LPDTM) which combines DTM with Laplace-Padé posttreatment greatly improves DTM’s truncated series solutions in convergence rate. In fact, the Laplace-Padé resummation method enlarges the domain of convergence of the truncated power series and often leads to an accurate approximation or the exact solution. It is worth mentioning that the constant input model is not treated here, because it was successfully solved by the DTM in .
The proposed method does not generate noise terms also known as secular terms in the solution as the homotopy perturbation based techniques . This property of DTM greatly reduces the volume of computation and improves the efficiency of LPDTM in comparison to the perturbation based methods. What is more, LPDTM does not require a perturbation parameter as the perturbation based techniques (including HPM). Finally, LPDTM is straightforward and can be programmed using computer algebra packages like Maple or Mathematica.
The rest of this paper is organized as follows. In the next section we describe how the DTM can be applied to solve systems of ordinary differential equations. The main idea behind the Padé approximant is given in Section 3. In Section 4, we give the basic concept of the Laplace-Padé resummation method. In Section 5, we give a description of the lakes pollution problem. In Section 6, we apply LPDTM to solve three pollution problems. In Section 7, we give a brief discussion. Finally, a conclusion is drawn in the last section.
2. Differential Transform Method (DTM)
The basic definitions and fundamental operations of differential transform are given in [1, 22–26]. For convenience of the reader, we will give a review of the DTM. We will also describe the DTM to solve systems of ordinary differential equations.
Definition 1. If a function is analytical with respect to in the domain of interest , then is the transformed function of .
Definition 2. The differential inverse transforms of the set is defined by Substituting (1) into (2), one deduces that From Definitions 1 and 2, it is easy to see that the concept of the DTM is obtained from the power series expansion. To illustrate the application of the proposed DTM to solve systems of ordinary differential equations, one considers the nonlinear system