Abstract

A new scheme, deduced from Khan’s homotopy perturbation transform method (HPTM) (Khan, 2014; Khan and Wu, 2011) via optimal parameter, is presented for solving nonlinear differential difference equations. Simple but typical examples are given to illustrate the validity and great potential of Khan’s homotopy perturbation transform method (HPTM) via optimal parameter in solving nonlinear differential difference equation. The numerical solutions show that the proposed method is very efficient and computationally attractive. It provides more realistic series solutions that converge very rapidly for nonlinear real physical problems. The results reveal that the method is very effective and simple. This method gives more reliable results as compared to other existing methods available in the literature. The numerical results demonstrate the validity and applicability of the method.

1. Introduction

Nonlinear differential difference equations (NDDEs) play a crucial role in many branches of applied physical sciences such as condensed matter physics, biophysics, atomic chains, molecular crystals, and discretization in solid-state and quantum physics. They also play an important role in numerical simulation of solution dynamics in high-energy physics because of their rich structures. Therefore, researchers have shown a wide interest in studying NDDEs since the original work of Fermi et al. [1] in the 1950s and [2, 3]. Contrary to difference equations that are being fully discretized, NDDEs are semidiscretized, with some (or all) of their space variables being discretized, while time is usually kept continuous. As far as we could verify, little work has been done to search for exact solutions of NDDEs. Hence, it would make sense to do more research on solving NDDEs. The approximate solution involves series of small parameters which poses difficulty since the majority of nonlinear problems have no small parameters at all. Although appropriate choices of small parameters sometime lead to ideal solutions in most cases, unsuitable choices lead to serious effects in the solutions. Therefore, an analytical method is welcome which does not require a small parameter in the equation modeling the phenomenon. The homotopy perturbation method (HPM) was first introduced by He [4]. The HPM was also used by many researchers to investigate various linear and nonlinear equations arising in physics and engineering. Many analytic approximate approaches for solving nonlinear differential equations have been proposed and the most outstanding one is the homotopy analysis method (HAM). In recent years, many authors have paid attention to studying the solutions of nonlinear partial differential equations and nonlinear differential difference equations by various methods [5–25].

The objective of the present paper is to extend the application of the HPTM to obtain an analytic approximate solution of the following nonlinear difference differential equations in mathematical physics:(i)the general lattice equation [1–3]: (ii)the relativistic Toda lattice system [1–3]: where , , and are arbitrary constants. The HPTM provides the solutions in terms of convergent series with easily computable components in a direct way without using linearization, perturbation, or restrictive assumptions. It is worth mentioning that the proposed approach is capable of reducing the volume of computational work compared to classical methods while still maintaining the high accuracy in the numerical result; the size reduction amounts to an improvement of the performance of the approach. The technique is based on the application of the Laplace transform via the homotopy method to solve nonlinear differential difference models. The main advantage of this problem is that we can accelerate the convergence rate, minimize iterative times, accordingly save the computation time, and evaluate the efficiency. Several examples are given to access the reliability of HPTM via optimal parameter.

2. Description of the Homotopy Perturbation Transform Method

We consider a general nonlinear, nonhomogenous partial differential equation where is a linear operator, is a nonlinear operator, and is a source function. The initial conditions are also as

Applying the Laplace transforms on both sides, we get

Using the differentiation property of Laplace transform, we get

Now we embed the homotopy in Laplace transform method. Hence we may write any equation in the form where is a real function of , , and . We construct a homotopy as where is an embedding parameter, is a nonzero auxiliary parameter, is a nonzero auxiliary function, is an auxiliary linear operator, is an initial guess, and is an unknown function. It is important that one has great freedom to choose auxiliary things in homotopy deformation. Obviously, when and , it holds

The embedding parameter increases from to . Using Taylor’s theorem, can be expanded in a power series of as follows: where

If the auxiliary linear operator, the initial guess, the auxiliary parameter , and the auxiliary function are so properly chosen, series (10) converges at ; then we have which must be one of the solutions of the original nonlinear equation. The governing equation can be deduced from zero-order deformation equation (8).

Define the vector

Differentiating (8)  -times with respect to the embedding parameter , then setting , and finally dividing them by , we obtain the th-order deformation equation:

Applying inverse Laplace transform we get where th-order deformation equation (15) is a linear iteration problem and thus can be easily solved, especially by means of symbolic computation software such as Mathematica or Maple.

3. Numerical Results

To demonstrate the effectiveness of the HPTM algorithm discussed above [19–28], several examples of variational problems will be studied in this section.

Example 1. Consider the following nonlinear difference differential equation: with the initial condition with the exact solution where , , and are Jacobian functions and , , , , and are arbitrary constants and
To solve (17) by means of the transform method we consider the following linear equation: with the property that which implies that
Taking Laplace transform of (17) with both sides subject to the initial condition, we get
We now define the nonlinear operator as and then the th-order deformation equation is given by
Taking inverse Laplace transform of (26), we get where
Let us take the initial approximation as the other components are given by other components of the approximate solution can be obtained in the same manner. Therefore, the approximate solution is We calculate the numerical results for different values of ; , , , , and are presented in Figures 1 and 2. The comparison between HPTM via optimal parameter and the exact solution is performed in Figures 3 and 4. A very good agreement is achieved between the results obtained by the presented method HPTM via optimal parameter and the exact solution (Figure 6) for different values of .

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Figure 1 

The 2nd-order approximate solution (31), respectively, when and 1 at .

Figure 2 

The 2nd-order approximate solution (31) when .

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Figure 3 

The behavior of the approximate solution using HPTM via optimal parameter when and 1.

Figure 4 

The behavior of the approximate solution using HPTM via optimal parameter when .

The Jacobi elliptic functions are generated into hyperbolic functions when as follows, and into trigonometric functions when as follows:

Example 2. Let us consider the following problem: with the initial condition with the exact solution where , , , , , and are arbitrary constants. Let with the property that which implies that
Taking Laplace transform of (34) with both sides subject to the initial condition, we get
We now define the nonlinear operator as and then the th-order deformation equation is given by
Taking inverse Laplace transform of (42), we get where where and .
Let us take the initial approximation as the other components (Figure 5) are given by

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Figure 5 

The 4th-order approximate solution (46) and (47), respectively, at .