The discrete homotopy perturbation Sumudu transform method for solving partial difference equations

In this paper, we introduce a combined form of the discrete Sumudu transform method with the discrete homotopy perturbation method to solve linear and nonlinear partial difference equations. This method is called the discrete homotopy perturbation Sumudu transform method(DHPSTM). The results reveal that the introduced method is very efficient, simple and can be applied to other linear and nonlinear difference equations. The nonlinear terms can be easily handled by use of He's polynomials.

1. Introduction. Partial difference equations are types of difference equations that involve functions of two or more independent variables. Such equations arise in applications involving population dynamics with spatial migrations, chemical reactions, the approximation of solutions of partial differential equations by finite difference methods, random walk problems, the study of molecular orbits, dynamical systems, economics, biology and other fields.
The homotopy perturbation method was first proposed by He in 1998 [19]. Recently, the discrete homotopy perturbation method(DHPM) was used to obtain the numerical solution of Burgers' equation and heat equation [33].
In this paper, we use discrete homotopy perturbation Sumudu transform method (DHPSTM) including discrete STM and DHPM in order to find solution of linear and nonlinear partial difference equations. Although the HPSTM has been widely applied to solve partial differential equations, to the best of our knowledge until 616 FIGENÖZPINAR AND FETHI BIN MUHAMMAD BELGACEM now, the HPSTM was regarded only for the continuous equations. There is no discrete version HPSTM ever used to solve linear or nonlinear PDEs. This paper proposed the DHPSTM, and finds the DHPSTM has similar advantages of continuous HPSTM. To illustrate the method we apply the DHPSTM to partial difference equations.

Preliminaries and notations.
2.1. Discrete Homotopy Perturbation Method(DHPM). Consider the following general partial difference equation where A is general difference operator. f m,n is a given discretized function. U m,n and f m,n denote discrete approximation of U (x, t) and f (x, t) at the mesh point (mh, nτ ), respectively. h is the space step in x direction, and τ represents the increment in time. In this work we will take h = τ = 1.
The operator A can be divided into two parts, which are L and N , where L is a linear and N is non-linear operator. Therefore Eq(1) can be rewritten as follows: By constructing the homotopy technique to Eq(2), we have a homotopy in the form where p ∈ [0, 1]is an embedding parameter, U m0,n is an initial approximate solution of the original equation, which satisfies the boundary conditions. From Eq(3) we have H(υ m,n (0), 0) =L(υ m,n (0)) − L(U m0,n ) = 0 H(υ m,n (1), 1) =A(υ m,n (1)) − f m,n = 0 changing the process of p from zero to unity is just change of υ m,n (p) from U m0,n to U m,n .

Definition 2.1 ([23]
). If f : N 0 → C is a function, then the discrete Sumudu transform is defined by for all values of u = −1 such that the series converges.
Below there are some properties of the discrete Sumudu transform: and in general

Discrete Homotopy Perturbation Sumudu Transform Method (DHPSTM).
To illustrate this method we consider the partial difference equation of the form with subject to initial condition where φ m,n and f m are given discrete functions. g m,n is the source term. The forward partial differences ∆ m and ∆ n are defined as usual, i.e., ∆ m U m,n = U m+1,n − U m,n and ∆ n U m,n = U m,n+1 − U m,n . Second order partial difference is defined by ∆ 2 m U m,n = ∆ m (∆ m U m,n ) and ∆ 0 m U m,n = U m,n , ∆ 0 n U m,n = U m,n . By applying discrete Sumudu transform on both sides of Eq(7) with respect to n, we get Taking the inverse discrete Sumudu transform to Eq(9) we have where G m,n represents the term arising from the source term and the prescribed initial conditions. According to DHPM, we construct a homotopy in the as following Substituting (12) into (11) and comparing the coefficients of the term with identical powers of p, lead to When the limit get for p → 1, the solutions obtain as following: 3. Applications of DHPSTM. In this section, we shall examine some applications of our newly developed method through the following examples. Example 1. We consider the following partial difference equation with the initial condition U m,0 = 5 m .