A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations

doi:10.1016/S0378-4754(01)00438-4

Mathematics and Computers in Simulation

Volume 60, Issue 6, 15 October 2002, Pages 507-512

https://doi.org/10.1016/S0378-4754(01)00438-4 Get rights and content

Abstract

In this study, the decomposition method for solving the linear heat equation and nonlinear Burgers equation is implemented with appropriate initial conditions. The application of the method demonstrated that the partial solution in the x-direction requires more computational work when compared with the partial solution developed in the t-direction but the numerical solution in the x-direction are performed extremely well in terms of accuracy and efficiency.

Introduction

In this paper, we consider the Burgers equation $u_{t} +ϵuu_{x} −νu_{xx} =0,$ where ϵ and ν are parameters and the subscripts t and x denote differentiation. The Burgers’ equation is a model of flow through a shock wave in a viscous fluid [1] and in the Burgers’ model of turbulence [2]. It is one of a few nonlinear partial differential equations which can be solved exactly for an appropriate initial conditions [3]. However, in [4], the authors are considered and made some comments concerning the singularities in solutions of Burgers’ equation. In their study some arguments are put forward in favor of the suggestion that nonsingular initial conditions cannot produce the singularity in the process of time evolution in real time, in particular the initial condition t=0.

The partial solutions were investigated in [5], [6]. In [5], the authors have shown that these solutions can be obtain in an identical form when the boundary conditions are in a specific type, and asymptotically equal when the boundary conditions are in other type. Recent work by Wazwaz [6] has extended the work [5] and made further progress and improvements beyond the results obtained. The author proved that partial solutions are identical for all types of boundary conditions. In this paper, we will consider and solve this problem in the x-direction and in the t-direction explicit and numerical solutions by using the Adomian decomposition method (in short ADM) [7], [8]. Before considering the Burgers’ equation we will consider the heat equation. This can be obtained from the Burgers’ equation by either rewriting Eq. (1) for ϵ=0 or using a Hopf–Cole transformation as a test problem by implementing the ADM for the exact solution and approximate solution of a linear heat equation and nonlinear Burgers’ equation. The nonlinear problem are solved easily and elegantly without linearizing the problem by using the ADM. The technique has many advantages over the classical techniques, mainly, it avoids linearization and perturbation in order to find solutions of given nonlinear equations. It also avoids discretization and provides an efficient numerical solution with high accuracy, minimal calculation, avoidance of physically unrealistic assumptions.

Section snippets

Analysis of the method

In the preceding section we have discussed particular devices of the general type of the Burgers equation. For the purposes of illustration of the ADM, in this study we shall consider Eq. (1) in an operator form $L_{t} (u(x,t))+ϵuu_{x} −νL_{x} (u(x,t))=0,$ with the initial conditions.

Following [7], [8] we define for the above Eq. (1) the linear operators $L ̷_{t} =∂/∂t$ , $L ̷_{x} =∂^{2} /∂x^{2}$ and the definite integration inverse operators $L ̷^{−1}_{t}$ and $L ̷^{−1}_{x}$ . Therefore, the solutions in the t- and x-directions can be written as

Application and results

Let us consider, for simplicity, the Burgers Eq. (1) with ϵ=0 and ν=1, linear one-dimensional heat conditional parabolic partial differential equation, namely, $u_{t} −u_{xx} =0, u(x,0)= sin (πx), u(0,t)=0, u_{x} (0,t)=π exp (−π^{2} t).$ For the solution of this equation in the t-direction, we simply take the equation in a operator form exactly in the same manner as the form of the Eq. (3) and using the specified initial condition u(x,0) in Eq. (8) to find the zeroth component of u₀ and obtained in succession u₁, u₂, u₃

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Short communicationA numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations

Abstract

Introduction

Section snippets

Analysis of the method

Application and results

Phys. Lett. A

Comp. Math. Appl.

J. Math. Anal. Appl.

Math. Comp. Model.

Appl. Math. Lett.

Math. Comp. Model.

Short communication
A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations