A numerical solution of Burgers’ equation by modified Adomian method

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Abstract

In this paper, the modified Adomian’s decomposition method for calculating a numerical solution of the one-dimensional quasi-linear, the Burgers’ equation, is presented. Time discretization has been used in decomposition, without using any transformation in the Burgers’ equation such as Hopf–Cole transformation. The numerical results obtained by this way for various values of viscosity have been compared with the exact solution to show the efficiency of the method.

Introduction

Recently, the Adomian’s decomposition method is emerging as an alternate method for solving a wide class of physically significant problems modeled by nonlinear partial differential equations. By Adomian’s method, the original nonlinear equation is directly solvable and does not require linearization. The Adomian’s method is very reliable and effective scheme that provides the solution in terms of rapid convergent series [1], [2], [3].

A well-known model is the one-dimensional Burgers’ equationut+uux=νuxx,a<x<b,t>0,where ν > 0 is the coefficient of the kinematics viscosity of the fluid and the subscripts x and t denote differentiation. This equation was intended as an approach to the study of turbulence, shock waves and continuous stochastic processes [4], [5]. The Eq. (1) involves nonlinearity, dissipation and is relatively simple. We consider (1) with the following initial and boundary conditionsu(x,0)=f(x),a<x<b,u(a,t)=g1(t),u(b,t)=g2(t),t>0.

In order to solve (1) numerically, Evans and Abdullah [7] alternating group explicit methods, Varoğlu and Finn [12] used a new finite element method based on a weighted residual formulation. Öziş and Özdeş [10] used a direct variational method to generate an approximation solution in the form of a sequence solution. Kutluay et al. [8] proposed the exact-explicit finite difference method to obtain numerical solution of adequate accuracy. Recently, Öziş et al. [9] used Hopf–Cole transformation in a finite element scheme.

In this paper, the Burgers’ equation was solved directly by the modified Adomian’s decomposition method which was constructed on the method of time discretization without using any transformation like as Hopf–Cole, which was used in [9].

Let us consider the Burgers’ equation (1) with the initial conditionu(x,0)=sin(πx),0<x<1,and homogeneous boundary conditionsu(0,t)=u(1,t)=0,t>0.The exact solution of (1) with conditions (3), (4) is obtained asu(x,t)=2πνn=1ane-n2π2νtnsin(nπx)a0+n=1ane-n2π2νtcos(nπx),where ais are Fourier coefficients anda0=01exp{-(2πν)-1[1-cos(πx)]}dx,an=201exp{-(2πν)-1[1-cos(πx)]}cos(nπx)dx,n1.

Section snippets

Analysis of the method

In method of time discretization, it is possible to convert the problem (1) with conditions (2), (3) to the system of ordinary differential equations with corresponding boundary conditions involves P equations. The interval [0, T], T is total and maximum time, is partitioned into P subintervals of equal length Δt such that 0 = t0 < t1 <  < tP = T and Δt = tj  tj−1 = (T  t0)/P for j = 1, 2,  , P. Hence, the derivative ut is replaced by the difference quotient (zj(x)  zj−1(x))/Δt [11]. The method of time

Numerical results

The obtained results for various values of ν were compared with the exact solution in Table 1, Table 2 and were shown graphically in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5. Moreover to show the efficiency of the method for our problem in comparison with the exact solution, E2 (norm of two) has been used which is defined byE2=i=0100[u(xi,T)-UP,M(xi)]21/2.

As seen in Table 1, Table 2, the solutions are in good agreement with each other. Moreover, when Δt is decreased, it was observed that E2 is

Conclusions

In this paper, we have proposed an efficient modification of the standard Adomian’s decomposition method, with high convergence and small errors, even for small values of viscosity, without using Hopf–Cole transformation.

References (12)

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