A numerical solution of Burgers’ equation by modified Adomian 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’ equationwhere 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 conditions
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 conditionand homogeneous boundary conditionsThe exact solution of (1) with conditions (3), (4) is obtained aswhere ais are Fourier coefficients and
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 by
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.
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