Numerical solutions for fractional KdV–Burgers equation by Adomian decomposition method

Abstract

Based on the symbolic computation system Maple, Adomian decomposition method, developed for differential equations of integer order, is directly extended to derive explicit and numerical solutions of the fractional KdV–Burgers equation. The fractional derivatives are described in the Caputo sense. According to my knowledge this paper represents the first available numerical solutions of the nonlinear fractional KdV–Burgers equation with time- and space-fractional derivatives. Finally, the solutions of our model equation are calculated in the form of convergent series with easily computable components.

Introduction

In the past decades, both mathematicians and physicists have devoted considerable effort to the study of explicit and numerical solutions to nonlinear evolution equation of integer order. Many powerful methods have been presented [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Among them, the Adomian decomposition method [11], [12], [13], [14], [15] (ADM) provides an effective procedure for explicit and numerical solutions of a wide and general class of differential systems representing real physical problems. Large classes of differential equations of integer order, both linear and nonlinear, ordinary as well as partial, can be solved by the ADM. Moreover no linearization or perturbation is required in the method.

In recent years, considerable interest in fractional differential equations has been stimulated due to their numerous applications in the areas of physics and engineering [16]. Many important phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry and material science are well described by differential equations of fractional order [17], [18], [19]. The solution of differential equations of fractional order is much involved. In general, there exists no method that yields an exact solution for nonlinear fractional differential equations. Only approximate solutions can be derived using linearization or perturbation method.

The aim of this paper is to directly extend the ADM to consider the numerical solution of the nonlinear fractional KdV–Burgers equation with time- and space-fractional derivatives of the form

$$\frac{{\mathrm{\partial }}^{\alpha }u}{\mathrm{\partial }{t}^{\alpha }}+\epsilon u\frac{{\mathrm{\partial }}^{\beta }u}{\mathrm{\partial }{x}^{\beta }}+\eta \frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^2}+\vartheta \frac{{\mathrm{\partial }}^{3}u}{\mathrm{\partial }{x}^3}=0\text{,}t>0\text{,}0<\alpha \text{,}\beta ⩽1\text{,}$$

where ε, η and ϑ are parameters and α and β are parameters describing the order of the fractional time- and space-derivatives, respectively. The function u(x, t) is assumed to be a causal function of time and space, i.e. vanishing for t < 0 and x < 0. The fractional derivatives are considered in the Caputo sense [20]. The general response expression contains a parameter describing the order of the fractional derivative that can be varied to obtain various responses. In the case of α = 1 and β = 1, the fractional differential equation reduces to the classical nonlinear KdV–Burgers equation. More important, above procedure is just an algebraic algorithm and can be easily applied in the symbolic computation system Maple.

Although there are a lot of studies for the classical nonlinear KdV–Burgers equation and some profound results have been established, it seems that detailed studies of the nonlinear fractional differential equation are only beginning. According to my knowledge this paper represents the first available numerical solution of the nonlinear fractional KdV–Burgers equation with time- and space-fractional derivatives.

The paper is organized as follows: in Section 2, a brief review of the theory of fractional calculus will be given to fix notation and provide a convenient reference. They are no mean complete, but are sufficient for the purpose of this paper. In Section 3, we extend the application of the ADM to construct numerical solutions for the nonlinear fractional KdV–Burgers equation. In Section 4, we present two examples to show the efficiency and simplicity of the proposed method. Finally, conclusions are presented.