Approximate solutions for boundary value problems of time-fractional wave equation

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Abstract

In this article, we implement relatively a new analytical technique, the Adomian decomposition method, for solving the boundary value problems of time-fractional wave equation. The fractional derivative is described in the Caputo sense. The decomposition method is used to construct analytical approximate solutions of time-fractional wave equation subject to specified boundary conditions. The solutions are calculated in the form of a convergent series with easily computable components. Some examples are given. The results reveal that the Adomian method is very effective and convenient.

Introduction

Fractional differential equations have been the focus of many studies due to their frequent appearance in various fields such as physics, chemistry and engineering [10], [12], [14], [20]. Several methods have been introduced to solve fractional differential equations, the popular Laplace transform method [12], [20], the Fourier transform method [11], the iteration method [10] and the operational method [25]. However, most of these methods are suitable for special types of fractional differential equations, mainly the linear with constant coefficients.

The time-fractional wave equation, which is a mathematical model of a wide range of important physical phenomena, is a partial differential equation obtained from the classical wave equation by replacing the second time derivative by a fractional derivative of order α, 1 < α  2. It has been the subject of many papers by Nigmatullin [11], Mainardi [6], Wyss [22], Schneider and Wyss [21], Fujita [24], El-Sayed [1], Gorenflo and Mainardi [18] and Hanyga [2].

Wyss [22] and Schneider and Wyss [21] considered the time-fractional diffusion and wave equations and obtained the solution in terms of Fox functions. Gorenflo et al. [19] used the similarity method and the Laplace transform method to obtain the scale-invariant solution of the time-fractional diffusion-wave equation in terms of the Wright function. Agrawal [16] presented a general solution for a fractional diffusion-wave equation defined on a bounded space. Gorenflo and Mainardi [18] considered random walk models for space fractional diffusion processes. The space–time fractional-wave equation has also treated by Mainardi et al. [7] as a Cauchy problem, and its fundamental solution was investigated in terms of Green’s function.

The Adomian decomposition method for solving differential and integral equations, linear and non linear, has been the subject of extensive analytical and numerical studies. The method, well addressed in [3], [4], [5], [8], [9], has a significant advantage in that it provides the solution in a rapid convergent series with easily computable components. The method provides a solution without linearization, perturbation, or unjustified assumptions. In recent years, a large amount of literatures developed concerning the Adomian decomposition method by applying it to a large size of applications in applied sciences.

The object of present paper is to extend the application of the decomposition method to drive analytical approximate solutions for boundary value problem of the generalized wave equationDtαu(x,t)=Lxu(x,t)+h(x),0<t<a,-<x<with the boundary conditionsu(x,0)=f(x),u(x,a)=g(x),where Dtα is the Caputo fractional derivative of order α, 1 < α  2 with respect to time variable t and Lx is the linear differential operatorLx=f1(x)2x2+f2(x)x+f3(x).

Section snippets

Definitions

For the concept of fractional derivative we will adopt Caputo’s definition which is a modification of the Riemann–Liouville definition and has the advantage of dealing properly with initial value problems in which the initial conditions are given in terms of the field variables and their integer order which is the case in most physical processes.

Definition 1

A function f(t) (t > 0) is said to be in the space Cα(α  R) if it can be written as f(t) = tpf1(t) for some p > α where f1(t) is continuous in [0, ∞), and it

Analysis of the method

The principles of the Adomian decomposition method and its applicability for various kinds of differential equations are well known, see [5], [8], [9], [15]. We consider the boundary value problem of the time-fractional wave equation:Dtαu(x,t)=Lxu(x,t)+h(x),0<t<a,-<x<with the boundary conditionsu(x,0)=f(x),u(x,a)=g(x),where Lx is the linear differential operator (1.3), Dtα is the Caputo fractional derivative of order α, 1 < α  2 with respect to time variable t and g(x) has the Taylor series

Numerical examples

In this section we present numerical examples to give a clear overview of the effectiveness of the method and to demonstrate the behavior of the solution of the time-fractional wave equation (3.1).

Example 1

Consider the boundary value problemDtαu(x,t)=2x2u(x,t),0<t<1,1<α2,u(x,0)=sinx,u(x,1)=sinx.In view of (3.13) and using the Taylor series expansion of sin x, the exact solution of (4.1) is given byu(x,t)=n=0(-1)ntnαΓ(nα+1)(1-t)sinx+n=0(-1)ntnα+1x2n+1(2n+1)!,=(1-t)Eα(-tα)sinx+t1-α/2sin(tα/2x),

Conclusions

In this paper, the decomposition method was implemented to solve boundary value problems of time-fractional wave equation. It may be concluded that the decomposition method is very powerful efficient technique in finding exact and approximate solutions for ordinary and partial differential equations of fractional order. Although the method is well suited to solve the time-fractional wave equation in terms of a rapid convergent series with easily computable components, the method could lead to a

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