Solving multi-term linear and non-linear diffusion–wave equations of fractional order by Adomian decomposition method

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Abstract

Multi-term linear and non-linear diffusion–wave equations of fractional order are solved using Adomian decomposition method. Some numerical examples are presented.

Introduction

Adomian decomposition method (ADM) introduced by Adomian [1] in 1980, has proved to be a very useful tool while dealing with non-linear functional equations.

In the last two decades, extensive work has been done using ADM as it provides analytical approximate solutions for non-linear equations without linearization, perturbation or discretization. Wazwaz [2], [3], [4], [5] has applied ADM to solve a variety of differential equations. Biazar et al. [6], [7], [8] have applied this method for solving system of ordinary differential equations, integro-differential equations and system of Volterra integral equations. Babolian et al. have used ADM to solve system of non-linear equations [9], Volterra/Fredholm integral equations [10], [11], fuzzy differential equations [12]. Kaya and El-Sayed [13], [14], [15] have solved Klein–Gordon equations, KdV equation, coupled Schrödinger–KdV equation using ADM. Shawagfeh [16] has employed ADM for solving non-linear fractional differential equations. Daftardar-Gejji and Jafari [17], [18] have used this method for solving systems of fractional differential equations (linear and non-linear). Further they have utilized this method for obtaining positive solutions of non-linear fractional boundary value problems [19] and solving linear/non-linear fractional diffusion and wave equation [20]. This list is by no means complete and is presented here to give some idea of the wide range of applicability of ADM. In the present paper, we employ ADM to solve multi-term linear/non-linear fractional diffusion–wave equationP(D)u(x¯,t)=i=1nNi2uxi2+ϕ(x¯,t)um(x¯,t),whereP(D)Dts1-j=2rλjDtsj,r2,rN,0<sr<sr-1<<s2<s1<2,m=0,1,2,,Ni(x¯,t)Cα. Dtsi are Caputo fractional derivatives.

The paper has been organized as follows. Notations and preliminaries are given in Section 2. In Section 3 multi-term fractional diffusion–wave equation is studied. Some numerical examples are presented in Section 4.

Section snippets

Preliminaries and notations

In this section, we set up notation and present basic definitions from fractional calculus [21], [22], [23].

Definition 2.1

A real function f(x), x>0 is said to be in space Cα,αR if there exists a real number p(>α), such that f(x)=xpf1(x) where f1(x)C[0,).

Definition 2.2

A function f(x), x>0 is said to be in space Cαm,mN{0} if f(m)Cα.

Definition 2.3

Let fCα and α-1, then the expressionDt-μf(t,x)=1Γ(μ)0t(t-τ)μ-1f(τ,x)dτ,t>0,μ>0,is called as the (left-sided) Riemann–Liouville integral of order μ.

Definition 2.4

The (left-sided) Caputo fractional

Fractional multi-term diffusion–wave equation

Consider the boundary value problemP(D)u(x¯,t)=i=1nNi2uxi2+ϕ(x¯,t)um(x¯,t),u(x¯,0)=f(x¯),ut(x¯,0)=g(x¯),whereP(D)Dts1-j=2rλjDtsj,0<sr<sr-1<<sr-k<1<sr-(k+1)<<s2<s1<2.

Applying Dt-s1 on left hand side of (3.1) we get [21], [22]Dt-s1[P(D)u(x¯,t)]=Dt-s1Dts1u(x¯,t)-j=2rλjDt-(s1-sj+sj)Dtsju(x¯,t)=(u(x¯,t)-u(x¯,0)-t·ut(x¯,0))-j=2r-(k+1)λjDt-(s1-sj)Dt-sjDtsju(x¯,t)-j=r-krλjDt-(s1-sj)Dt-sjDtsju(x¯,t)=(u(x¯,t)-f(x¯)-t·g(x¯))-j=2r-(k+1)λjDt-(s1-sj)(u(x¯,t)-f(x¯)-t·g(x¯))-j=r-krλjDt-(s1-sj)(u(x¯,

Illustrative examples

To demonstrate the method we consider here some fractional diffusion and wave equations

(1) Consider the two-term fractional diffusion equation(Dts1-Dts2)U=-i=132Uxi2,-<xi<,t>0,U(x¯,0)=e-(x1+x2+x3),0<s2<s1<1.

This is equivalent to the integral equationU(x¯,t)=e-(x1+x2+x3)1-ts1-s2Γ(s1-s2+1)+Dt-(s1-s2)U-Dt-s1i=132Uxi2.

The Adomian decomposition leads tou0=e-(x1+x2+x3)1-ts1-s2Γ(s1-s2+1),uj+1=Dt-(s1-s2)uj-Dt-s1i=132ujxi2,j=0,1,.

Thus uj=e-(x1+x2+x3)ηj(t),j=0,1,2,, whereηj(t)=k=0j(-3)kjktj(

Acknowledgements

The authors are grateful to U.G.C. New Delhi, India for the funding through Major Research Project (F. No. 31-82/2005(SR)). Varsha Daftardar-Gejji acknowledges University of Pune for providing the research grant. Mathematica has been used for calculations in the present paper.

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