Positive solutions of nonlinear fractional boundary value problems using Adomian decomposition method

Abstract

We use Adomian decomposition method for solving the fractional nonlinear two-point boundary value problem

$$\begin{array}{cc}\hfill & D^{\alpha }u(x)+\mu F(x\text{,}u(x))=0\text{,}0<x<1\text{,}1<\alpha ⩽2\text{,}\hfill \\ \hfill & u(0)=0\text{,}u(1)=c\text{,}\hfill \end{array}$$

where D^α is Caputo fractional derivative, c is a constant, μ > 0, and $F\text{:}[0\text{,}1]\times [0\text{,}\infty )\to [0\text{,}\infty )$ a continuous function. The fractional Bratu problem is solved as an illustrative example.

Introduction

Nonlinear boundary value problems (BVP) have been studied in the literature extensively [4], [5], [6], [11], [12], [17], [19]. Several techniques, such as shooting method, finite difference method, Green’s function method, Adomian decomposition method (ADM) have been used to solve nonlinear BVP. In the present paper we study nonlinear fractional order BVP using Adomian decomposition.

In the last two decades, extensive work has been done using ADM, which provides analytical approximation for nonlinear equation without linearization, perturbation or discretization. This method has been implemented in several boundary value problems with all types of boundary conditions. Wazwaz [17], [18] has used Adomian’s method to solve BVP with Dirichlet and Neumann conditions. Deeba et al. [9] have employed Adomian’s method for solving Bratu equation. Bellomo et al. [8] have demonstrated that Adomian method is better than perturbation techniques and minimizes the volume of computational work. Hon [10] has made comparison between Green’s function method and Adomian method. He has shown that it is not always easy to find Green’s function and Green’s function method requires heavy computations compared to Adomian’s method. In the present paper we investigate the following fractional BVP:

$$\begin{array}{cc}\hfill & D^{\alpha }u(x)+\mu F(x\text{,}u(x))=0\text{,}0<x<1\text{,}1<\alpha ⩽2\text{,}\mu >0\text{,}\hfill \\ \hfill & u(0)=0\text{,}u(1)=c\text{,}\hfill \end{array}$$

where D^α is Caputo fractional derivative, c is a constant and $F\text{:}[0\text{,}1]\times [0\text{,}\infty )\to [0\text{,}\infty )$ is a continuous function. Existence and multiplicity results of positive solution of (1.1) have been obtained by means of fixed-point theorems on cone [7]. In the present work we employ ADM [2], [3] to obtain explicit solutions of (1.1).

The paper has been organized as follows: In Section 2 basic definitions are given. Section 3 deals with Adomian decomposition method. Some numerical examples have been presented in Section 4.