New Modified Adomian Decomposition Recursion Schemes for Solving Certain Types of Nonlinear Fractional Two-Point Boundary Value Problems

Sekson Sirisubtawee,Supaporn Kaewta

doi:10.1155/2017/5742965

Abstract

We apply new modified recursion schemes obtained by the Adomian decomposition method (ADM) to analytically solve specific types of two-point boundary value problems for nonlinear fractional order ordinary and partial differential equations. The new modified recursion schemes, which sometimes utilize the technique of Duan’s convergence parameter, are derived using the Duan-Rach modified ADM. The Duan-Rach modified ADM employs all of the given boundary conditions to compute the remaining unknown constants of integration, which are then embedded in the integral solution form before constructing recursion schemes for the solution components. New modified recursion schemes obtained by the method are generated in order to analytically solve nonlinear fractional order boundary value problems with a variety of two-point boundary conditions such as Robin and separated boundary conditions. Some numerical examples of such problems are demonstrated graphically. In addition, the maximal errors (MEn) or the error remainder functions (ERn(x)) of each problem are calculated.

Highlights

During the last three decades, fractional order differential equations (FDEs) have played an important role in modelling many phenomena in engineering [1, 2], applied sciences [3,4,5,6], and biological systems [7, 8]
The Adomian decomposition method (ADM) has been extensively utilized to solve initial value problems (IVPs) and boundary value problems (BVPs) for nonlinear ordinary or partial differential equations, integral equations, or integrodifferential equations since it can provide approximate analytic solutions without linearization, perturbation, discretization, guessing the initial term, or using Green’s functions which are quite difficult to determine in most cases
Modifications of the ADM have been developed for different purposes for IVPs for both integer order and fractional order differential equations

Summary

Introduction

During the last three decades, fractional order differential equations (FDEs) have played an important role in modelling many phenomena in engineering [1, 2], applied sciences [3,4,5,6], and biological systems [7, 8]. These examples include a Bratu-type fractional BVP, an oscillating base temperature equation, and an elastic beam problem

Mathematical Preliminaries

Numerical Examples

Conclusions