On the Solutions of Nonlinear Higher‐Order Boundary Value Problems by Using Differential Transformation Method and Adomian Decomposition Method

Che Haziqah Che Hussin,Adem Kiliçman

doi:10.1155/2011/724927

Abstract

We study higher‐order boundary value problems (HOBVP) for higher‐order nonlinear differential equation. We make comparison among differential transformation method (DTM), Adomian decomposition method (ADM), and exact solutions. We provide several examples in order to compare our results. We extend and prove a theorem for nonlinear differential equations by using the DTM. The numerical examples show that the DTM is a good method compared to the ADM since it is effective, uses less time in computation, easy to implement and achieve high accuracy. In addition, DTM has many advantages compared to ADM since the calculation of Adomian polynomial is tedious. From the numerical results, DTM is suitable to apply for nonlinear problems.

Highlights

Many researchers use ADM to approximate numerical solutions
We study higher-order boundary value problems HOBVP for higher-order nonlinear differential equation
In 17, Ayaz investigated initial value problem of partial differential equation PDE to solve two-dimensional differential transformation method, and we compare the results with Adomian decomposition method

Summary

Introduction

In 1 , Wazwaz proposed a modification of ADM method in series solution to accelerate its rapid convergence, and, in 2 , Wazwaz presented several numerical examples of higher-order boundary value problems for first-order linear equation and second-order nonlinear equation by applying modified decomposition method. In 17 , Ayaz investigated initial value problem of partial differential equation PDE to solve two-dimensional differential transformation method, and we compare the results with Adomian decomposition method. Erturk and Momani in 19 presented numerical solution by comparing the differential transformation method DTM and Adomian decomposition method ADM for solving linear and nonlinear fourth-order boundary value problems and proved that DTM is very accurate and efficient in numerical solution. We prove that DTM is more powerful technique than ADM and can be applied to nonlinear problems

Differential Transformation Method

General Differential Transformation

Adomian Decomposition Method

14 Then we get the series solution as follows:

Cx5 120

Conclusion