The use of Adomian decomposition method for solving the one-dimensional parabolic equation with non-local boundary specifications

Abstract

Over the last 20 years, the Adomian decomposition approach has been applied to obtain formal solutions to a wide class of stochastic and deterministic problems involving algebraic, differential, integro-differential, differential delay, integral and partial differential equations. This method leads to computable, efficient, solutions to linear and nonlinear operator equations. Furthermore in the past, only classical boundary value problems have been considered. The parabolic partial differential equations with non-classical conditions model various physical problems. The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the second-order linear parabolic partial differential equation with nonlocal boundary specifications replacing the standard boundary conditions. This scheme is employed for solving the heat equation in one space dimension with boundary conditions containing integrals over the interior of the interval. The Adomian decomposition method provides a reliable technique that requires less work when compared with the traditional techniques. This method is used by many researchers to investigate several scientific applications. Some experimental results using the newly proposed procedure are given to confirm our belief of the reliability of the approach.

Keywords:

• Adomian decomposition method
• One-dimensional parabolic equation
• Non-local boundary conditions
• Analytic solutions
• Specification of energy
• Closed form solutions