New methods for applying the Adomian method to partial differential equations with boundary conditions

Abstract

Purpose

To find methods for solving non‐linear partial differential equations. The decomposition method may be applied, but a difficulty arises when applied to non‐linear partial differential equations with initial and boundary conditions. In this work, two methods are described that take into account the boundary conditions.

Design/methodology/approach

The decomposition method whilst being a powerful tool for solving non‐linear functional equations encounters difficulties in finding solutions of partial differential equations with boundary conditions. In this paper, two methods are introduced which consist of setting boundary conditions to the equations so that the decomposition methods can be applied.

Findings

By using the two proposed methods the decomposition method can then be easily used. In this work the two methods taking account of the boundary conditions were found to be efficient and allows a solution to be found using the Adomian decomposition method.

Research limitations/implications

The two new methods provide solutions by the application of the decomposition method of George Adomian as extended by other researchers. Both are efficient: the first giving interesting results for linear and non‐linear problems; the second one is also efficient, but difficulties could arise from the calculations of the required series.

Practical implications

The research provides two efficient methods. The first method gives the demonstrated results for linear and non‐linear problems due to the use of symbolic software such as Mathematica or Maple.

Originality/value

Both methods illustrate the powerful use of the decomposition techniques pioneered by Adomian and as a result of their application may be applied to the solving of non‐linear functional equations of any kind. This paper tackles the problems by introducing new methods of applying the Adomian techniques to partial differential equations with boundary conditions.

Citation

Benneouala, T., Cherruault, Y. and Abbaoui, K. (2005), "New methods for applying the Adomian method to partial differential equations with boundary conditions", Kybernetes, Vol. 34 No. 7/8, pp. 924-933. https://doi.org/10.1108/03684920510605740