Solving integral equations of the first kind by decomposition method

Abstract

Purpose

The purpose of this paper is to find methods for solving linear or non‐linear integral equations. The decomposition method may be applied, but a difficulty arises when it is applied to integral equations of the first kind which are not in canonical form. This paper aims to describe some techniques to obtain canonical form of Adomian decomposition method (ADM) for solving integral equations of the first kind.

Design/methodology/approach

Among various numerical methods for solving integral equations, the ADM has shown a good skill for determining the solutions as polynomial series forms. In this paper, some methods based on regularization, derivation or substitution techniques are presented to permit the application of ADM to integral equations of the first kind.

Findings

By using the three proposed methods the decomposition method can then be easily used to solve integral equations of the first kind. The three methods allow to obtain the canonical form and a solution to find using the ADM.

Research limitations/implications

The three methods provide the solution by the application of the decomposition method of George Adomian as extended by other researchers. All are efficient: the derivation and substitution techniques giving interesting results for linear and non‐linear equations of Volterra type; the regularization technique is also efficient for the equations of Fredholm type, but difficulties could arise from the calculations of the required series.

Practical implications

The research provides three efficient methods for solving integral equations of first kind.

Originality/value

The three methods illustrate the powerful use of the decomposition techniques pioneered by Adomian and a result of their applications may be applied for solving of functional equations of any kind. This paper shows some methods for applying the Adomian techniques to integral equations of the first kind.

Citation

Ngarasta, N. (2009), "Solving integral equations of the first kind by decomposition method", Kybernetes, Vol. 38 No. 5, pp. 733-743. https://doi.org/10.1108/03684920910962632