A reliable approach to the Korteweg‐de Vries equation.: An application of the decomposition method

Abstract

Purpose

This is another application of the Adomian decomposition method (ADM). It is used to implement the linear homogeneous and the non‐homogeneous Korteweg‐de Vries equations (KdV).

Design/methodology/approach

The analytical solution of the equation is calculated in the form of a series with easily computable components. The design of the study is to form the decomposition series solutions of the linear homogeneous problem. This is quickly obtained by observing the existence of the self‐cancelling “noise” terms where the sum of components vanishes to the limit. The convergence criterion is then considered and examples included.

Findings

It was found that the ADM is a very powerful and efficient method for finding analytical solutions for wide classes of problems. This was particularly evident in comparison with the traditional methods where massive calculations are usually used.

Research limitations/implications

This research study in addition to illustrating the power of applying ADM also showed its advantages in providing a fast convergence of the solution which may be achieved by observing the self‐cancelling “noise” terms.

Practical implications

The convergence of the ADM applied to KdV equation has been proved. Many test modelling problems from mathematical physics, linear and non‐linear, have been presented and illustrate the effectiveness and the performance of the methodology.

Originality/value

Provides a reliable and new approach to studies of the KdV equation and illustrates through its examples the use of ADM.

Citation

Inc, M. and Cherruault, Y. (2005), "A reliable approach to the Korteweg‐de Vries equation.: An application of the decomposition method", Kybernetes, Vol. 34 No. 7/8, pp. 951-959. https://doi.org/10.1108/03684920510605777