An application for a modified KdV equation by the decomposition method and finite element method

A numerical solution to a Modified Korteweg-de Vries (MKdV) equation is obtained using a “lumped” Galerkin method with quadratic B-spline finite element method (FEM) and using the Adomian's Decomposition Method (ADM). Test problems concerning the motion and interaction of soliton solutions are used to compare the FEM with the ADM. Classical problems concerning the development, motion and interaction of solitons are used to validate the methods. The present methods extremely well in terms of accuracy, efficiency, simplicity, stability and reliability.