APPROXIMATE SOLUTION OF MODIFIED CAMASSA–HOLM AND DEGASPERIS–PROCESI EQUATIONS USING WAVELET OPTIMIZED FINITE DIFFERENCE METHOD

Abstract

In this paper, we apply wavelet optimized finite difference method to solve modified Camassa–Holm and modified Degasperis–Procesi equations. The method is based on Daubechies wavelet with finite difference method on an arbitrary grid. The wavelet is used at regular intervals to adaptively select the grid points according to the local behaviour of the solution. The purpose of wavelet-based numerical methods for solving linear or nonlinear partial differential equations is to develop adaptive schemes, in order to achieve accuracy and computational efficiency. Since most of physical and scientific phenomena are modeled by nonlinear partial differential equations, but it is difficult to handle nonlinear partial differential equations analytically. So we need approximate solution to solve these type of partial differential equation. Numerical results are presented for approximating modified Camassa–Holm and modified Degasperis–Procesi equations, which demonstrate the advantages of this method.