A novel approach via mixed Crank–Nicolson scheme and differential quadrature method for numerical solutions of solitons of mKdV equation

Abstract

The purpose of the present study is to obtain numerical solutions of the modified Korteweg–de Vries equation (mKdV) by using mixed Crank–Nicolson scheme and differential quadrature method based on quintic B-spline basis functions. In order to control the effectiveness and accuracy of the present approximation, five well-known test problems, namely, single soliton, interaction of double solitons, interaction of triple solitons, Maxwellian initial condition and tanh initial condition, are used. Furthermore, the error norms $$L_{2}$$ and $$ L _{\infty }$$ are calculated for single soliton solutions to measure the efficiency and the accuracy of the present method. At the same time, the three lowest conservation quantities are calculated and also used to test the efficiency of the method. In addition to these test tools, relative changes of the invariants are calculated and presented. After all these processes, the newly obtained numerical results are compared with results of some of the published articles.