The general analytical and numerical solution for the modified KdV equation with convergence analysis

Abstract

Communicated by S. GeorgievWe investigate the analytical and numerical solutions of the modified Kortweg de Vries equation by applying the idea of commutative hypercomplex mathematics, He's homotopy perturbation method as a simple particular procedure, and the Runge-Kutta discontinuous Galerkin methods. Moreover, we discuss at great length the convergence conditions for this equation by using the Banach fixed point theory, which could provide a good iteration algorithm. Finally, we compare the homotopy perturbation method with some standard ideas same as the Runge-Kutta discontinuous Galerkin method by some numerical illustrations.where a and b are constants. The mKdV equation plays a crucial role in applied mathematics and physics as a nonlinear convection and alike diffusion problem. It has been used to consider acoustic waves in certain anhurmonic lattices, Alfven waves in collision less plasma, Schottky barriers transmission lines, models of traffic congestion, electric circuits, and many different phenomena. Some scientists worked about the mKdV equation and obtained solutions for this equation [7][8][9][10][11]. Wazwaz used three methods for solving the mKdV equation in [12] and obtained three sets of complex and soliton solutions of this equation. Yan obtained a new binary traveling wave periodic solutions of the mKdV equation that are based on some possible Jacobi elliptic functions in [8]. In [13,14], Geyikli and Kaya applied decomposition method for solving this equation, and also, Yan extended the Adomian decomposition method to the mKdV equation and obtained numerical Jacobi elliptic function solution with a assumed initial condition in [9].During the last years, the Runge-Kutta discontinuous Galerkin methods (RKDG) have become a very popular technique for the solution of the nonlinear convection diffusion problem [15][16][17][18][19]. For a survey on discontinuous Galerkin method, see [20].In this work, there is an emphasis on assessing the convergence analysis for the HPM as a simple scheme rather than the RKDG methods on the aforementioned problem. Hence, we try to obtain the classical solutions by a new idea that is the commutative hypercomplex mathematics [21][22][23]. Therefore, this paper is organized as follows:In Section 2, we introduce hypercomplex mathematics, and in Section 3, we use hypercomplex mathematics to obtain analytical solutions of the mKdV equation. In Section 4, we give a brief idea of how to construct the RKDG. In section 5, we recall the idea of In the aforementioned, we have that˛,ˇ, , ı, z 0 are the arbitrary constants, the function SnOE: : : is the Jacobi elliptic function that is defined in [24], k is a 4D algebraic basis element, and c is a scale factor.