Numerical solutions of KDV and mKDV equations: Using sequence and multi-core parallelization implementation

Abstract

In this study, we use the explicit difference method and also the combination of the explicit difference method with the implicit method to numerically solve the third order of Korteweg–de Vries (KDV) and modified Korteweg–de Vries (mKDV) equations. This method is applied to obtain soliton solutions on several problems and the results obtained are compared with each other and the exact solution of the problem. Also, due to that these problems are very ill-posed, when the time increases, the waveform of solitons and the input parameters of these methods are carefully examined. The L∞, L2 and mean-square errors of the solutions show that these methods can be applied to different problems and have very good accuracy. These methods are widely used in combination with other algorithms, but their very high execution time, especially in this category of problems, is always a big limitation. In this paper, a parallel approach for implementing these methods is presented and very good results have been obtained compared to sequential implementation. This is a reference to further study solutions of other models of KDV and mKDV problems as well as solving their inverse problems.