Numerical solutions of Korteweg de Vries and Korteweg de Vries-Burger's equations using computer programming

Abstract

In this paper, numerical and solitonic solutions of Korteweg de Vries(KdV) and Korteweg de Vries-Burger's (KdVB) equations with initial and boundary conditions are calculated by sinc-collocation method. The basis of method is sinc functions. First, discretizing time derivative of KdV and KdVB's equations using a classic finite difference formula and space derivatives by θ- weighted scheme between successive two time lev- els is applied, then Sinc functions are used to solve these two equations. Mathematica programming is used to solve matrix representation of these equations. KdV equation describes behavior of traveling waves which is a third order non-linear partial differential equation (PDE). Maximum absolute errors are given in Tables. The figures show approximate solutions of these two equations. Three conservation laws for KdV's equation are obtained.