Sine numerical solution for solitons and solitary waves

A numerical scheme using Sine-Galerkin method is developed to approximate the solution for the Korteweg-de Vries model equation. Sine approximation to both derivatives and indefinite integral reduce the integral equation to an explicit system of algebraic equations, then using various properties of Sine functions, it is shown that the Sine solution produce an error of order script O sign(exp(-c/h)) for some positive constants c, h. The method is applied to a few test examples to illustrate the accuracy and the implementation of the method.