Numerical solution of KdV equation using modified Bernstein polynomials

摘要

Here we present an algorithm for approximating numerical solution of Korteweg–de Vries (KdV) equation in a modified B-polynomial basis. A set of continuous polynomials over the spatial domain is used to expand the desired solution requiring discretization with only the time variable. Galerkin method is used to determine the expansion coefficients to construct initial trial functions. For the time variable, the system of equations is solved using fourth-order Runge–Kutta method. The accuracy of the solutions is dependent on the size of the B-polynomial basis set. We have presented our numerical result with an exact analytical result. Excellent agreement is found between exact and approximate solutions. This procedure has a potential to be used in more complex system of differential equations where no exact solution is available.