Numerical solutions of Burgers' equation in a B-polynomial basis

A new algorithm for approximating numerical solutions of Burgers' equation in a modified B-polynomial basis is presented. The desired solutions are expanded in terms of a set of continuous polynomials over the entire interval for the spatial part with only the time variable requiring discretization. The algorithm makes use of the Galerkin method to determine the expansion coefficients to construct initial values for expansion coefficients. Matrix formulation is used to construct system of equations in time variable which are solved using a fourth-order Runge–Kutta method. The accuracy of the solutions is dependent on the size of the B-polynomial basis set. The current algorithm is implemented in solving two cases of homogeneous and non-homogeneous boundary conditions. Excellent agreement is found between exact and approximate solutions. This procedure has great potential to be implemented in more complex systems of differential equations where no exact solution is available.