Orthogonal collocation on finite elements using Quadratic and Cubic B-Splines.

Abstract

We discuss the application of orthogonal collocation on finite elements (OCFE) method using quadratic and cubic B-spline basis functions to ordinary, partial and fractional differential equations. Collocation is performed at Gaussian points to obtain optimal solutions, hence the name orthogonal collocation. The OCFE method based on quadratic spline is used to solve Burgers’ equation, modified Burgers’ equation and the nonlinear Schrödinger equation in non-uniform subintervals with different cases of soliton solutions. It is also extended to obtain numerical solutions of fractional differential equations taking the fractional diffusion equation as a case study. In addition, we apply orthogonal collocation based on modified quadratic B-spline functions to solve time-dependent and time independent two-dimensional partial differential equations. The numerical results are in good agreement with previous ones in the literature. Furthermore, KdV-Burgers’ equation that does not have an exact solution and the one with exact solution, Burgers’ equation, KdV equation and fractional differential equations are considered as test cases for the OCFE method using cubic B-spline basis functions. The results compare favourably with exact solutions and existing results in the literature. Generally, the numerical schemes are proven to be convergent, unconditionally stable and utilizes minimal memory storage due to the sparse matrix systems associated with B-spline basis functions.