Graded mesh B-spline collocation method for two parameters singularly perturbed boundary value problems

Abstract

The solutions of two parameters singularly perturbed boundary value problems typically exhibit two boundary layers. Because of the presence of these layers standard numerical methods fail to give accurate approximations. This paper introduces a numerical treatment of a class of two parameters singularly perturbed boundary value problems whose solution exhibits boundary layer phenomena. A graded mesh is considered to resolve the boundary layers and collocation method with cubic B-splines on the graded mesh is proposed and analyzed. The proposed method leads to a tri-diagonal linear system of equations. The stability and parameters uniform convergence of the present method are examined. To verify the theoretical estimates and efficiency of the method several known test problems in the literature are considered. Comparisons to some existing results are made to show the better efficiency of the proposed method. Summing up: • The present method is found to be stable and parameters uniform convergent and the numerical results support the theoretical findings. • Experimental results show that the present method approximates the solution very well and has a rate of convergence of order two in the maximum norm. • Experimental results show that cubic B-spline collocation method on graded mesh is more efficient than cubic B-spline collocation method on Shishkin mesh and some other existing methods in the literature.

Keywords

• Cubic B-spline collocation method