Trigonometric quintic B-spline collocation method for singularly perturbed turning point boundary value problems

Abstract

A trigonometric quintic B-spline method is proposed for the solution of a class of turning point singularly perturbed boundary value problems (SP-BVPs) whose solution exhibits either twin boundary layers near both endpoints of the interval of consideration or an interior layer near the turning point. To resolve the boundary/interior layer(s) trigonometric quintic B-spline basis functions are used with a piecewise-uniform mesh generated with the help of a transition parameter that separates the layer and regular regions. The proposed method reduces the problem into a system of algebraic equations which can be written in matrix form with the penta-diagonal coefficient matrix. The well-known fast penta-diagonal system solver algorithm is used to solve the system. The method is shown almost fourth-order convergent irrespective of the size of the diffusion parameter ϵ. The theoretical error bounds are verified by taking some relevant test examples computationally.

Keywords:

• Boundary layers
• interior layers
• parameter-uniform convergence
• Shishkin mesh
• trigonometric quintic B-splines

AMS subject classifications (2000 MSC):

• 65L10
• 65L11
• 65L20
• 65L50
• 65L70

Acknowledgments

The authors are indebted to the anonymous reviewers for their helpful, valuable comments and suggestions in the improvement of this manuscript. The first author is thankful to Council of Scientific and Industrial Research, Govt of India for providing JRF (award letter No. 09/466(0193)/2018-EMR-I).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author is thankful to Council of Scientific and Industrial Research, India for providing JRF (award letter No. 09/466(0193)/2018-EMR-I).