A parameter uniform B-spline collocation method for solving singularly perturbed turning point problem having twin boundary layers

Abstract

A numerical scheme is proposed to solve singularly perturbed two-point boundary value problems with a turning point exhibiting twin boundary layers. The scheme comprises B-spline collocation method on a non-uniform mesh of Shishkin type. Asymptotic bounds are established for the derivative of the analytical solution of a turning point problem. The present method is boundary layer resolving as well as second-order accurate in the maximum norm. A brief analysis has been carried out to prove the uniform convergence with respect to the singular perturbation parameter ϵ by decomposing the solution into smooth and singular components. Some relevant numerical examples are also illustrated to verify computationally the theoretical aspects.

Keywords:

• singular perturbation problem
• turning points
• B-spline collocation
• Shishkin mesh
• boundary layer
• uniform convergence

2000 AMS Subject Classifications :

• 65L10
• 65L20
• 65L50
• 65L70

Acknowledgements

Authors are thankful to the referees for their valuable suggestions and careful corrections in the presentation.