A parameter-uniform method for singularly perturbed turning point problems exhibiting interior or twin boundary layers

ABSTRACT

A numerical scheme for a class of two-point singularly perturbed boundary value problems with an interior turning point having an interior layer or twin boundary layers is proposed. The solution of this type of problem exhibits a transition region between rapid oscillations and the exponential behaviour. The problem with interior turning point represents a one-dimensional version of stationary convection–diffusion problems with a dominant convective term and a speed field that changes its sign in the catch basin. To solve these problems numerically, we consider a scheme which comprises quintic B-spline collocation method on an appropriate piecewise-uniform mesh, which is dense in the neighbourhood of the interior/boundary layer(s). The method is shown to be parameter-uniform with respect to the singular perturbation parameter ϵ. Some relevant numerical examples are illustrated to verify the theoretical aspects computationally. The results compared with other existing methods show that the proposed method provides more accurate solutions.

Keywords:

• Singular perturbation problems
• interior layers
• boundary layers
• turning points
• fitted-mesh
• collocation method

2010 AMS Subject Classifications:

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

Acknowledgements

The author would like to express thanks to the anonyms referees for their valuable comments and suggestions which have improved the presentation of this work.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Devendra Kumar http://orcid.org/0000-0002-4832-8481