A parameter uniform orthogonal spline collocation method for singularly perturbed semilinear reaction-diffusion problems in one dimension

Abstract

An orthogonal spline collocation method with C¹ splines of degree $\ge 3$ on a Shishkin mesh is formulated and analyzed for the approximate solution of a singularly perturbed semilinear reaction diffusion problem. A convergence analysis yields parameter uniform error estimates of order ${(N^{-1}\hspace{0.17em}\text{log}\hspace{0.17em}N)}^{r+1}$ in an L² discrete norm and of order ${(N^{-1}\hspace{0.17em}\text{log}\hspace{0.17em}N)}^{r+1-m}$ in weighted H^m norms, m = 1, 2, where N is the number of mesh subintervals. Results of numerical experiments support the analytical results, and also exhibit parameter uniform error estimates in other norms for which an analysis has yet to be derived.

Keywords:

• Parameter uniform
• singularly perturbed differential equation
• orthogonal spline collocation
• semilinear; Shishkin mesh

Acknowledgments

The authors thank Prof. Amiya Pani, IIT Bombay, for his valuable comments and suggestions. Support was received by GF from IIT Bombay while a Distinguished Guest Professor at that institution.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This study was funded by the Department of Science and Technology, Government of India, through the National Program on Differential Equations: Theory, Computation and Applications, DST Project No. SERB/F/1279/2011-2012.