Fitted cubic spline scheme for two-parameter singularly perturbed time-delay parabolic problems

Abstract

In this work, we investigated a numerical solution for a two-parameter singularly perturbed time-delayed problems. Due to the presence of small parameters, the solution of these problems exhibits twin boundary layers in the neighborhood of the end of the spatial domain, depending on the size of the parameters. In the development of the scheme, we have used the θ-method in the discretization of the temporal variable on a uniform mesh, whereas in the spatial variable discretization cubic spline scheme is applied by introducing a suitable fitting factor on a uniform mesh. Stability and error analysis of the proposed scheme is performed. It is proved that the developed scheme is uniformly convergent. Computational experiments have been performed, verifying that the results are in agreement with the theoretical findings. Moreover, proposed scheme provides a more accurate solution as compared to other methods available in the literature.

Keywords

• Singular perturbation
• Two-parameter
• Cubic-spline
• θ−method
• Time-delay