Cubic Non-Polynomial Spline on Piecewise Mesh for Robin Type Singularly Perturbed Reaction Diffusion Problem

Abstract

This thesis introduces a novel approach to solving Robin type singularly perturbed reaction diffusion problems through cubic non-polynomial spline approximation on a piecewise mesh. The methodology involves discretizing the solution domain using a piecewise mesh size, followed by defining a cubic non-polynomial spline function and obtaining its derivatives. Subsequently, the derivatives in the differential equations are transformed into difference approximations, leading to a linear system of algebraic equations expressed in a three-term recurrence relation, solvable through an elimination algorithm. The stability and consistency of the method are thoroughly investigated to ensure its convergence. Furthermore, numerical model examples are employed to validate the proposed method, with results compared against other methods cited in the literature. The maximum absolute error and order of convergence for each model example are presented to demonstrate the significant advancements offered by the present method.