Trigonometric B-spline based ε-uniform scheme for singularly perturbed problems with Robin boundary conditions

Abstract

In this paper, a non-polynomial-based trigonometric cubic B-spline collocation method is developed to solve the reaction-diffusion singularly perturbed problems with Robin boundary conditions. These problems are more tedious to solve than those with Dirichlet and Neumann boundary conditions. The parameter ε in the differential equation results in a rapid change in the solution over a small region. A piecewise uniform mesh is constructed to handle this difficulty. Also, a modification of the proposed mesh is suggested to improve the accuracy of the numerical results by introducing a change in the transition parameter. Through rigorous analysis, it has been shown that the method is almost second-order uniformly convergent. The performance and theoretical findings of the proposed scheme are validated through numerical experiments presented for two test problems. The accuracy of the method is measured in the discrete maximum norm. The tabular results demonstrate that the newly added mesh produces better results.