Numerical solution of time-dependent two-parameter singularly perturbed problems via trigonometric quintic B-spline collocation technique

Abstract

Abstract This paper introduces a novel algorithm for solving time-dependent two-parameter singularly perturbed parabolic convection-diffusion-reaction equations with Dirichlet boundary conditions. The algorithm is formulated using the Crank-Nicolson (CN) scheme for the temporal derivative discretization. Then, the Trigonometric Quintic B-spline (TQBS) is applied to approximate the state variable and its spatial derivatives on nonuniform collocation points. We conducted a comprehensive convergence analysis and stability of the proposed method and proved that the scheme achieved a parameter uniform convergence of approximately fourth order in space and second order in time. To make additional evidence to support the theoretical findings and further assess the proposed method, we implemented the numerical algorithm to solve three test examples. Furthermore, using these test examples, we demonstrated the parameter-uniform convergence of the proposed numerical scheme.