A Study on Time-Dependent Two Parameter Singularly Perturbed Problems via Trigonometric Quintic B-Splines on an Exponentially Graded Mesh

This paper presents a fresh perspective on solving two-parameter singularly perturbed parabolic convection-diffusion-reaction equations with Dirichlet boundary conditions. The methodology integrates the Crank–Nicolson (CN) scheme for discretizing temporal derivatives and applies the Trigonometric Quintic B-splines (TQBS) approach to approximate both the state variable and its spatial derivatives on an exponentially graded mesh. Through a meticulous convergence analysis, the study establishes a parameter-uniform convergence of fourth order in space and second order in time. To verify the theoretical claims and evaluate the method's efficacy, four test examples are solved using the numerical algorithm, offering tangible evidence of the parameter-uniform convergence of the proposed numerical scheme. Additionally, the paper includes a graphical comparison of the proposed method with Shishkin meshes, providing empirical evidence of its efficacy and parameter-uniform convergence.