Accurate Computational Approach on Piecewise Mesh for Singularly Perturbed Differential Equations with Mixed Shifts

Abstract

This thesis presents an accurate computational approach utilizing piecewise mesh for solving singularly perturbed differential equations with mixed shifts. The proposed method begins by discretizing the solution domain using a defined piecewise mesh size. The singularly perturbed differential problems are first transformed into asymptotically equivalent singularly perturbed boundary value problems. Subsequently, the finite difference approximation technique is employed to convert the derivatives in the differential equations into a linear system of algebraic equations, represented as a three-term recurrence relation that can be effectively solved using the Thomas algorithm. The stability and consistency of the method are rigorously investigated, ensuring the convergence of the proposed approach. Additionally, the influence of delay and advance parameters on the solution profile is explored. To validate the method's applicability, various examples are solved for different perturbation parameters and mesh sizes. The numerical results obtained are compared with existing findings in the literature, demonstrating that the proposed method significantly enhances accuracy. Furthermore, graphical representations illustrate the solution's behavior concerning varying shifts and boundary layer dynamics.