Grid Equidistribution Method For Numerical Solutions To A Class Of Singularly Perturbed Integro-Differential Equations

Abstract

Numerical solutions to singular perturbation problems are well known to be difficult due to the inherent presence of a small parameter ε, which drastically amplifies errors. These errors are highly dependent on the mesh distribution, which demands extremely fine discretizations in specific regions, leading to a significant computational burden. This study resolved this challenge by presenting numerical schemes for a class of sin- gularly perturbed integro-differential equations that emancipate their solutions from the effect of ε. These schemes demonstrated remarkable accuracy, irrespective of the per- turbation parameter’s value. This work focused on three classes of integro-differential equations: Fredholm, Volterra, and the hybrid Volterra-Fredholm type. Each equation presents a unique challenge, as they are all not only singularly perturbed but also ex- hibit a variety of complexities. These complexities include discontinuities present in the source term for some equations, and the presence of a delay parameter in another. To construct the required parameter uniform numerical schemes, the finite difference method was used for the differential component and the composite trapezoidal rule for the integral portion of the equations. To mitigate the perturbation parameter’s influence, the grid equidistribution method was primarily relied upon, an adaptive mesh technique that strategically positions grid nodes to achieve uniform error distribution within the domain. Additionally, for problems with discontinuities, a novel grid distribution tech- nique was introduced that demonstrated superior performance. Rigorous convergence and stability analysis were employed to establish the theoretical foundation for the effec- tiveness of the proposed schemes. Compelling numerical simulations strongly endorse the accuracy and stability of the proposed schemes. A comparative analysis with exist- ing approaches revealed their distinct advantages in terms of accuracy, as illustrated by compelling numerical examples.