Stability and convergence analysis of an exact finite difference scheme for Fredholm-Volterra integro-differential equations

Singularly perturbed integro-differential equations frequently arise in the mathematical modelling of physical phenomena such as boundary layer theory, plasma transport processes and quantum systems. This work considers a boundary value problem for a second-order linear singularly perturbed Fredholm-Volterra integro-differential equation (SPVFIDE). An exact finite difference method is proposed to overcome stability difficulties associated with small perturbation parameters. The stability and ϵ-uniform convergence of the scheme are analysed in detail. Numerical experiments confirm the theoretical results and demonstrate first-order uniform convergence, showing that the method efficiently captures perturbation effects.