Numerical analysis for second order differential equation of reaction-diffusion problems in viscoelasticity

This study uses numerical methods to solve a specific type of reaction-diffusion problem arising in viscoelasticity (singularly perturbed Fredholm integro-differential equations). These equations are challenging because they exhibit ‘boundary layers’ near the edges of the area of interest. To approximate solutions, a technique known as a second-order scheme is used for derivatives, and the trapezoidal rule is used for integral terms. This is done on non-standard grids known as Shishkin-type meshes. We found that this numerical method and its rate of improvement (or convergence) are both of the second order, which means they improve consistently as the calculations continue. This improvement rate remains consistent even when dealing with small parameters in the equations. In addition, a post-processing method is used to enhance the rate of convergence from second order to almost fourth order, indicating a significant improvement in the speed and accuracy of the solutions. The practical effectiveness of these methods is confirmed through performance testing of the numerical scheme.