Search

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.

This paper presents a numerical solution for a time delay parabolic problem (reaction-diffusion) containing a small parameter. The numerical method combines the implicit Crank–Nicolson scheme for the time derivative on the uniform mesh and the central difference scheme for the spatial derivative on the Shishkin type meshes. It is shown to be second-order uniformly convergent in time and space. Then Richardson extrapolation technique is applied to enhance the accuracy from second-order to fourth-order. The error analysis is carried out, and the method is proved to be uniformly convergent. These two methods are applied to two test examples in support of the theoretical results.