The numerical methods for the singularly perturbed delay nonlinear sobolev initial-boundary value problems

In this study, we consider a singularly perturbed nonlinear delay Sobolev initial-boundary value problem by means of a numerical approach. Firstly, the asymptotic estimates for the stability of the solution of the presented problem are derived using the energy inequalities. Subsequently, an exponentially fitted difference scheme is constructed on a uniform mesh by employing interpolating quadrature rules with a basis function and remainder term in integral form. Furthermore, a finite difference scheme is constructed on non-uniform meshes by employing interpolating quadrature rules with a linear basis function. An error analysis is conducted in the discrete norm, and the order of uniform convergence is determined for the proposed exponentially fitted difference scheme. The error analysis is performed for non-uniform meshes, taking into account the node points of the adaptive and piecewise-equidistant mesh, in order to acquire the convergence rates. Finally, the theoretical results are tested on numerical examples with the help of computer codes, and the order of convergence rates is shown in tables. In this regard, the maximum-pointwise errors are considered in order to obtain the orders of convergence. It is observed that the computed results agree with the theoretical findings.