Numerical algorithms based on splines for singularly perturbed parabolic partial differential equations with mixed shifts

In this paper, we discuss singularly perturbed time-dependent convection-diffusion problems that arise in computational neuroscience. Specifically, we provide approaches for one-dimensional singularly perturbed parabolic partial differential difference equations (SPPPDDEs) with mixed shifts in the spatial variable using fitted operator spline in compression and adaptive spline. Temporal discretization is done by backward Euler's method, and spline methods with exponential fitting on uniform mesh are implemented in the spatial domain. For better approximations, the Richardson extrapolation technique is used, which is demonstrated by two numerical examples. The convergence of the proposed methods is investigated and found to be uniform with respect to the perturbation parameter. Graphical representations are provided to show how the shifts affect the proposed solution to the problem.