A Trigonometric Spline Method for a Singularly Perturbed Parabolic Time-Dependent Partial Differential - Difference Equations Arising in Computational Neuroscience

Abstract We examine a computational technique for a singularly perturbed parabolic partial differential equation with mixed small shifts arguments, whose solution displays parabolic boundary layer behaviour. To derive the scheme, backward Euler approach was used for temporal discretization, and a trigonometric spline method was used for spatial discretization. The Taylor series is utilized to estimate the shift terms, resulting in a singularly perturbed parabolic differential equation with a nearby singularly perturbed parabolic differential equation. The existence and uniqueness of a solution for the proposed technique are investigated. The scheme is proven to be fourth order convergent in space and first order convergent in time. The accuracy of the suggested spline numerical approach is demonstrated by comparing numerical results with those obtained using other approaches. 2010mathematics Subject Classification. 65L10, 65L11, 65L12