Analysis of Time-Fractional Delay Partial Differential Equations Using a Local Radial Basis Function Method

Delay partial differential equations have significant applications in numerous fields, such as population dynamics, control systems, neuroscience, and epidemiology, where they are required to efficiently model the effects of past states on current system behavior. This work presents an RBF-based localized meshless method for the numerical solution of delay partial differential equations. In the suggested numerical scheme, the localized meshless method is combined with the Laplace transform. The main attractive features of the localized meshless method are its simplicity, adaptability, and ease of implementation for complex problems defined on complex shaped domains. In a localized meshless scheme, a linear system of equations is solved. The Laplace transform, which is one of the most powerful techniques for solving integer- and non-integer-order problems, is used to represent the desired solution as a contour integral in the complex plane, known as the Bromwich integral. However, the analytic inversion of contour integral becomes very laborious in many situations. Therefore, a contour integration method is utilized to numerically approximate the Bromwich integral. The aim of utilizing the Laplace transform is to handle the costly convolution integral associated with the Caputo derivative and to avoid the effects of time-stepping techniques on the stability and accuracy of the numerical solution. We also discuss the convergence and stability of the suggested scheme. Furthermore, the existence and uniqueness of the solution for the considered model are studied. The efficiency, efficacy, and accuracy of the proposed numerical scheme have been demonstrated through numerical experiments on various problems.