Numerical and analytical investigations for neutral delay fractional damped diffusion-wave equation based on the stabilized interpolating element free Galerkin (IEFG) method

Abstract

A delay PDE is different from a PDE in which it depends not only on the solution at a present stage but also on the solution at some past stage(s). In the current paper, we develop interpolating stabilized EFG method for a neutral delay PDE with fractional derivative in terms of Caputo fractional derivatives. The considered model in this paper is a generalized form of the other equations such that it contains a delay term. Thus, the existence and uniqueness of its analytic solution have been proven by the separation variable technique. At first, the temporal direction is discretized with a finite difference scheme. Then the EFG method has been employed to discrete the spatial direction. The stability of time-discrete scheme is studied by using the energy method. Also, we show the convergence order of the developed technique is O(τ3−α) in the temporal direction and the convergence order in spatial direction is O(rp+1). Finally, two test problems have been considered to demonstrate the capability and efficiency of the proposed numerical technique.