Robust spectral treatment for time-fractional delay partial differential equations

Fractional delay differential equations (FDDEs) and time-fractional delay partial differential equations (TFDPDEs) are the focus of the present research. The FDDEs is converted into a system of algebraic equations utilizing a novel numerical approach based on the spectral Galerkin (SG) technique. The suggested numerical technique is likewise utilized for TFDPDEs. In terms of shifted Jacobi polynomials, suitable trial functions are developed to fulfill the initial-boundary conditions of the main problems. According to the authors, this is the first time utilizing the SG technique to solve TFDPDEs. The approximate solution of five numerical examples is provided and compared with those of other approaches and with the analytic solutions to test the superiority of the proposed method.