On new approximations of Caputo–Prabhakar fractional derivative and their application to reaction–diffusion problems with variable coefficients

Abstract

This article is devoted to constructing and analyzing two new approximations (CPL2-1 (Formula presented.) and CPL-2 formulas) for the Caputo–Prabhakar fractional derivative. The error bounds for the CPL2-1 (Formula presented.) and CPL-2 formulas are proved to be of order (Formula presented.) and (Formula presented.), respectively, where (Formula presented.) is the order of time-fractional derivative. The newly developed approximations are then used in the numerical treatment of a reaction–diffusion problem with variable coefficients defined in the Caputo–Prabhakar sense. Moreover, the space variable in the developed numerical schemes, CFD1 and CFD2, is discretized using a fourth-order compact difference operator. Both schemes' stability and convergence analysis are demonstrated thoroughly using the discrete energy method. It is shown that the convergence orders of CFD1 and CFD2 schemes are (Formula presented.) and (Formula presented.), respectively, where (Formula presented.) and (Formula presented.) represent the mesh spacing in time and space directions, respectively. In addition, numerical results are obtained for three test problems to confirm the theory and demonstrate the efficiency and superiority of the proposed schemes. © 2023 John Wiley & Sons, Ltd.