High-order schemes and their error analysis for generalized variable coefficients fractional reaction–diffusion equations

Abstract

In this manuscript, we develop and analyze two high-order schemes, CFD (Figure presented.) and PQS (Figure presented.), for generalized variable coefficients fractional reaction–diffusion equations. The generalized fractional derivative is characterized by a weight function and a scale function. We approximate it using generalized Alikhanov formula ((Figure presented.)) of order (Figure presented.), where (Figure presented.) (Figure presented.) denotes the order of the generalized fractional derivative. Moreover, for spatial discretization, we use a compact operator in CFD (Figure presented.) scheme and parametric quintic splines in PQS (Figure presented.) scheme. The stability and convergence analysis of both schemes are demonstrated thoroughly using the discrete energy method in the (Figure presented.) -norm. It is shown that the convergence orders of the CFD (Figure presented.) and PQS (Figure presented.) schemes are (Figure presented.) and (Figure presented.), respectively, where (Figure presented.) and (Figure presented.) represent the mesh spacing in the time direction and (Figure presented.) is the mesh spacing in the space direction. In addition, numerical results are obtained for three test problems to validate the theory and demonstrate the efficiency and superiority of the proposed schemes.