Mathematical analysis and efficient finite element approximation for variable‐order time‐fractional reaction–diffusion equation with nonsingular kernel

Abstract

The fractional derivative with nonsingular kernel has been widely used to many physical fields which was shown to offer a new insight into the mathematical modeling of natural phenomena. In this paper, we first study the well-posedness and regularity of the multidimensional variable-order time-fractional reaction-diffusion equation with Caputo-Fabrizio fractional derivative and then present and analyze a Galerkin finite element approximation to the proposed model based on the proved smoothing properties of the solutions. To improve the computational efficiency of the variable-order Caputo-Fabrizio derivative, we approximate the kernel of the fractional derivative by the K-term truncation of its Taylor expansion at a fixed fractional order to develop a fast evaluation scheme which significantly reduces the memory requirement from (N) to (K) and the computational complexity from (N 2 ) to (K 2 N) where N refers to the number of time steps. We accordingly develop a fast Galerkin finite element method to the proposed model. Numerical experiments are carried out to substantiate the theoretical findings and to show the performance of the fast method. KEYWORDSfast method, fractional derivatives and integrals, reaction-diffusion equations, variable-order Caputo-Fabrizio fractional derivative, well-posedness and regularity MSC CLASSIFICATION35R11; 26A33