Numerical Approximation and Error Estimation of a Time Fractional Order Diffusion Equation

Finite element method is used to approximately solve a class of linear time-invariant, time-fractional-order diffusion equation formulated by the non-classical Fick law and a “long-tail” power kernel. In our derivation, “long-tail” power kernel relates the matter flux vector to the concentration gradient while the power-law relates the mean-squared displacement to the Gauss white noise. This work contributes a numerical analysis of a fully discrete numerical approximation using the space Galerkin finite element method and the approximation property of the Caputo time fractional derivative of an efficient fractional finite difference scheme. Both approximate schemes and error estimates are presented in details. Numerical examples are included to validate the theoretical predictions for various values of order of fractional derivatives.