Higher Order Computational Approach for Generalized Time-Fractional Diffusion Equation

The present article is devoted to developing new finite difference schemes with a higher order of the convergence for the generalized time-fractional diffusion equations (GTFDEs) that are characterized by a weight function w(t). Three different discrete analogs with different orders of approximations are designed for the generalized Caputo derivative. The major contribution of this paper is the development of an L2 type difference scheme that results in the \((3-\alpha )\) $(3-\alpha )$ order of convergence in time. The spatial direction is discretized using a second-order difference operator. Fundamental properties of the coefficients of the L2 difference operator are examined and proved theoretically. The stability and convergence analysis of the developed L2 scheme are established theoretically using the energy method. An efficient algorithm is developed and implemented on numerical test problems to prove the numerical accuracy of the scheme.