A Time-Fractional Diffusion Equation with Generalized Memory Kernel in Differential and Difference Settings with Smooth Solutions

In this paper, an a priori estimate for the corresponding differential problem is obtained by using the method of the energy inequalities. We construct a difference analog of the Caputo fractional derivative with generalized memory kernel (L1 formula). The basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and fourth order in space and the (2 - alpha)-th order in time for the generalized time-fractional diffusion equation with variable coefficients are considered. Stability of the suggested schemes and also their convergence in the grid L-2-norm with the rate equal to the order of the approximation error are proved. The obtained results are supported by numerical calculations carried out for some test problems.