A second order difference scheme for time fractional diffusion equation with generalized memory kernel

In the current work we build a difference analog of the Caputo fractional derivative with generalized memory kernel ($_\lambda$L2-1$_\sigma$ formula). The fundamental features of this difference operator are studied and on its ground some difference schemes generating approximations of the second order in time for the generalized time-fractional diffusion equation with variable coefficients are worked out. We have proved stability and convergence of the given schemes in the grid $L_2$ - norm with the rate equal to the order of the approximation error. The achieved results are supported by the numerical computations performed for some test problems.