Stability and convergence of difference schemes for the multi-term time-fractional diffusion equation with generalized memory kernels

In this paper, 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 multi-term Caputo fractional derivative with generalized memory kernels (analog of 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_0) $-th order in time for the generalized multi-term 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.