Numerical methods with weight for a fractional diffusion equations with functional delay and drift term

A fractional diffusion equation with the presence of a drift and a delay of a general form is considered. For this problem, a family of grid schemes with weights is constructed based on the L1-method for approximating the fractional derivative and applying the piecewise constant interpolation of discrete history. The algorithm is reduced to solving linear systems with a tridiagonal matrix. The order of the residual of the method without interpolation and the order of the residual of the method with interpolation are investigated. The stability conditions of the algorithm are obtained. A theorem is proved that under stability conditions the method has the first order with respect to the time-step and the second order with respect to the space-step.