Difference schemes for the equation of heat conduction with a fractional derivative in boundary conditions

In this work the author presented the boundary value problem for a heat conduction equation with a fractional derivative in boundary conditions. The equation of parabolic type with variable factors and a fractional derivative on time in boundary conditions (the concept of a fractional derivative of Riemann - Liouville is used at $0<\alpha<1$) is considered. For this problem the a priori estimation is obtained from which the solution stability on input data and uniqueness follows. The discrete analogue of a problem is constructed, the approximation error is investigated, and also the stability and convergence of the difference scheme are proved.