Stability and convergence of difference schemes approximating the first boundary value problem for integral-differential parabolic equations in a multidimensional domain

Integral-differential parabolic equations are studied in a multidimensional domain with boundary conditions of the first kind. For each problem, a difference scheme is constructed with the order of approximation $O(|h|^2+\tau^{m_\sigma})$, where $m_\sigma = 1$ if $\sigma\neq0.5$ and $m_ \sigma = 2$, if $\sigma=0.5$, an a priori estimate is obtained by the method of energy inequalities for solving the difference problem. The obtained estimates imply the uniqueness and stability of the solution with respect to the right-hand side and initial data, as well as the convergence of the solution of the difference problem to the solution of the corresponding original differential problem at a rate of $O(|h|^2+\tau^2)$ for $\sigma = 0.5$.