Grid method for approximate solution of initial-boundary value problems for generalized convection-diffusion equations

In a rectangular domain, we study an initial-boundary value problems for one-dimensional generalized convection-diffusion equations with the Bessel operator and fractional derivatives in the sense of Riemann–Liouville and Caputo of order $\alpha$ ($0<\alpha<1$) with boundary conditions of the first and third kind. The fractional-order convection-diffusion equation with the Bessel operator arises when passing from the three-dimensional fractional-order convection-diffusion equation to cylindrical (spherical) coordinates, in the case when the solution $u=u(r)$ does not depend on either $z$ or $\varphi$. For the numerical solution of the problems under consideration, monotone difference schemes of the second order of accuracy with respect to the grid parameters are constructed, which approximate these problems on uniform grids. Using the method of energy inequalities for solving initial-boundary value problems, a priori estimates are obtained in differential and difference interpretations under the assumption of the existence of a regular solution to the original differential problem. The obtained a priori estimates imply the uniqueness and stability of the solution with respect to the right-hand side and the initial data, as well as, due to the linearity of the difference problems, the convergence of the solution of the corresponding difference problem to the solution of the original differential problem with the rate $O(h^2 + \tau^2)$.