Numerical methods for solving nonlocal boundary value problems for generalized loaded Hallaire equations

The work is devoted to nonlocal boundary value problems for one-dimensional space-loaded Hallaire equations with variable coefficients and two Caputo fractional differentiation operators with orders $\alpha$ and $\beta$. Similar problems arise in the practice of regulating the salt regime of soils, when the stratification of the upper layer is achieved by draining the water layer from the surface of a site flooded for some time. Difference schemes are constructed for the numerical solution of the problems posed on a uniform grid. Using the method of energy inequalities for various relations between the orders of the fractional Caputo derivative $\alpha$ and $\beta$, we obtain a priori estimates in differential and difference interpretations for solutions of nonlocal boundary value problems. The obtained a priori estimates imply uniqueness and stability of the solution with respect to the right-hand side and the initial data, as well as the convergence of the solution of the difference problem to the solution of the corresponding original differential problem (assuming the existence of a solution to the differential problem in the class of sufficiently smooth functions) at a rate of $O(h^2+ \tau^2)$ for $\alpha=\beta$ and $O(h^2+\tau^{2-\max\{\alpha,\beta\}})$ for $\alpha\neq\beta $. The paper also presents an algorithm for the numerical solution of a nonlocal boundary value problem for a loaded Hallaire equation with variable coefficients and a Bessel operator.