Difference Methods of the Solution of Local and Non-local Boundary Value Problems for Loaded Equation of Thermal Conductivity of Fractional Order

Abstract

We study local and non-local boundary value problems for a one-dimensional space-loaded differential equation of thermal conductivity with variable coefficients with a fractional Caputo derivative, as well as difference schemes approximating these problems on uniform grids. For the solution of local and non-local boundary value problems by the method of energy inequalities, a priori estimates in differential and difference interpretations are obtained, which implies the uniqueness and stability of the solution from the initial data and the right side, as well as the convergence of the solution of the difference problem to the solution of the corresponding differential problem at the rate of \(O(h^2+\tau ^2)\).