Устойчивость и сходимость разностной схемы, аппроксимирующей нелокальную краевую задачу для нагруженного уравнения влагопереноса с производными дробного порядка

Abstract

We study a boundary value problem for a loaded moisture transfer equation with two Gerasimov - Caputo fractional derivatives of different orders (α, β), variable coefficients, and a nonlocal integral boundary condition. On a uniform grid a difference scheme of order of approximation 𝑂(ℎ2 + 𝜏2) for 𝛼 = 𝛽 and 𝑂(ℎ2 + 𝜏2−𝑚𝑎𝑥⁡{𝛼,𝛽}) for 𝛼 ≠ 𝛽 is constructed. The study is carried out by the method of energy inequalities. For different ratios between the orders of fractional derivatives α and β, for solving the posed nonlocal boundary value problem, a priori estimates in the difference interpretation are obtained, from which the uniqueness of the solution of the posed problem, the continuous and uniform dependence of the solution on the input data, and the conv ergence of the solution of the difference problem to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme.