Grid Approximation of the Subdiffusion Equation with Variable Order Time Fractional Derivative

Abstract: A grid approximation of the one-dimensional Dirichlet boundary value problem for an equation with a fractional time derivative with a variable order α(x,t) is studied. The existence of a unique solution is proved, and an a priori estimate for the grid solution in the uniform norm is established. An a priori estimate for this problem is used in the study of a grid scheme that approximates a problem with a fractional derivative, the order of which α(u(x,t)) is a function of the desired solution u(x,t). An existence theorem for a solution to the grid problem is proved based on the Schauder theorem. Under the assumption that the derivative of α'(u) is sufficiently small, the uniqueness of the solution and the convergence of the iterative solution method based on the contraction mapping theorem are proved. © 2023, Pleiades Publishing, Ltd.