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

• Published in: Lobachevskii journal of mathematics Vol. 44; no. 1; pp. 387 - 393
• Main Author: Lapin, A.
• Format: Journal Article
• Language: English
• Published: Moscow Pleiades Publishing 2023; Springer Nature B.V
• Subjects: Algebra; Analysis; Approximation; Boundary value problems; Dirichlet problem; Existence theorems; Geometry; Iterative solution; Mathematical analysis; Mathematical Logic and Foundations; Mathematics; Mathematics and Statistics; Probability Theory and Stochastic Processes; a priori estimates; variable order fractional time derivative; solvability; finite difference scheme; subdiffusion equation
• ISSN: 1995-0802; 1818-9962
• DOI: 10.1134/S1995080223010286

A grid approximation of the one-dimensional Dirichlet boundary value problem for an equation with a fractional time derivative with a variable order 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 is a function of the desired solution . An existence theorem for a solution to the grid problem is proved based on the Schauder theorem. Under the assumption that the derivative of is sufficiently small, the uniqueness of the solution and the convergence of the iterative solution method based on the contraction mapping theorem are proved.