STUDY ON THE INITIAL BOUNDARY VALUE PROBLEM FOR A FRACTIONAL DIFFERENTIAL EQUATION WITH A FRACTIONAL DERIVATIVE OF VARIABLE ORDER

Alimbekova N.B. Bakishev A.K. Berdyshev A.S.
20 September 2024 al-Farabi Kazakh State National University

KazNU Bulletin. Mathematics, Mechanics, Computer Science Series
2024 #123 Issue 3 121 - 134 pp.

This article studies the convergence of a numerical method for solving an initial-boundary value problem of a fractional differential equation with a variable order of the fractional derivative. In the generalized fractional differential filtration equation with a transitional filtration law in heterogeneous porous media, it is assumed that the order of the fractional derivative depends on the spatial variable. The main attention is paid to the development and theoretical justification of a method that provides high accuracy and efficiency of calculations with a variable order of the fractional derivative. For the numerical solution, an approximation was developed that combines the finite difference method for the time derivative and the finite element method for the spatial variable. The fractional derivative of variable order in the sense of Caputo is approximated by a formula of second order in time. The convergence of the constructed method is proven with order O⁽τ² + h^k+1) for the case α (x) ∈ (0, 1). The results of computational experiments for various functions of the order of the fractional derivative are presented, confirming the reliability of the theoretical analysis. The conclusions drawn emphasize the importance and relevance of the further development of numerical methods for fractional differential equations of variable order in modern mathematics and applied sciences, including the modeling of complex processes.

Filtration problem , Fractional derivative , Fractional differential problem , Heterogeneous medium , Variable order of fractional derivative

Text of the article Перейти на текст статьи

S. Amanzholov East Kazakhstan University, Department of Mathematics, Ust-Kamenogorsk, Kazakhstan
Abai Kazakh National Pedagogical University, Department of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

S. Amanzholov East Kazakhstan University
Abai Kazakh National Pedagogical University

10 лет помогаем публиковать статьи Международный издатель

Книга Публикация научной статьи Волощук 2026 Book Publication of a scientific article 2026