Якісний аналіз нелокальної задачі щодо одновимірного аналога біпараболічного рівняння з похідними Капуто

• Datum: 2025
• 1. Verfasser: Bulavatsky, Volodymyr
• Format: Artikel
• Sprache: Ukrainian
• Veröffentlicht: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2025
• Schlagworte: нелінійні дробово-диференціальні рівняння; дробово-ди¬ференціальний аналог біпараболічного рівняння; похідна Капуто; якісний аналіз; нелокальні умови; існування розв’язку; єдиність; UH-стійкість
• Online Zugang: https://jais.net.ua/index.php/files/article/view/520
• Назва журналу: Problems of Control and Informatics

A nonlinear problem with a nonlocal condition for a one-dimensional version of the fractional-differential analogue of the biparabolic evolutionary equation is considered. It is noted that classical mathematical models of the dynamics of transport processes, which are based on linear equations of the parabolic type, predict an infinite speed of propagation of disturbances, which leads to a number of well-known paradoxes. In the works of V.I. Fuschich and his students, a gen-eralization of the classical parabolic Fourier equation was proposed and a new (biparabolic) evolutionary equation with fourth-order partial derivatives was introduced. This equation is invariant with respect to the Galileo group and can therefore be used to describe transfer processes independent of in which inertial systems they are observed. Compared to the classical linear parabolic equation, this equation describes evolutionary processes more correctly and allows us to study special regimes, in particular, with a finite perturbation propagation rate. It is worth noting that the biparabolic equation has been repeatedly used to model the dynamics of various evolutionary processes. At present (with the significant development of studies of the dynamics of anomalous transport processes based on the ideas of fractional order integro-differentiation), some fractional differential analogues have been introduced for this equation and a number of model boundary value problems have been solved (in particular, for geofiltration and filtration-consolidation processes). For a one-dimensional version of the fractional-differential analogue of a biparabolic evolutionary equation with derivatives of the Caputo type, this work considers a nonlinear problem with a non-local condition. Some questions of correctness the problem with a nonlocal condition regarding the specified one-dimensional fractional differential equation are studied. The problem solution is reduced to the solution of the corresponding nonlinear integral equation and, on the basis of the classical methodology of the theory of fixed points of nonlinear operators, some conditions for the correctness of this problem are established. In particular, the question of the existence and uniqueness of its solution is highlighted and the conditions of UH-stability are determined.