Period of project implementation: 01.07.2024-31.12.2026
Project relevance: The main idea is that to model processes described by hyperbolic equations, take into account the influence of memory. That is, to introduce into consideration not only the current state of the process, but also its previous states, the so-called memory. Next, such a problem is studied for correctness. A computational model for its numerical solution is being developed.
Project purpose: To create a new numerical method that takes into account not only the current state of the process, but also its history, which will help us obtain more accurate and reliable results when solving a mixed problem for a hyperbolic equation with fractional differentiation Caputo memory.
Project objectives: To achieve the project goal, it is necessary to solve the following important tasks, which are interconnected and form an integral research plan:
1) Conduct an extensive analysis of existing works and research related to the formulation and study of a mixed problem for a hyperbolic equation with memory in the form of fractional differentiation in the sense of Caputo. This will provide us with an understanding of the current state of the field and help us identify our unique contribution to the project.
2) Creation of a numerical method for solving a mixed problem of a hyperbolic equation with memory in the form of fractional differentiation in the sense of Caputo. Development of a robust memory-aware difference scheme.
3) Formulation of the concept of stability of a numerical solution of a difference scheme, taking into account the memory of the system. This is a task that requires specific considerations and analysis of the influence of memory on the stability of numerical results.
4) Formulate and prove a theorem on the stability and convergence of a numerical solution to a difference problem taking into account memory. This stage represents the most difficult part of the project and requires in-depth mathematical analysis.
5) Finally, conducting practical computational experiments using the developed numerical method and testing it on real data. This will help demonstrate the applicability and effectiveness of the developed methods. This project plan represents a systematic and focused approach to achieving a given goal, including both theoretical research and practical calculations.
Research group composition:
1) Abdiramanov Zhanars Alda-ongarovich – PhD, project manager, chief researcher, Scopus Author ID: 57219802387; ORCID: 0000-0003-3820-7253
2) Berdyshev Abdumauvlen Suleimanovich – d.phy-math sci., professor, scientific consultant, leading researcher, Scopus Author ID: 8916505200; ORCID: 0000-0002-1228-8246
Results // Expected results According to the requirements of the competition documentation, at least 2 (two) articles will be published in journals from the first three quartiles by impact factor in the Web of Science database or having a CiteScore percentile in the Scopus database of at least 50.
