Qualitative analysis for nonlocal and fractional models

Collaborative Research Program 2023-2025

The basic idea of this project is to do qualitative analysis for general nonlocal and fractional models as well as to study their consequences and applications. Thus, the main purpose is to construct new methods for qualitative analysis for general nonlocal and fractional models. Our unifying methods can be readily used to recover most of the previously known results as well as to construct other new techniques which will be useful to study/understand new nonlocal and fractional models in different fields.
Here we can only say about the main results that will be obtained in the framework of the project. Naturally, we will talk about the results in general terms without specific wording of conditions and approvals.
We will obtain:
• [Objective 1] Explicit solutions of generalized (linear) space-time fractional models with variable coefficients, which will be extensively and efficiently applied for analytic and computational goals.
• [Objective 2] Algorithms for inverse problems of finding coefficients or/and sources for concrete fractional models, which involve the explicit representation of the solution of the direct problems and a recent method to recover variable coefficients and sources for the considered inverse problems.
• [Objective 3] Applications/interpretations of generalized space-time fractional models in real-world phenomena from physics, biology, and engineering. We will develop efficient numerical schemes which will inherit the real-world properties as the original models and investigate numerical properties (stability, uniqueness, and convergence) as well as convergent to the exact solution(s).
• [Objective 4] New function inequalities for nonlocal operators, which are crucial tools to obtain existence/stability and non-existence/blow-up results for space-time fractional models.
Research on the topic is mainly theoretical and fundamental, their scientific novelty is due to the use of deep, modern results of function inequalities for nonlocal operators, fractional differential equations and fractional numerical analysis. It allows to discover new applicable results for nonlocal and fractional models in different fields. Thus, along with applications it contributes significantly to the development of the theory of nonlocal and fractional models. All applicants have strong and complementary skills in the theory of fractional and nonlocal operators, applied mathematics/engineering, and numerical methods needed to address these problems. Note that our research accomplishments in the field of function inequalities have been publicized in worldwide publications such as the European Mathematical Society's Newsletters, the American Mathematical Society's Notices, the-steppe.com, physics.org, and YouTube, among others.
The project's most significant contribution to Kazakhstani society and science will be the training of at least three graduate students who will earn their Ph.D. and MS degrees while working on it. They will become Kazakhstan's future generation of academicians by performing sophisticated research and publishing in prestigious journals. It indeed meets with NU Research Strategy Thrusts Alignment I. Socio-economic Transformation and Human Capital Development.
The project's another significant contribution to Kazakhstani society and science will be the publication of important discoveries in leading international scientific journals, which helps to improve Kazakhstan's reputation in the scientific community. The impact of the expected results will be in the development of fundamental science in Kazakhstan. It completely meets with Kazakhstani Research Priorities Alignment: Research priorities approved by the Government, Higher Scientific and Technical Commission #10 Research in natural sciences.