Contributions to the Study of Qualitative Properties of Time and Space Fractional Differential Equations

Abstract

In this thesis, we focus on the study of the qualitative behavior of dynamical systems governed by fractional derivatives in time and/or space. We analyze various evolution equations exhibiting different dynamics. We begin with a linear system with time-fractional derivatives that models the diabetes in the United Arab Emirates (UAE); we establish the well-posedness in the sense of Hadamard; and we present numerical simulations based on UAE data to compare our result with the classical models, which demonstrates the better suitability of fractional derivative compared to the classical derivative. Next, we investigate a nonlinear system with fractional in time derivatives which involve power-type nonlinearities. We establish sufficient conditions for blowup, asymptotic growth near blowing up time, and graphical profiles of solutions. To study the role of fractional Laplacian in dynamical systems, we study a regional optimal control problem on spatio-temporal epidemiological model involving Fractional Laplacian. We prove global existence of positive solutions, necessary optimality condition, and establish optimal control strategies minimizing infections while optimizing costs. The last model that we consider is a non-autonomous nonlinear Volterra equation involving both fractional derivative in time and fractional Laplacian. We prove global existence, positivity, and the large-time behavior of global solutions. More precisely, we show global attractivity of the constant positive solution. Finally, in the last chapter, we explore the tools for stability analysis of fractional-in-time semi-linear parabolic equations with the Neumann boundary conditions. We establish the corresponding stability results that apply in the abstract semi-linear parabolic equations with classical derivative to the fractional in time derivative regime. The results include the linearization principle together with their various applications; including instability of non-constant stationary solutions in one dimensional and Turing instability criteria.

Date of Award	2025
Original language	American English
Supervisor	Mokhtar Kirane (Supervisor)