EXISTENCE AND UNIQUENESS OF SOLUTIONS TO NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

Abstract

Our understanding of the fundamental processes of the natural world is based, to a large extent, on di erential equations. Fractional di erential equations (FDEs), that is, di erential equations involving non-integer order derivatives, have gained signi cant attention in almost all branches of science and engineering. It has been shown that new fractionalorder models are often more adequate than previously used integer-order models. The countless applications of FDEs rely both on the mathematical theory and on the precise numerical computation of solutions. The theory of FDEs is a vast and fast-growing topic in the realm of fractional calculus (FC). As the title suggests, this thesis is devoted to the analytical study, supplemented by numerical investigations, of various classes of nonlinear FDEs of order between 0 and 1 and of order between 1 and 2. Even though there are several di erent de nitions of the fractional di erential operators in the literature of FC, we only focus on three well-known fractional derivatives: the Riemann-Liouville derivative, the modi ed Caputo derivative, and the Atangana-Baleanu derivative. These three de nitions are, in general, non-equivalent. The fundamental result for FDEs is an existence and uniqueness theorem. The main objective of this thesis is to study the existence and uniqueness of solutions to the proposed FDEs on nite non-negative intervals. We derive the so-called fundamental theorem of FC for the modi ed Caputo fractional derivative of order 2 (0; 1) in Lp spaces, where 1 p < 1. Our main results rely on the tools of nonlinear functional analysis and some standard xed point theorems, particularly the Banach contraction principle, the Leray- Schauder alternative, and the Krasnosel'ski xed point theorem. From a mathematical perspective, a key insight when studying an FDE using the xed point technique, theequivalence between the FDE and the corresponding integral equation. Due to this fact, the equivalences for all the considered problems are rigorously obtained in appropriate function spaces by applying the mathematical tools of FC (expounded in Chapter 2). The study of initial or boundary value problems for FDEs of order 2 (0; 1) is always tricky but quite interesting due to the complexities presented by the weakly singular Volterra equations. The rst aim of this thesis is to derive the existence and uniqueness results for fractional Cauchy problems involving the modi ed Caputo derivatives in Lp spaces, where 1 p < 1. The novelty of this work is that the nonlinearity f lies in Lp spaces instead of the conventional space of continuous functions. Our main results are achieved by appealing to the Banach contraction principle. We employ the Adams-type predictor-corrector method for the numerical solution. The derived results are illustrated with examples, presenting the corresponding numerical simulations. The second goal is to enhance the previous Cauchy problems by replacing initial conditions with integral conditions and to obtain analogous results, as previously demonstrated in Chapter 3, in a parallel approach. Based on the Banach contraction principle, we establish many new existence and uniqueness theorems, imposing as few weaker assumptions as possible. We provide some examples to illustrate the theoretical results, along with numerical simulations of the implicit solutions. The next goal of this thesis is to study modi ed Caputo FDEs that include the p- Laplacian operator and a weakly singular source term of the form t􀀀 (with 0 < 1), equipped with integral boundary conditions. Our main results, namely the existence and uniqueness of solutions, rely on the Leray-Schauder alternative and the Banach contraction principle. It is also of great interest to discuss the regularity of the solutions. We present several constructive examples to validate the obtained results. The fourth aim of this thesis is to establish the existence and uniqueness results for nonlinear fractional Langevin equations that incorporate the p-Laplacian operator and the singular term t􀀀 (with 0 < 1), equipped with nonlocal conditions. New kinds of nonlocal conditions are introduced. The analysis techniques rely on the Krasnosel'ski xed point theorem and the Banach contraction principle, each serving a di erent purpose. Examples are constructed to illustrate the theory. Our obtained results extend some previous theorems in the literature while incorporating a couple of new ones.The next goal is to develop the mathematical theory of a fully fractional thermostat model involving the Riemann-Liouville fractional derivatives. By constructing a special Banach space and employing xed-point theorems, we obtain some new su cient conditions to ensure the existence and uniqueness of blow-up solutions. Moreover, an implicit numerical scheme based on the right product rectangle rule is presented, providing a numerical approximation of the solution. We give some examples, along with the corresponding numerical solution, to support the theoretical results obtained. The nal aim of this thesis is to study a class of nonlinear FDEs involving the Atangana-Baleanu derivatives with fractional boundary conditions. Fractional boundary conditions in the framework of Atangana-Baleanu derivatives are considered for the rst time. Many important properties of higher-order Atangana-Baleanu operators are introduced. The existence theorem is based on the Leray-Schauder alternative, while the existence and uniqueness theorem relies on the Banach contraction principle. We develop an implicit numerical scheme based on the trapezoidal method to yield an e ective numerical solution. We provide illustrative examples for the theory alongside numerical simulations of the solutions. At the end of the thesis, we present the conclusions of the entire work and propose several potential future research directions. It is noted that all the numerical simulations presented in this thesis are carried out using the \MATLAB" software. most prominent and ubiquitous analytical approach, is the establishment of a precise