Date of Award

22-5-2024

Document Type

Thesis

School

School of Arts, Sciences, Humanities & Education

Programme

Ph.D.-Doctoral of Philosophy

First Advisor

Dr.S.Raja Balachandar

Keywords

Deformable Fractional Derivative, Fixed Point Theorems, Approximate Controllability, Finite Difference Approximations, Time Fractional Diffusion Equations

Abstract

Fractional calculus and fractional differential equations are considered to be the valuable tools in modeling many phenomena in various fields of science and engineering. In the literature, many definitions for fractional order derivatives, such as Riemann-Liouville, Caputo, Jumarie, Hadamard, Weyl, and more, were developed to study the fractional differential equations that govern various phenomena in science and engineering. But these definitions have their own limitations, such as derivatives of constants, product of two functions, quotient of two functions, assertion laws, and limiting values of the derivatives at zero and negative numbers.

To overcome the deficiencies, researchers have recently introduced some new derivatives, namely conformable and deformable fractional derivatives. In particular, a deformable derivative has a special property called the intrinsic property, where the derivative is linearly related to the integer-order derivative. Fractional calculus stands out in modeling the problems involving the concepts of non-locality and memory effect that are not well explained by integer-order calculus. Indeed, fractional calculus tackles the concept of the derivative operator, where in integer-order calculus, the operator has a local nature, whereas in fractional calculus, it has a non-local nature.

Another important aspect is the concept of mild solutions, which are strongly associated with fractional calculus and fractional integral operators. Mild solutions involve integral operators that account for the past behaviour of the system, making them well-suited for modelling phenomena with long- term memory effects. Controllability is one of the basic problems in control theory for the fractional dynamical system represented by linear or non-linear fractional differential equations. Most importantly, obtaining the solution of linear or nonlinear fractional differential equations in analytical or numerical form is very tedious, as the integration of nonlinear fractional order terms is very difficult. The researchers keep studying to develop new methods to tackle these difficulties.

The utilisation of the special property of deformable derivatives, the interesting behaviours of fractional differential equations, and the different kinds of solutions to some fractional differential equations are studied in this thesis as a main objective, and it can be divided into four specific aims. In the first three specific aims, the research reported in this thesis deals with the problem of existence and approximate controllability results for the various types of perturbed fractional differential and integro-differential systems with deformable derivatives in Banach spaces.

First, we study the existence of solutions for perturbed fractional neutral differential and integro-differential equations using the deformable derivative in Banach spaces. Next, the existence of mild solutions for perturbed fractional neutral differential and integro-differential equations using the deformable derivative in Banach spaces is examined. Finally, the existence, uniqueness, and approximate controllability of mild solutions for fractional neutral differential equations using the deformable derivative in Hilbert spaces are discussed.

The stability analysis is also discussed. The approach that is considered here is based on fixed-point techniques such as Banach’s, Krasnoselskii’s, Leray Schauder’s alternative, Schauder’s fixed-point theorem and Ulam-Hyer stability. Several abstract fractional differential equations are provided to illustrate the obtained results. The fourth specific aim is concerned with the methods for solving linear and nonlinear fractional-order differential equations with deformable derivatives. First, the new fractional differential equations governing some models, namely the relaxation equation, the population growth equation, and the one-dimensional diffusion equation, are obtained by replacing the existing fractional order derivative of Caputo type with the deformable derivatives.

Next, the analytical, semi-analytical, and numerical methods such as the Laplace transform, the Homotopy perturbation method, and the finite difference method are developed and applied to the modified models to study the solution quality at different deformable fractional orders. The solutions are also compared with the exact solution as well as the numerical solutions. The methods that are developed are simple, easy, and convenient to handle fractional- order differential equations with deformable derivatives.

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