Study of existence, uniqueness, stability and numerical resolutions of differential equations and inclusions of fractional order

Abstract

This thesis comprehensively addresses theoretical and numerical aspects of a specific class of fractional differential equations. Initially, we discuss results related to the uniqueness, existence and stability in the Hyers-Ulam sense for a category of initial value problems concerning nonlinear implicit fractional differential equations with non-instantaneous impulses including the Caputo-Fabrizio fractional derivative. Furthermore, our focus extends to exploring existence and uniqueness results for a class of fractional integrodifferential equations (FIDEs) with non-instantaneous impulses under the Caputo fractional derivative. To achieve the existence and uniqueness results,we employed the fixed point theorems of Krasnoselskii, Darbo combined with the Kuratowski’s measure of noncompactness as well as the Banach contraction principle. Adding a numerical dimension, we delve into the resolution of linear Fredholm fractional integro-differential equations, where the fractional derivative is considered in the Caputo sense. To establish this, we utilize the least squares method (LSM) alongside spectral approximation, employing acompact combination of shifted Chebyshev polynomials (SCP) of the first kind. Throughout the thesis, various examples are provided to validate and elucidate both theoretical and numerical results discussed in each chapter.