Fractional Differential Equations: New Advancements for Generalized Fractional Derivatives

Description

This book covers problems involving a variety of fractional differential equations, as well as some involving the generalized Hilfer fractional derivative, which unifies the Riemann-Liouville and Caputo fractional derivatives. The authors highlight the existence, uniqueness, and stability results for various classes of fractional differential equations based on the most recent research in the area. The book discusses the classic and novel fixed point theorems related to the measure of noncompactness in Banach spaces and explains how to utilize them as tools. The authors build each chapter upon the previous one, helping readers to develop their understanding of the topic. The book includes illustrated results, analysis, and suggestions for further study. Preface Contents 1 Introduction 2 Preliminary Background 2.1 Notations and Functional Spaces 2.1.1 Space of Continuous Functions 2.1.2 Spaces of Integrable Functions 2.1.3 Spaces of Continuous Functions with Weight 2.2 Special Functions of the Fractional Calculus 2.2.1 Gamma Function 2.2.2 k-Gamma and k-Beta Functions 2.3 Elements from Fractional Calculus Theory 2.3.1 Fractional Integrals 2.3.2 Fractional Derivatives 2.3.3 Necessary Lemmas, Theorems and Properties 2.4 Kuratowski Measure of Noncompactness 2.5 Fixed Point Theorems 3 Hybrid Fractional Differential Equations 3.1 Introduction and Motivations 3.2 Initial Value Problem for Hybrid Generalized Hilfer Fractional Differential Equations 3.2.1 Introduction 3.2.2 Existence Results 3.2.3 Examples 3.3 Boundary Value Problem for Hybrid Generalized Hilfer Fractional … 3.3.1 Introduction 3.3.2 Existence Results 3.3.3 Examples 3.4 Nonlocal Initial Value Problem for Hybrid Generalized Hilfer-type … 3.4.1 Existence Results 3.4.2 Example 3.5 Initial Value Problem for Hybrid ψ-Hilfer Fractional … 3.5.1 Existence Results 3.5.2 Examples 3.6 Notes and Remarks 4 Fractional Differential Equations with Retardation and Anticipation 4.1 Introduction and Motivations 4.2 On k-Generalized ψ-Hilfer Boundary Value … 4.2.1 Existence Results 4.2.2 Examples 4.3 Nonlocal k-Generalized ψ-Hilfer Terminal Value Problem … 4.3.1 Existence Results 4.3.2 Examples 4.4 Notes and Remarks 5 Impulsive Fractional Differential Equations with Retardation and Anticipation 5.1 Introduction and Motivations 5.2 On k-Generalized ψ-Hilfer Impulsive Boundary Value … 5.2.1 Existence Results 5.2.2 An Example 5.3 On k-Generalized ψ-Hilfer Impulsive Boundary Value Problem … 5.3.1 Existence Results 5.3.2 An Example 5.4 Nonlocal k-Generalized ψ-Hilfer Impulsive Initial … 5.4.1 Existence Results 5.4.2 An Example 5.5 Notes and Remarks 6 Coupled Systems for Fractional Differential Equations 6.1 Introduction and Motivations 6.2 On Coupled Systems for k-Generalized ψ-Hilfer … 6.2.1 Existence Results 6.2.2 An Example 6.3 Implicit Coupled k-Generalized … 6.3.1 Existence Results 6.3.2 An Example 6.4 Notes and Remarks Bibliography Index