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Über dieses Buch

​​​ Topics in Fractional Differential Equations is devoted to the existence and uniqueness of solutions for various classes of Darboux problems for hyperbolic differential equations or inclusions involving the Caputo fractional derivative. ​​Fractional calculus generalizes the integrals and derivatives to non-integer orders. During the last decade, fractional calculus was found to play a fundamental role in the modeling of a considerable number of phenomena; in particular the modeling of memory-dependent and complex media such as porous media. It has emerged as an important tool for the study of dynamical systems where classical methods reveal strong limitations. Some equations present delays which may be finite, infinite, or state-dependent. Others are subject to an impulsive effect. The above problems are studied using the fixed point approach, the method of upper and lower solution, and the Kuratowski measure of noncompactness. This book is addressed to a wide audience of specialists such as mathematicians, engineers, biologists, and physicists. ​

Inhaltsverzeichnis

Chapter 1. Introduction

Fractional calculus is a generalization of the ordinary differentiation and integration to arbitrary non-integer order. The subject is as old as the differential calculus and goes back to times when Leibniz and Newton invented differential calculus. One owes to Leibniz in a letter to L’Hôspital, dated September 30, 1695 [181], the exact birthday of the fractional calculus and the idea of the fractional derivative.
Saïd Abbas, Mouffak Benchohra, Gaston M. N’Guérékata

Chapter 2. Preliminary Background

In this chapter, we introduce notations, definitions, and preliminary facts that will be used in the remainder of this book.
Saïd Abbas, Mouffak Benchohra, Gaston M. N’Guérékata

Chapter 3. Partial Hyperbolic Functional Differential Equations

In this chapter, we shall present existence results for some classes of IVP for partial hyperbolic differential equations with fractional order.
Saïd Abbas, Mouffak Benchohra, Gaston M. N’Guérékata

Chapter 4. Partial Hyperbolic Functional Differential Inclusions

In this chapter, we shall present existence results for some classes of initial value problems for partial hyperbolic differential inclusions with fractional order involving the Caputo fractional derivative, when the right-hand side is convex as well as nonconvex valued. Some results rely on the nonlinear alternative of Leray–Schauder type. In other results, we shall use the fixed-point theorem for contraction multivalued maps due to Covitz and Nadler.
Saïd Abbas, Mouffak Benchohra, Gaston M. N’Guérékata

Chapter 5. Impulsive Partial Hyperbolic Functional Differential Equations

In this chapter, we shall present existence results for some classes of initial value problems for fractional order partial hyperbolic differential equations with impulses at fixed or variable times impulses.
Saïd Abbas, Mouffak Benchohra, Gaston M. N’Guérékata

Chapter 6. Impulsive Partial Hyperbolic Functional Differential Inclusions

In this chapter, we shall present existence results for some classes of initial value problems for impulsive partial hyperbolic differential inclusions with fractional order.
Saïd Abbas, Mouffak Benchohra, Gaston M. N’Guérékata

Chapter 7. Implicit Partial Hyperbolic Functional Differential Equations

In this chapter, we shall present existence results for some classes of initial value problems for partial hyperbolic implicit differential equations with fractional order.
Saïd Abbas, Mouffak Benchohra, Gaston M. N’Guérékata

Chapter 8. Fractional Order Riemann–Liouville Integral Equations

In this chapter, we shall present existence results for some classes of Riemann–Liouville integral equations of two variables by using some fixed-point theorems.
Saïd Abbas, Mouffak Benchohra, Gaston M. N’Guérékata

Backmatter

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