Fractional Calculus with Applications in Mechanics

The books Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes and Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles contain various applications of fractional calculus to the fields of classical mechanics. Namely, the books study problems in fields such as viscoelasticity of fractional order, lateral vibrations of a rod of fractional order type, lateral vibrations of a rod positioned on fractional order viscoelastic foundations, diffusion-wave phenomena, heat conduction, wave propagation, forced oscillations of a body attached to a rod, impact and variational principles of a Hamiltonian type. The books will be useful for graduate students in mechanics and applied mathematics, as well as for researchers in these fields. Part 1 of this book presents an introduction to fractional calculus. Chapter 1 briefly gives definitions and notions that are needed later in the book and Chapter 2 presents definitions and some of the properties of fractional integrals and derivatives. Part 2 is the central part of the book. Chapter 3 presents the analysis of waves in fractional viscoelastic materials in infinite and finite spatial domains. In Chapter 4, the problem of oscillations of a translatory moving rigid body, attached to a heavy, or light viscoelastic rod of fractional order type, is studied in detail. In Chapter 5, the authors analyze a specific engineering problem of the impact of a viscoelastic rod against a rigid wall. Finally, in Chapter 6, some results for the optimization of a functional containing fractional derivatives of constant and variable order are presented.

Preface xi Part 1. Mathematical Preliminaries, Definitions and Properties of Fractional Integrals and Derivatives 1 Chapter 1. Mathematical Preliminaries 3 Chapter 2. Basic Definitions and Properties of Fractional Integrals and Derivatives 17 Part 2. Mechanical Systems 49 Chapter 3. Waves in Viscoelastic Materials of Fractional-Order Type 51 Chapter 4. Forced Oscillations of a System: Viscoelastic Rod and Body 149 Chapter 5. Impact of Viscoelastic Body Against the Rigid Wall 243 Chapter 6. Variational Problems with Fractional Derivatives 279 Bibliography 379 Index 403