Skip to main content

Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes

Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes

Teodor M. Atanackovic, Stevan Pilipovic, Bogoljub Stankovic, Dusan Zorica

ISBN: 978-1-118-57753-0

Feb 2014

336 pages

Description

This book contains mathematical preliminaries in which basic definitions of fractional derivatives and spaces are presented. The central part of the book contains various applications in classical mechanics including fields such as: viscoelasticity, heat conduction, wave propagation and variational Hamilton–type principles. Mathematical rigor will be observed in the applications. The authors provide some problems formulated in the classical setting and some in the distributional setting. The solutions to these problems are presented in analytical form and these solutions are then analyzed numerically. Theorems on the existence of solutions will be presented for all examples discussed. In using various constitutive equations the restrictions following from the second law of thermodynamics will be implemented. Finally, the physical implications of obtained solutions will be discussed in detail.

Preface ix

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. Restrictions Following from the Thermodynamics for Fractional Derivative Models of a Viscoelastic Body 51

Chapter 4. Vibrations with Fractional Dissipation 83

Chapter 5. Lateral Vibrations and Stability of Viscoelastic Rods 123

Chapter 6. Fractional Diffusion-Wave Equations 185

Chapter 7. Fractional Heat Conduction Equations 257

Bibliography 289

Index 311

“The book will be useful to researchers and students looking for applications of fractional calculus in applied mechanics and engineering.”  (Zentralblatt MATH, 1 November 2014)