Fractional Calculus with Applications in Mechanics
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
352 pages, Hardcover