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Full 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.
Contents
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
