Fractional Calculus with Applications in Mechanics Wave Propagation, Impact and Variational Principles

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

Teodor Atanackovic is Full Professor at the University of Novi Sad, Serbia. He has authored or coauthored 8 books and more than 170 articles for journals and proceedings.

Stevan Pilipovic is Full Professor at the University of Novi Sad, Serbia. He has written more than 180 scientific papers in refereed international journals, and more than 35 contributions to special volumes and international conference proceedings.

Bogoljub Stankovic is a retired Professor at the University of Novi Sad, Serbia. His interests include classical theory of integral transforms and operational calculus, special functions, functional analysis, generalized functions and hyperfunctions. He has authored several books and more than 200 articles for journals and proceedings.

Duaan Zorica is Assistant Research Professor at the Mathematical Institute, Serbian Academy of Arts and Sciences in Belgrade, Serbia. He has authored or co-authored over 30 journal and conference papers. His current research interest is in various aspects of fractional calculus and its application to physical problems.