Fractional Derivative Modeling in Mechanics and Engineering

This book highlights the theory of fractional calculus and its wide applications in mechanics and engineering. It describes research findings in using fractional calculus methods for modeling and numerical simulation of complex mechanical behavior. It covers the mathematical basis of fractional calculus, the relationship between fractal and fractional calculus, unconventional statistics and anomalous diffusion, typical applications of fractional calculus, and the numerical solution of the fractional differential equation. It also summaries the latest findings, such as variable order derivative, distributed order derivative, and its applications. The book avoids lengthy mathematical demonstrations and presents the theories related to the applications in an easily readable manner. This textbook intends for students, researchers, and professionals in applied physics, engineering mechanics, and applied mathematics. It is also of high reference value for those in environmental mechanics, geotechnical mechanics, biomechanics, and rheology.

Auteur

Dr. Wen Chen is a Distinguished Professor and former Dean of the College of Mechanics and Materials at Hohai University, China. His research covers computational mechanics, hydrodynamics, and acoustics. His research interests include RBF-based numerical simulation, anomalous diffusion, and non-local statistics of soft matter mechanics. He also serves as Associate Director of the Chinese Society of Environmental Mechanics and the TC member on Linear Control Systems of the International Federation of Automatic Control. He is former TC Chair of the sector in computational mechanics software, China Mechanics Society.

Dr. Hongguang Sun works as a Professor in the College of Mechanics and Materials, Director of Sino-US Joint Research Center of Groundwater and Environmental Fluid Mechanics, and Deputy Director of the Institute of Hydraulics and Fluid Mechanics, Hohai University, China. His main research interests include simulation and remediation of groundwater and soil pollution, sediment transport, and high-precision computational mechanics.

Dr. Xicheng Li works as an Associate Professor at the School of Mathematical Sciences, the University of Jinan, China. He has been engaged in theoretical and applied research of fractional calculus, especially fractional derivative modeling of anomalous diffusion. He is also done much exploration in modeling heat and mass transfer and solving fractional differential equations.

Contenu

Preface Chapter 1 Introduction

1.1 History of fractional calculus

1.2 Geometric and physical interpretation of fractional derivative equation

1.3 Application in science and engineering

Chapter 2 Mathematical foundation of fractional calculus

2.1 Definition of fractional calculus

2.2 Properties of fractional calculus

2.3 Fourier and Laplace transform of the fractional calculus

2.4 Analytical solution of fractional-order equations

2.5 Questions and discussions

Chapter 3 Fractal and fractional calculus

3.1 Fractal introduction and application

3.2 The relationship between fractional calculus and fractal

Chapter 4 Fractional diffusion model

4.1 The fractional derivative anomalous diffusion equation

4.2 Statistical model of the acceleration distribution of turbulence particle

4.3 Lévy stable distributions

4.4 Stretched Gaussian distribution

4.5 Tsallis distribution

4.6 Ito formula

4.7 Random walk model

Chapter 5 Typical applications of fractional differential equations

5.1 Power-law phenomena and non-gradient constitutive relation

5.2 Fractional Langevin equation

5.3 The complex damped vibration

5.4 Viscoelastic and rheological models

5.5 The power law frequency dependent acoustic dissipation

5.6 The fractional variational principle of mechanics

5.7 Fractional Schrödinger equation

5.8 Other application fields

5.9 Some applications of fractional calculus in biomechanics

5.10 Some applications of fractional calculus in the modeling of drug release process

Chapter 6 Numerical methods for fractional differential equations

6.1 Time fractional differential equations

6.2 Space fractional differential equations 6.3 Open issues of numerical methods for FDEs

Chapter 7 Current development and perspectives of fractional calculus

7.1 Summary and Discussion

7.2 Perspectives

Appendix I Special Functions

Appendix II Related electronic resources of fractional dynamics