Description
Book SynopsisThis textbook highlights the theory of fractional calculus and its wide applications in mechanics and engineering. It describes in details the 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 includes latest findings, such as variable order derivative, distributed order derivative and its applications. Different from other textbooks in this subject, the book avoids lengthy mathematical demonstrations, and presents the theories in close connection to the applications in an easily readable manner. This textbook is intended 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.
Table of ContentsPreface
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
