Fractional Derivative Modeling in Mechanics and Engineering

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 L憝y 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 Schr鐰inger 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

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