Preface
Chapter 1 Physics Background
1.1 Origin of the fractional derivative
1.2 Anomalous diffusion and fractional advection-diffusion
1.2.1 The random walk and fractional equations
1.2.2 Fractional advection-diffusion equation
1.2.3 Fractional Fokker-Planck equation
1.2.4 Fractional Klein-Framers equation
1.3 Fractional quasi-geostrophic equation
1.4 Fractional nonlinear SchrSdinger equation
1.5 Fractional Ginzburg-Landau equation
1.6 Fractional Landau-Lifshitz equation
1.7 Some applications of fractional differential equations
Chapter 2 Fractional Calculus and Fractional Differential Equations
2.1 Fractional integrals and derivatives
2.1.1 Riemann-Liouville fractional integrals
2.1.2 R-L fractional derivatives
2.1.3 Laplace transforms of R-L fractional derivatives
2.1.4 Caputo's definition of fractional derivatives
2.1.5 Weyl's definition for fractional derivatives
2.2 Fractional Laplacian
2.2.1 Definition and properties
2.2.2 Pseudo-differential operator
2.2.3 Riesz potential and Bessel potential
2.2.4 Fractional Sobolev space
2.2.5 Commutator estimates
2.3 Existence of solutions
2.4 Distributed order differential equations
2.4.1 Distributed order diffusion-wave equation
2.4.2 Initial boundary value problem of distributed order
2.5 Appendix A: the Fourier transform
2.6 Appendix B: Laplace transform
2.7 Appendix C: Mittag-Leffler function
2.7.1 Gamma function and Beta function
2.7.2 Mittag-Leffler function
Chapter 3 Fractional Partial Differential Equations
3.1 Fractional diffusion equation
3.2 Fractional nonlinear SchrSdinger equation
3.2.1 Space fractional nonlinear SchrSdinger equation
3.2.2 Time fractional nonlinear Schr5dinger equation
3.2.3 Global well-posedness of the one-dimensional fractional nonlinear SchrSdinger equation
3.3 Fractional Ginzburg-Landau equation
3.3.1 Existence of weak solutions
3.3.2 Global existence of strong solutions
3.3.3 Existence of attractors
3.4 Fractional Landau-Lifshitz equation
3.4.1 Vanishing viscosity method
3.4.2 Ginzburg-Landau approximation and asymptotic limit
3.4.3 Higher dimensional case-Galerkin approximation
3.4.4 Local well-posedness
3.5 Fractional QG equations
3.5.1 Existence and uniqueness of solutions
3.5.2 Inviscid limit
3.5.3 Decay and approximation
3.5.4 Existence of attractors
3.6 Fractional Boussinesq approximation
3.7 Boundary value problems
Chapter 4 Numerical Approximations in Fractional Calculus
4.1 Fundamentals of fractional calculus
4.2 G-Algorithm for Riemann-Liouville fractional derivative
4.3 D-Algorithm for Riemann-Liouville fractional derivative
4.4 R-Algorithm for Riemann-Liouville fractional integral
4.5 L-Algorithm for fractional derivative
4.6 General form of fractional difference quotient approximations
4.7 Extension of integer-order numerical differentiation and integration
4.7.1 Extension of backward and central difference quotient schemes
4.7.2 Extension of interpolation-type integration quadrature formulas
4.7.3 Extension of linear multi-step method: Lubich fractional linear multi-step method
4.8 Applications of other approximation techniques
4.8.1 Approximation of fractional integral and derivative of periodic function using Fourier Series
4.8.2 Short memory principle
Chapter 5 Numerical Methods for the Fractional Ordinary Differential Equations
5.1 Solution of fractional linear differential equation
5.2 Solution of the general fractional differential equations
5.2.1 Direct method
5.2.2 Indirect method
Chapter 6 Numerical Methods for Fractional Partial Differential Equations
6.1 Space fractional advection-diffusion equation
6.2 Time fractional partial differential equation
6.2.1 Finite difference scheme
6.2.2 Stability analysis: Fourier-von Neumann method
6.2.3 Error analysis
6.3 Time-space fractional partial differential equation
6.3.1 Finite difference scheme
6.3.2 Stability and convergence analysis
6.4 Numerical methods for non-linear fractional partial differential equations
6.4.1 Adomina decomposition method
6.4.2 Variational iteration method
Bibliography
