走出去：Fractional Partial Differential Equations and Their Numerical Solutions

定　价：¥158.00

作　者：	郭柏灵，蒲学科，黄凤辉著
出版社：	科学出版社
丛编项：	走出去
标　签：	数学微积分自然科学

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ISBN：	9787030432704	出版时间：	2015-04-01	包装：	精装
开本：	16开	页数：	335	字数：

内容简介

《走出去：Fractional Partial Differential Equations and their Numerical Solutions》 mainly concerns the partial differential equations of fractional order and their numerical solutions. In Chapter 1, we briefly introduce the history of fractional order derivatives and the background of some fractional partial differential equations, in particular, their interplay with random walks. Chapter 2 is devoted to the definition of fractional derivatives and integrals from different points of view, from the Riemann-Liouville type, Caputo type derivatives and fractional Laplacian, to several useful tools in fractional calculus, including the pseudo-differential operators, fractional order Sobolev spaces, commutator estimates and so on. In chapter 3, we discuss some partial differential equations of wide interests, such as the fractional reaction-diffusion equation, fractional Ginzburg-Landau equation, fractional Landau-Lifshitz equations, fractional quasi-geostrophic equation, as well as some boundary value problems, especially the harmonic extension method. The local and global well-posedness, long time dynamics are also discussed. Last three chapters are devoted to the numerical aspects of fractional partial differential equations, mainly focusing on the finite difference method, series approximation method, Adomian decomposition method, variational iterative method, finite element method, spectral method and meshfree method and so on.

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图书目录

Preface
Chapter1 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 Schrǒodinger equation
1.5 Fractional Ginzburg-Landau equation
1.6 Fractional Landau-Lifshitz equation
1.7 Some applications of fractional differential equations

Chapter2 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

Chapter3 Fractional Partial Differential Equations
3.1 Fractional diffusion equation
3.2 Fractional nonlinear Schrǒodinger equation
3.2.1 Space fractional nonlinear Schrǒodinger equation
3.2.2 Time fractional nonlinear Schrǒodinger equation
3.2.3 Global well-posedness of the one-dimensional fractional nonlinear Schrǒodinger 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

Chapter4 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

Chapter5 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

Chapter6 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