This book 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.
目錄：
Preface
Chapter1 Physics Background1
1.1 Origin of the fractional derivative1
1.2 Anomalous diffusion and fractional advection-diffusion5
1.2.1 The random walk and fractional equations6
1.2.2 Fractional advection-diffusion equation10
1.2.3 Fractional Fokker-Planck equation11
1.2.4 Fractional Klein-Framers equation15
1.3 Fractional quasi-geostrophic equation16
1.4 Fractional nonlinear Schrǒodinger equation20
1.5 Fractional Ginzburg-Landau equation23
1.6 Fractional Landau-Lifshitz equation27
1.7 Some applications of fractional differential equations29
Chapter2 Fractional Calculus and Fractional Differential Equations34
2.1 Fractional integrals and derivatives34
2.1.1 Riemann-Liouville fractional integrals34
2.1.2 R-L fractional derivatives42
2.1.3 Laplace transforms of R-L fractional derivatives48
2.1.4 Caputo’s definition of fractional derivatives50
2.1.5 Weyl’s definition for fractional derivatives53
2.2 Fractional Laplacian56
2.2.1 Definition and properties56
2.2.2 Pseudo-differential operator63
2.2.3 Riesz potential and Bessel potential69
2.2.4 Fractional Sobolev space71
2.2.5 Commutator estimates76
2.3 Existence of solutions83
2.4 Distributed order differential equations89
2.4.1 Distributed order diffusion-wave equation91
2.4.2 Initial boundary value problem of distributed order94
2.5 Appendix A: the Fourier transform96
2.6 Appendix B: Laplace transform104
2.7 Appendix C: Mittag-Leffler function105
2.7.1 Gamma function and Beta function105
2.7.2 Mittag-Leffler function107
Chapter3 Fractional Partial Differential Equations108
3.1 Fractional diffusion equation108
3.2 Fractional nonlinear Schrǒodinger equation112
3.2.1 Space fractional nonlinear Schrǒodinger equation112
3.2.2 Time fractional nonlinear Schrǒodinger equation124
3.2.3 Global well-posedness of the one-dimensional fractional nonlinear Schrǒodinger equation128
3.3 Fractional Ginzburg-Landau equation137
3.3.1 Existence of weak solutions137
3.3.2 Global existence of strong solutions142
3.3.3 Existence of attractors150
3.4 Fractional Landau-Lifshitz equation154
3.4.1 Vanishing viscosity method154
3.4.2 Ginzburg-Landau approximation and asymptotic limit161
3.4.3 Higher dimensional case-Galerkin approximation169
3.4.4 Local well-posedness184
3.5 Fractional QG equations198
3.5.1 Existence and uniqueness of solutions199
3.5.2 Inviscid limit208
3.5.3 Decay and approximation212
3.5.4 Existence of attractors220
3.6 Fractional Boussinesq approximation228
3.7 Boundary value problems246
Chapter4 Numerical Approximations in Fractional Calculus.257
4.1 Fundamentals of fractional calculus258
4.2 G-Algorithm for Riemann-Liouville fractional derivative261
4.3 D-Algorithm for Riemann-Liouville fractional derivative266
4.4 R-Algorithm for Riemann-Liouville fractional integral269
4.5 L-Algorithm for fractional derivative272
4.6 General form of fractional difference quotient approximations274
4.7 Extension of integer-order numerical differentiation and integration276
4.7.1 Extension of backward and central difference quotient schemes276
4.7.2 Extension of interpolation-type integration quadrature formulas279
4.7.3 Extension of linear multi-step method: Lubich fractional linear multi-step method280
4.8 Applications of other approximation techniques283
4.8.1 Approximation of fractional integral and derivative of periodic function using Fourier Series283
4.8.2 Short memory principle284
Chapter5 Numerical Methods for the Fractional Ordinary Differential Equations286
5.1 Solution of fractional linear differential equation286
5.2 Solution of the general fractional differential equations287
5.2.1 Direct method289
5.2.2 Indirect method292
Chapter6 Numerical Methods for Fractional Partial Differential Equations299
6.1 Space fractional advection-diffusion equation301
6.2 Time fractional partial differential equation305
6.2.1 Finite difference scheme306
6.2.2 Stability analysis: Fourier-von Neumann method307
6.2.3 Error analysis308
6.3 Time-space fractional partial differential equation310
6.3.1 Finite difference scheme310
6.3.2 Stability and convergence analysis312
6.4 Numerical methods for non-linear fractional partial differential equations318
6.4.1 Adomina decomposition method318
6.4.2 Variational iteration method320
Bibliography322
內容試閱：
Chapter 1
Physics Background
Fractional differential equations have profound physical backgrounds and rich related theories， and are noticeable in recent years. They are equations containing fractional derivative or fractional integrals， which have applied in various disciplines such as physics， biology and chemistry. More specifically， they are widely used in dynamical systems with chaotic dynamical behavior， quasi-chaotic dynamical systems， dynamics of complex material or porous media and random walks with memory. The purpose of this chapter is to introduce the origin of the fractional derivative， and then some physical backgrounds of fractional differential equations. Due to space limitations， this chapter only gives some brief introductions. Even so， these are sufficient to show that the fractional differential equations， including fractional partial differential equations and fractional integral equations， are widely employed in various applied fields. However， further mathematical theories and numerical algorithms of fractional differential equations need to be studied. Interested readers can refer to more monographs and literatures.
1.1 Origin of the fractional derivative
The concepts of integer order derivative and integral are well known. The derivative dnydxn describes the changes of variable y with respect to variable x， supported by profound physical backgrounds. Now the problem is how to generalize n into a fraction， even a complex number.
The long-standing problem can be traced back to the letter from L’H.opital to Leibniz in 1695， in which it is asked like what the derivative dnydxn is when n = 12. In the same year， the derivative of general order was mentioned in the letter from Leibniz to J. Bernoulli as well. The problem was also considered by Euler1730， Lagrange1849 et al， who gave some relevant insights. In 1812， by using the concept of integral， Laplace provided a definition of fractional derivative. When y = xm， using the gamma function， was derived by Lacroix， who gives . This is consistent with the Riemann-Liouville fractional derivative in the present.
Soon later， Fourier 1822 gave the definition of fractional derivative through the Fourier transform. The function fx can be expressed as a double integral By replacing n with ν， and calculating the derivative under the integral sign， one can generalize the integer order derivative into the fractional order derivative Consider the Abel integral equation where f is to be determined. The right-hand side defines a definite integral of fractional integral with order 12. Abel wrote √π dx.12 fx for the righthand
component， then ， which indicates that the fractional derivative of a constant is no longer zero.
In 1930s， Liouville， possibly inspired by Fourier and Abel， made a series of work in the field of fractional derivative， and successfully applied them into the potential theory. Since Dmeax = ameax，
1.1 Origin of the fractional derivative 3
the order of the derivative was generalized to be arbitrary by Liouville ν can be a rational number， an irrational number， even a complex number If a function f can be expanded into an infinite series its fractional derivative can be obtained as How can we obtain the fractional derivative if f cannot be written in the form of equation 1.1.5? Liouville probably had noticed this problem， and he gave another expression by using the Gamma function. In order to make use of the basic assumptions 1.1.4， noting that one then obtains So far， we have introduced two different definitions of fractional derivatives.
One is the definition 1.1.1 with respect to xaa 0 given by Lacroix， the other is the definition 1.1.7 with regard to x.aa 0 given by Liouville.
It can be seen that， Lacroix’s definition shows that the fractional derivative of a constant x0 is no longer zero. For instance， when m = 0， n = However， in Liouville’s definition， since Γ0 = ∞， the fractional derivative of a constant is zero despite Liouville’s assumption a 0. As far as which of the two definitions is the correct form of fractional derivative， Willian Center pointed out it can be attributed to how to determine dνx0dxν; and as DeMorgan indicated 1840， both of them may very possibly be parts of a more general system.
The present Riemann-Liouville’s definition R-L of fractional derivative may be derived from N. Ya Sonin1869 whose starting point was the Cauchy integration formula， from which the nth derivative of f can be defined as Using contour integration， the following generalization can be obtained in which， Laurent’s work were contributed! where the constant c = 0 is commonly used. It is known as the Riemann- Liouville fractional derivative， i.e.， In order to make the integral convergent， a sufficient condition is f1x = Ox1.ε， ε 0. An integrable function with this property is often referred to as belonging to the function of the Riemann class. When c = .∞， .∞D.ν In order to m
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