Fractional Partial Differential Equations and Their Numerical Solutions

The following sections are included:

• Preface
• Contents

Fractional differential equations have profound physical background and rich theory, and are particularly noticeable in recent years. They are equations containing fractional derivative or fractional integrals, which have received great interest across disciplines such as physics, biology and chemistry. More specifically, they are widely used in dynamical systems with chaotic dynamical behavior, quasi-chaotic dynamical systems, the 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, then introduce some physical background of fractional differential equations. Due to space limitations, this chapter only gives some brief introductions, but 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, the mathematical theory and the numerical algorithms of fractional differential equations need to be further studied. Interested readers can refer to the monographs and literature.

This Chapter mainly introduces definitions and basic properties of fractional derivatives, including Riemann-Liouville fractional derivative, Caputo fractional derivative and fractional Laplace operator, etc. For the fractional Laplace operator, some basic tools of partial differential equations are introduced, such as pseudo-differential operators, fractional Sobolev spaces and commutators estimates, etc. Also, some existence results of fractional ordinary equations are obtained by iteration. For readers' convenience, some basics of Fourier transform, Laplace transform and Mittag-Leffler function are given at the end of the chapter.

The following sections are included:

• Fractional diffusion equation
• Fractional nonlinear Schrödinger equation
  • Space fractional nonlinear Schrödinger equation
  • Time fractional nonlinear Schrödinger equation
  • Global well-posedness of the one-dimensional fractional nonlinear Schrödinger equation
• Fractional Ginzburg-Landau equation
  • Existence of weak solutions
  • Global existence of strong solutions
  • Existence of attractors
• Fractional Landau-Lifshitz equation
  • Vanishing viscosity method
  • Ginzburg-Landau approximation and asymptotic limit
  • Higher dimensional case—Galerkin approximation
  • Local well-posedness
• Fractional QG equations
  • Existence and uniqueness of solutions
  • Inviscid limit
  • Decay and approximation
  • Existence of attractors
• Fractional Boussinesq approximation
• Boundary value problems

Recent decades have witnessed a fast growing applications of fractional calculus to diverse scientific and engineering fields regarding anomalous diffusion, constitutive modelling in viscoelasticity, signal processing and control, fluid mechanics, image processing, and researches on soft matter behaviors, to just mention a few. Compared to integer-order calculus, fractional calculus has the capacity of providing a more simple and accurate description of complex mechanical and physical processes featuring history dependency and space nonlocality, and has thus induced the occurrences of a series of fractional differential equations. Although the analytical solutions of some of fractional differential equations are obtainable, yet these solutions are expressed in terms of special functions which are usually difficult for numerical evaluation, and the solutions are even inaccessible for some of fractional nonlinear equations. These naturally lead to a rapid increasing developments of numerical methods for fractional differential equations. Due to the history dependency and space nonlocality of fractional calculus, numerical solution of fractional differential equations usually characterizes extremely high computational cost and memory requirements. Even though a high-performed computer is employed, it is still difficult to perform a long-time or largedomain simulation, whose operations are found to increase exponentially with time. Up to now, the “short memory principle” has been proposed to reduce the computational effort, but this principle, as pointed out by Ford et al, will yield instability in numerical computations for some specific problems, which implies the applicability of the “short memory principle” seems not very appealing. It is therefore an open issue how to successfully implement the long-time simulation for fractional calculus. On the other hand, less are now known about the systematic analyzes on the stability of numerical methods concerning fractional calculus, together with the solution techniques for high-dimensional fractional differential equations, especially for nonlinear equations…

This chapter presents the numerical methods for fractional ordinary differential equations. It is noted that many researchers have studied the numerical solution of Abel-Volterra integral equations of the first kind and the second kind. These equations are also termed fractional integral equations in which Riemann-Liouville fractional integral are considered (see the references [97] and [91]). On the other hand, research on numerical methods for fractional derivative equations merely commences in recent decades. Here we investigate the numerical solution of fractional ordinary differential equation and fractional integral equation based on the approximation schemes for fractional derivative and fractional integral, respectively.

Finite difference methods and series approximation methods (mainly including Adomian decomposition method and variational iteration method) are the dominant numerical methods for solving fractional partial differential equations. The corresponding theoretically analyzing methods include Fourier methods, energy estimation, matrix eigenvalue method and mathematical induction. There still exist other types of methods but with either somewhat weaker applicability or absence of relatively sound theoretical analyses. The developments of the correlative numerical methods can be briefly reviewed as follows. From the end of 20th century, Gorenflo et al have published a series of papers [93-95, 149, 150] regarding the finite difference schemes for solving time, space, and time-space fractional diffusion equations. These schemes are formulated by using the equivalence of the Riemann-Liouville and Grünwald-Letnikov fractional derivatives, and are further interpreted as the discrete random walk models in terms of time, space and time-space levels. To guarantee the stability of the schemes, shifted Grünwald-Letnikov approximation schemes are constructed in place of the standard schemes. It should be also noted these finite difference schemes can be easily extended to solution of the generic fractional partial differential equations. In studying the saltwater intrusion into aquifer systems, Liu et al presented the “Method of Lines” [137] which transforms the fractional partial differential equation to a system of fractional ordinary differential equations. Their approach takes the backward difference scheme with variable-order and variable-step, and has been widely accepted and extensively used to solution of space fractional partial differential equations. In 2004, Meerschaert and Tadjeran [157] presented the finite difference scheme for space advection-dispersion equation with variable coefficients, together with its error analysis. Afterwards, Tadjeran et al, in 2006, derived a temporally second-order, spatially first-order accurate and unconditionally stable finite difference scheme by combining Grünwald-Letnikov formula with Crank-Nicolson method. Space extrapolation was used to increase the spatial convergence rate to the second-order, and the proposed finite difference scheme is further applied to solving other types of space fractional partial differential equations [75, 158, 202, 208]. It is worth noting that the finite difference schemes mentioned above are all based on the Grünwald-Letnikov approximation schemes. In addition to that, L-algorithms can also be utilized to construct the finite difference schemes [46, 81, 137, 199, 224], yet in the absence of rigorous theoretical analyses for most of schemes…

The following section is included:

• Bibliography