Space- And Time- Fractional Partial Difference Equations

Fractional differential equation have been used to simulate many phenomena in engineering, physics, chemistry and other science. However numerical methods and theoretical analysis of fractional equations are very difficult tasks. Theoretical analysis is different with classical numerical method. At present, though a growing number of works from various fields of science and application deal with dynamical systems described by fractional order equations, very few papers describe the numerical methods for fractional differential equations, especially for fractional partial differential equations.In this paper, we consider the Riesz space fractional reaction-dispersion equation, time fractional telegraph equation and time fractional wave equation with damping.In Chapter 1, surveys of the history of the theory of fractional calculus are introduced. Furthermore some related knowledge about fractional derivatives are presented.In Chapter 2, we mainly study Riesz space fractional reaction-dispersion(RSFR DE). Using the method of the Laplace and Fourier transform, we firstly obtain the fundamental solution (Green function) for RSFRDE in an infinite domain. The solution for the fractional partial difference equation is difficult to solve. So we are interested to develop numerical methods for fractional partial difference equations. We consider RSFRDE in a bounded space domain. We construct both explicit finite difference approximation and implicit difference approximation for RSFRDE in a bounded space domain by discretizing fractional derivative with second-order center difference. We conclude that the explicit difference scheme is conditional stable and convergent but the implicit difference scheme is unconditionally stable and convergent. Finally, some numerical examples are presented. We also compare the numerical results and the results of Method Of Line (MOL) to prove the numerical methods are practical and efficient computational methods. The techniques can also be applied to deal with other fractional order problems. Furthermore we consider RSFRDE with Dirichlet boundary in a bounded space domain. Using the relationship between the Riemann-Liouville definition and the Grünwald-Letnikov definition and applying shift G-L technique, we propose an explicit numerical approximation. The stability and convergence of the approximation are analyzed. Finally, we also give some numerical examples.In Chapter 3, a method of separating variables is effectively implemented for solving time-fractional telegraph equation (TFTE). We discuss and derive the analytical solution of the TFTE with three kinds of nonhomogeneous boundary conditions, namely, Dirichlet, Neumann and Robin boundary conditions.In Chapter 4, we consider a time fractional wave equation with damping. The equation is obtained from the standard wave equation with damping by replacing the second-order time derivative by a Caputo fractional derivative. An implicit difference approximation is constructed. Stability and convergence are proved by the energy method. Finally, a numerical example is presented to show the difference method is effective. The analytical solutions are expressed by the multi-Mittag-Leffler function.In Chapter 5, we conclude the works in the paper.