Finite Difference Methods For Fractional Partial Differential Equations

In recent decades, fractional calculus and fractional diferential equations are un-dergoing rapid development due to their wide applications in physics, engineering, eco-nomics and some other research realms. Till now, several methods have been introducedfor some special fractional diferential equations to seek their analytical solutions, such asintegral transformation method (Laplace’s transform, Fourier’s transform, and Mellin’stransform), Adomian decomposition method, the method of separating variables and soon. However, the exact solutions of the most fractional diferential equations cannot beobtained. So it becomes important to develop numerical methods for these equations.This dissertation mainly includes five parts. The main research jobs are given asfollows.The first chapter briefly introduces the development of fractional calculus and thedefinitions and properties of fractional calculus. Numerical methods and research situa-tion for fractional calculus and fractional diferential equations.Chapter II discusses high-order numerical methods for Riemann-Liouvile and Rieszderivatives. Firstly, a fourth-order compact scheme is proposed for Riemann-Liouvilederivative, Next, we give another method to calculate the coefcients of fractional linearmultisteps method. Finally, we get a series of high-order methods for Riesz derivative.Chapter III studies the higher-order methods for reaction-subdifusion equation. wedevelop two classes of finite diference schemes for the reaction-subdifusion equationsby using a mixed spline function in space direction, forward and backward diferencein time direction, respectively. It has been shown that the previousrelated diferenceschemes can be derived from our schemes if we suitably choose the spline parameters.By Fourier method, we prove that one class of diference scheme is unconditionallystable and convergent, the other is conditionally stable and convergent. Finally, somenumerical results are provided to demonstrate the efectiveness of the proposed diferenceschemes.Chapter IV discusses the higher-order methods for the wave equation with reactionterm. Firstly, using the relations between Caputo and Riemann-Liouville derivatives,we get two equivalent forms of the original equation, where we approximate Riemann- Liouville derivative by a second-order diference scheme. Secondly, for second-orderderivative in space dimension, we construct a fourth-order diference scheme in termsof the hyperbolic-trigonometric spline function. The stability analysis of the derivednumerical methods are given by means of the fractional Fourier method. Finally, anillustrative example is presented to show that the numerical results are in good agreementwith the theoretical analysis.In the last chapter, we propose a temporal second order numerical method for thefractional reaction-dispersion equation where we discrete the Riesz fractional derivativeby using two numerical schemes. We prove that the numerical methods for the spatialRiesz fractional reaction dispersion equation are both unconditionally stable and conver-gent, and the orders of convergence are O(τ2+h6) and O(τ2+h8), in which τ and h arespatial and temporal step sizes, respectively. Finally, we test our numerical schemes andobserve that the numerical results are in line with the theoretical analysis.