Analysis Of Numerical Methods For Several Fractional Differential Equations

The topic of this dissertation is to study the numerical methods for fractional dif-ferential equations and their stability and convergence. This dissertation is composed ofthree parts: finite diference method, finite element method and collocation method.The introduction of the dissertation deals with the background and significance offractional calculus. Three kinds of definitions for fractional derivatives are shown. Theresearch history and recent advancement of numerical methods for fractional diferentialequations are recalled. The main contents of the dissertation are also given.In Chapter2, we investigate a finite diference method for the fractional partial difer-ential equations, which Diethelm first used to solve one variable left fractional diferentialequation in1997. Here, we extend this method to right fractional diferential equationwith initial value and boundary value conditions and obtain the discretization scheme.The results of stability condition and convergence order for this method are given. Thenumerical results are given to validate the stability and convergence of this method.The second main part of this dissertation, including Chapter3,4,5, is the finiteelement methods for fractional partial diferential equations.In Chapter3, we study a finite element method for the time-space Riesz fractionaladvection-dispersion equations. The initial value and boundary value problem of the time-space Riesz fractional advection-dispersion equations has left Caputo fractional deriva-tive in time and Riesz fractional derivative in space. First, based on the definitions ofSobolev spaces, we prove the existence and uniqueness results of the weak formulationfor the time-space Riesz fractional advection-dispersion equations by Lax-Milgram the-orem. Then, we present a finite diference method for time discretization and a finiteelement method for space discretization. Furthermore, the convergence rate is given andthe unconditional stability is proved for fully-discretization scheme. Finally, some nu-merical examples are given for matching well with the theoretical results.The main purpose of the Chapter4is to study the finite element method for a mul-titerm time fractional partial diferential equations, which have the multiterm left Caputofractional derivative in time. Similar to the method in Chapter3, we consider the weakformulation of the multi-term time fractional partial diferential equation and prove theexistence and uniqueness of weak solution. By using Diethelm fractional discretization method, we make the time discretization for the multiterm time fractional partial difer-ential equation and prove the convergence rate. In space, we consider the finite elementmethod and obtain the convergence rate. Then, we prove the unconditional stability ofthe method. Finally, some numerical examples are given to check the convergence of thenumerical solution and show the efectiveness of such method by comparing with otherexisting methods for fractional diferential equations.In Chapter5, we investigate the finite element method for the multiterm time-spaceRiesz fractional advection-difusion equations, which have multiterm left Caputo frac-tional diferential operators in time and Riemann-Liouville Riesz fractional diferentialoperators in space. In theory, we show the weak formulation of such equation and provethe uniqueness and existence of the weak solution. In numerical analysis, we consider thetime and space discretization respectively. The finite diference method is used for timediscretization and the convergence rate is obtained; the finite element method is used forspace discretization and the convergence rate is obtained. Then, the unconditional stabil-ity of the method is proved. The numerical examples are given to show the efectivenessof the method.As part three, Chapter6shows the collocation method for the fractional integro-diferential equations with weakly singular kernel. Firstly, we prove that a suitable trans-formation can be used to convert the fractional integro-diferential equations to a linearsecond kind Volterra integral equations with weakly singular kernels. Secondly, we showthat if the exact solution is smooth, the piecewise polynomial collocation method withuniform meshes yields optimal convergence rates; if the exact solution is not smooth,the solution can be made smooth by a variable substitution, and the collocation methodfor such solution with uniform meshes also yields optimal convergence rate. Finally, weconsider some numerical experiments to check the convergence rate by using the trape-zoidal collocation method, the Simpson’s collocation method, the Newton’s three-eightcollocation method and the Cotes collocation method.