Numerical Solution Of Several Classes Of Fractional Partial Differential Equations

Because of the non-locality property of the fractional derivatives,fractional differential equations can simulate many natural physical processes and dynamic system processes better than classical differential equations,therefore they are becoming more and more widespread in various fields,especially in engineering, physics,finance and hydrology.Unfortunately,most of the analytical solution for fractional differential equation are complicated owing to complex series or special functions.So it is extremely important to resort to numerical solutions.However, numerical methods and theoretical analysis for fractional differential equations are hard tasks.At present,there are some papers discussing the numerical methods for fractional differential equations,involving few classes of equations,in some of which no theoretical analysis are given or theoretical analysis are not completed. Therefore,a great deal of work need to be done in this field.In this paper,numerical solutions of three kinds of fractional partial differential equations are considered.The equations describe anomalous diffusion of the particles.In Chapter 1,the meaning of the fractional derivatives is discussed and the previous works about the numerical methods are introduced,then the definitions and properties of some common used fractional operators are presented.In Chapter 2,numerical methods for solving a class of initial-boundary value problems of space fractional partial differential equations with variable coefficients are considered.We generalize the finite difference methods proposed by Meerschaert and Tadjeran and present a class of weighted finite difference methods. Stability analysis of the methods is performed.It is shown that the weighted finite difference methods are unconditionally stable for 0≤r≤1/2,and conditionally stable for 1/2＜r＜1.The order of convergence is given.Some examples are provided and compared numerically in order to prove the validity of the methods with different weighting parameters.In Chapter 3,a class of space fractional advection-dispersion equations is discussed,the advection term of the equation isα-order R-L derivative,the dispersion term isβ-order R-L derivative.In order to solve this equation class,we use Grǖnwald formula to approximate advection term and shifted Grǖnwald formula to the dispersion term.Then we have an effective fractional Cranck-Nicolson (CN) scheme.The detailed analysis of the stability and convergence of the fractional CN scheme is given using mathematical induction and the properties of fractional dispersion coefficients.The convergence result is second order in time but only first order in space,Because of the property of the Grǖnwald formula and the shifted Grǖnwald formula which can be written in the form of series like Taylor expansion series,we use extrapolation method to improve the convergence accuracy and a second-order precision is got.Finally,we provide some numerical experiments to verify the feasibility of the numerical methods and the consistency between the numerical convergence-order and theoretical convergence-order.In Chapter 4,an implicit finite difference method for fractional dispersion advection equation is proposed,we prove that the method is unconditionally stable and convergent by using Fourier analysis.Numerical experiments are provided for the validity of the methods.