Numerical Solution Of The Space Fractional Advection-Dispersion Equations And Applications

Fractional-order differential equation is obtained from the Standard differential equation by replacing the integer-order derivative with fractional-order derivative. It describes more efficiently the property of memory and heredity of much material. It plays a more and more important role in various fields,especially in engineering, physics,finance and hydrology.Unfortuenately,most of the analytical solution for the fractional differential equation are complicated,i.e.with complicated series or especial function.So it becomes more important to solve those numerical solutions: The numerical solution of fractional-order partial differential equation is considered mainly about parabolic differential equation up to the present.The finite difference method was studied usually.In this paper we also use the finite difference method to discuss the parabolic differential equation.But in our paper,we discuss two kind of more complicated parabolic differential equations.The first kind of parabolic differential equation discussed in this paper is the Lévy-Feller advection-dispersion equation.There is a complicated fractional deriva-tive-Riesz-Feller fractional derivative.It is a non-symmetirc fractional derivative, with two-side fractional derivatives and skewness parameter,which results in the complicated disposal of the discrete form.The second kind of parabolic differential equation,multi-dimensional fractional equation,is generalized from the seepage flow.The two- and three- dimensional cases are discussed in this paper.The calculation of numerical solution is intense large,if the traditional difference method is used.We fall back on the alternation direction method,according to the multi-dimensional equation with integral deriva-tive.Unfortunatly,the added disturbed term becomes more complicated because of the fractional derivative,which results in the theoretical analysis more complicated.In the first chapter,the background and significance of this dissertation are dicussed,the former research is introduced,and finally the framework of this disser-tation is proposed.In the second chapter,the definitions and properties of the fractional derivatives used in this paper,are listed.Then the Lévy-Feller advection-dispersion equation discussed in the third chapter,is deduced from the random walk and Lévy flight. And the multi-dimensional fractional advection-dispersion equation discussed in the forth and fifth chapters,are deduced from the seepage flow.In the third chapter,the Lévy-Feller advection-dispersion equation describing anomalous diffusion,with asymmetric fractional derivative,is considered.Firstly, the fundamentional solution is derived from Fourier and Lapalace transform.And secondly,the discrete scheme is obtained after discreting the Riesz-Feller fractional derivative in the Lévy-Feller advection-dispersion equation by Grünwald-Letnikov shift operators,then the scheme is proved to be considered as discrete randon walk model.And we show that random walk model converges to the stable law of Lévy-Feller advection-dispersion equation by use of a properly scaled transition to vanish-ing space and time steps,We propose an explicit finite difference approximation for Lévy-Feller advection-dispersion equation.As a result of the interpretation of the discrete random walk model,the stability and convergence of the numerical method in a bounded domain are discussed.In the forth chapter,the numerical simulation of two-dimensional seepage flow with fractional derivatives in porous media,is considered under two special cases with different boundary conditions:a modified alternating direction implicit scheme, fractional Euler implicit scheme,for the non-continued seepage flow in uniform me-dia,is proposed with the stability and consistency analysis.And the modified Peaceman-Rachford scheme for the continued seepage flow in non-uniform media are proposed with the stability and consistency.A method for improving the order of convergence by Richardson extrapolation for the Peaceman-Rachford scheme is also presented,which obtains the two order approximating accuracy.In the fifth chapter,the numerical simulation of three-dimensional seepage flow with fractional derivatives in porous medias,is considered under two special cases: a modified alternating direction implicit scheme for the non-continued seepage flow in uniform media,is proposed with the stability and consistency analysis.And a modified Douglas scheme for the continued seepage flow in non-uniform media are proposed with the stability and consistency.To improve the order of convergence for the modified Douglas scheme,Richardson extrapolation is also presented.Some numerical results are presented to support our theoretical analysis in every chapter.