Finite Difference Methods For Several Kinds Of Fractional Differential Equations

Fractional-order computation is an old and fresh research field.Especially in the last more than 10 years,fractional differential equations and their applications have been widely concerned.It is mainly attributed to the rapid development of frac-tional differential theory and its extensive applications in mathematics,fluid mechanics,material mechanics,biology and other disciplines.Fractional differential equations are generalized and non integer-order differential equations.It can obtain a non local re-lationship between the power law memory kernel in time and space,which provides a powerful tool for the description of the memory inheritance of different substances.Because the overwhelming majority of fractional partial differential equations are im-possible to obtain analytical solutions,researchers have paid more and more attention to the numerical methods of fractional partial differential equations.In this doctoral dis-sertation,we study finite difference methods for several kinds of fractional differential equations.Firstly,we sketch the historical background,status,the up-to-date progress for the discussed problems,our main work,preliminary facts on fractional partial differential equations and the main tools used in the dissertation and the algorithm of fractional derivative.Secondly,we introduce two parts.In the first part,we consider the following space fractional percolation equation:subject to the initial condition:p(x,0)? s(x),0?x?L,and the Dirichlet boundary conditions:p(0,t)= 0,p(L,t)=u(l),0?t?T.We give the Crank-Nicolson difference scheme.Its solvability,consistency,stability and convergence of the method are proven.And we combine Richardson extrapola-tion to improve the convergence precision,which makes the convergence order of the numerical solution of the equation achieve second-order convergence in time and space.The numerical example is used to verify the convergence accuracy of the scheme.In the second part,we consider the finite difference method of the two-dimensional space fractional percolation equation:subject to the initial condition:p(x,y,0)=?(x,y),(x,y)??and the Dirichlet boundary conditions:p(a1,Y,0)=p(x,b1,t)= 0,p(a2,y,O)= u(y,t),p(x,b2,t)= v(x,t),0?t?T,(x,y)??We give the ADI-CN difference method.Its solvability,stability and convergence of the method are proven.And we combine Richardson extrapolation to improve the convergence precision,which makes the convergence order of the numerical solution of the equation achieve second-order convergence in time and space.We use a numerical experiment to verify the efficiency of the convergence accuracy of the method.Then,we consider the following two-sided space-fractional diffusion equation:+f(x,y,t),(x,y)??,t>0,subject to the initial condition:x(x,y,0)= ?(x,y),(x,y)??and the Dirichlet boundary conditions:u(al,y,l)=u(a2,y,l)=u(x,b1,t)=0,l?0,(x,y)??.We give an alternating direction Euler implicit difference method,(ADI).Its solvability,stability and convergence of the method are proven.We use a numerical experiment to show the efficiency of the convergence accuracy of the method.Then,we consider the following one-dimensional Riesz fractional advection-dispersion equation with fractional boundary conditions:subject to the initial condition:u(x,0)=q(x),0<x<R and the fractional boundary conditions:u(0,t)= 0,?u(R,t)+(d(x)0RDx?-1(x,t))x=R=?(t),t?0,In order to propose an implicit finite difference method,we use the fractional cen-tered derivative approach to approximate the Riesz fractional derivative and use the standard Grunwald-Letnikov fractional order operator to discrete Riemann-Liouville fractional derivative in fractional boundary conditions.Its solvability,stability and convergence of the method are proven.We present a fast iterative method for the im-plicit finite difference scheme,which only requires storage of O(N)and computational cost of O(Nlog2N),where N is the number of spatial grids.Traditionally,the Gaus-sian elimination method requires storage of O(N2)and computational cost of O(N3),because the format coefficient matrix is a dense full rank matrix.The efficiency of the method is checked with a numerical example.Finally,we summarize the main works and shortcomings of this paper,and prospect the future research.