High Order Finite Difference Methods For Fractional Partial Differential Equations Based On The Polynomial Interpolation

In recent decades,due to the fact that the fractional derivatives are nonlocal,they are more appropriate for the description of memorial and hereditary properties of various materials and processes than the traditional integer-order models.They can model many phenomenon more accurately and have attracted more and more attention of the scholars.The numerical methods for solving the fractional partial differential equations are of great value both in the field of the engineering and technology and in mathematics itself.This thesis is devoted to constructing high order difference schemes for the fractional wave equation,the time multi-term fractional wave equation,space-time fractional Bloch-Torrey equation and the nonlinear time-fractional fourth-order reaction-diffusion equation and presenting the strict theoretical analysis of the difference schemes.Firstly,we consider the numerical methods for the one-dimensional and two-dimensional fractional wave equations.By the reduction method,we obtain an equivalent system for frac-tional wave equation.And then,using L2-la formula,which was proposed in J.Comput.Phys.280(2015),424-438,we construct the difference schemes for the equivalent system which can achieve second-order accuracy in time and second-order accuracy and fourth-order accuracy,respectively in space.The difference schemes are proved to be unconditionally stable and con-vergent in H1-norm.The difference scheme for three-dimensional problem is also presented.Numerical examples illustrate the efficiency of the schemes.Secondly,we focus on the time multi-term fractional wave equation.An equivalent system is obtained by the reduction method.Then,we present the difference schemes for the time multi-term fractional wave equation.The schemes can achieve second-order accuracy in time,and second-order accuracy and fourth-order accuracy in space,respectively.The two schemes are uniquely solvable and unconditionally stable and convergent in L?-norm with the conver-gence order of O(?2 + h2)and O(?2 + h4),respectively.The numerical examples are shown to demonstrate the numerical accuracy and efficiency of the difference schemes.Thirdly,we investigate the difference schemes for both one-dimensional and two-dimension-al space and time fractional Bloch-Torrey equations.We apply L2-la formula to approximate temporal Caputo derivative.Simultaneously,we discretize Riesz derivative by the fractional center difference operator(Celik,J.Comput.Phys.231(2012),1743-1750)with the approx-imation order of O(h2)and the fourth-order compact operator(Zhao,SIAM J.Sci.Comput.36(2014),A2865-A2886)with approximation order of O(h4),respectively.We determine the lower bound of the tail of the sum of fractional center difference weights,which is the key to proving the stability and convergence of the difference schemes.By the discrete energy method,we show the unconditional stability and convergence of the difference schemes.We also give two ADI schemes for the two-dimensional problem.Two numerical examples are presented to verify the theoretical results.Finally,we consider the numerical methods for the nonlinear time-fractional fourth-order reaction-diffusion equation.We get an equivalent system by the reduction method.The Caputo derivative is approximated by L2-1? formula.Then,a three-level linearized difference scheme is constructed for solving the nonlinear time-fractional fourth-order reaction-diffusion equation.The unconditional stability and convergence in L2-norm are proved by energy method with the convergence order of O(?2+h12+h22).The major difficulty is in the analysis on convergence of the difference scheme.The convergence of the difference scheme is proved by an embedding theorem in multidimensional Sobolev space.A numerical example is given to verify the numerical accuracy and efficiency of the difference scheme.