Fast Solutions For Time/Time-Space Fractional Partial Differential Equations

Fractional calculus has been successfully used in many fields such as viscoelastic mechanics,system control,image processing and finance.Unfortunately,only a few fractional partial differential equations can obtain analytical solutions.Thus,numerical methods for solving fractional partial differential equations have been concerned by many researchers.Since fractional differential operators are nonlocal,the discretized systems of fractional partial differential equations are often dense,which greatly reduce the efficiency of traditional methods.Thus,developing efficient and reliable algorithms to solve these systems has great significance.In this dissertation,for the discretized systems of several fractional partial differential equations,their properties are employed for designing fast algorithms.The main contents are as follows:1.Two finite difference schemes are constructed to solve one-and two-dimensional time fractional reaction-diffusion equations with variable coefficients and time drift term,respectively.Then,the stabilities and convergences of them are proved.Based on the structure of the discretized system of the two-dimensional problem,several fast algorithms are proposed.Numerical examples are provided to verify the validity of our numerical schemes and fast algorithms.2.The all-at-once system from the time fractional mobile/immobile advectiondiffusion equation is studied.Based on a finite difference scheme of the equation,the all-at-once system is obtained by stacking all time steps into a column vector.This means that the numerical solutions of the considered equation can be obtained simultaneously by solving such the system.According to the structure of the system,two preconditioners are designed to accelerate the convergence speed of the chosen Krylov subspace methods.Moreover,some properties of them are discussed,such as the nonsingularity.Numerical results are reported to show the effectiveness of our method.3.A finite difference scheme is proposed to solve time-space fractional advectiondiffusion equations,and also its stability and convergence are proved.Moreover,the proposed method is extended to solve nonlinear time-space fractional advection-diffusion equations.In order to fast obtain the numerical solution,suitable Krylov subspace methods are chosen,and a circulant preconditioner is designed to accelerate their convergence.Numerical results show that our fast algorithm is more efficient than direct methods.4.We study the all-at-once system arising from time-space fractional diffusion equations.Based on the structure of the system,we apply two Krylov subspace methods to solve it,and a preconditioner is designed to accelerate the convergence of the chosen Krylov subspace methods.In computing the inverse of the proposed preconditioner,a Toeplitz matrix inversion formula is used and a preconditioner is designed to accelerate it.Numerical results indicate that our fast algorithm for solving such the system is reliable and efficient.