Finite Difference Methods For Some Fractional Anomalous Diffusion Equations

As the generalization of the classical calculus,fractional calculus is recognized a useful tool for describing hereditary materials and processes.In recent years,it has become an important issue for solving the fractional differential equations with the establishment and application of numerous fractional anomalous diffusion models.In most instances,analytical solutions are not readily available,and even impossibly.Thus it is imperative to develop numerical algorithms for approximate solutions.However,because of non-locality of fractional derivatives,numerically solving fractional partial differential equations usually causes the trouble of excessive memory storage and expensive computational cost.To overcome the hurdles,it is effective to develop high accuracy numerical methods,since the storage and computational cost can be reduced.On the other hand,for the high-dimensional fractional differential equations which is of widespread use in modeling,by applying the technique of alternating direction implicit(ADI)decomposition,the efficiency can also be improved effectively.This thesis focuses on three kinds of fractional models which describe the process of anomalous diffusion.The aim of this thesis is to study the efficient numerical methods for high-dimensional problems and investigate highly accurate numerical algorithms.The main work of this thesis contains the following four parts.The first part studies the numerical methods for solving the two-and three-dimensional time-fractional wave equation of distributed-order with a nonlinear source term.For each case,based on the composite mid-point quadrature rule and composite two-point Gauss-Legendre quadrature rule,two finite difference schemes are established.The unique solvability,unconditional stability and convergence of the difference schemes are theoretically proved.To resolve the difficulty of large computational cost and its rapid increase,the technique of ADI decomposition is also adopted.Finally,numerical experiments are carried out to verify the effectiveness and accuracy of the algorithms for both the two-and three-dimensional cases.In the second part,the high accuracy numerical methods for one-dimensional time-fractional diffusion equation of distributed-order are presented.Based on the composite Simpson quadrature rule and composite two-point Gauss-Legendre quadrature rule,two high-order finite difference schemes are proposed.The presented two schemes both have 3rd-order convergence rate in the temporal direction,and 4th-order convergence rate in distributed-order and spatial directions,respectively.The unique solvability,unconditional stability and convergence of the difference schemes are theoretically proved.Such numerical stability and high-order convergence rates are verified by numerical examples.In the third part,compact finite difference schemes for the backward fractional FeynmanKac equation with fractional substantial derivative are proposed.Based on the newly developed approximation operators for fractional substantial derivative,and the compact difference operator applied to space derivative,the compact finite difference schemes are established.The proposed schemes have the q-th(q=1,2,3,4)order accuracy in temporal direction and 4th-order accuracy in spatial direction,respectively.The numerical stability and convergence in the maximum norm are proved for the 1st-order time discretization scheme by the discrete energy method,where an inner product in complex space is introduced.Finally,extensive numerical experiments are carried out to verify the availability of the algorithms.Also,by using the algorithm of numerical inversion of Laplace transforms and composite trapezoidal formula,simulations of the backward fractional Feynman-Kac equation with Dirac delta function as the initial condition are performed.In the last part,two ADI difference schemes for solving three-dimensional space fractional advection diffusion equation are presented,i.e.,predictor-corrector scheme and Douglas-Gunn scheme.The derived schemes have the 2nd-order accuracy in both time and space directions,respectively.As the generalization of the ADI schemes for solving integer order parabolic equations,the necessary analysis of stability and convergence are carried out by the matrix method.The efficiency of the two algorithms are demonstrated by some numerical examples finally.