Operator Splitting Method For Fractional Differential Equations

The history of fractional calculus can goes back to more than three hun-dred years ago, almost the same as classical calculus.Nowadays it has be-come more and more popular among various scientific fields, materials、signal pro-cess、rheology、chemistry、biology、finance et al.The exact solutions of most frac-tional differential equations cannot be expressed by the practical analytical solutions, that is,the analytical solutions for fractional differential equations are special functions of most quite difficult to calculate.This motivates us to develop effective numerical algorithms.This thesis consists of five chapters.In the first chapter, we briefly review the history of fractional calculus, the signifi-cance and background of this thesis,and the previous works about the operator splitting method of fractional differential equations,and the definitions of fractional calculus and existing discrete schemes.In Chapter2,we discuss the practical alternating directions implicit method to solve two-dimensional two-sided space fractional convectional diffusion equation with variable coefficients on a finite domain.We theoretically prove and numerically verify that the present finite difference scheme is unconditionally stable and second order convergent in both space and time directions.Chapter3,we consider the locally one-dimensional numerical methods for effi-ciently solving the N-dimensional fractional differential equations.As concrete exam-ple, we focus on the two-dimensional and three-dimensional two-sided space fractional differential equations with variable coefficients on a finite domain.We theoretically prove and numerically verify that the present finite difference scheme is uncondition-ally stable and second order convergent in both space and time directions.Additionally, some simulations of three-dimensional fractional systems are performed to show the observed physical phenomena and further confirm the effectiveness of the methods.In Chapter4, we discuss the time-space fractional diffusion-wave equation with variable coefficients on a finite domain.The equation is obtained from the classical diffusion-wave equation by replacing the first-order time direction derivative by frac-tional derivative of order β∈(0,1), and the second-order space direction derivative by fractional derivative of order α∈(1,2].Using mathematical induction, we theo-retically prove and numerically verify that the present finite difference scheme is un-conditionally stable and2-β order convergent in time direction, and second order convergent in space direction.Chapter5,N-dimensional fractional wave equation model on a finite domain is studied, and detailedly discusses the two-dimensional and three-dimensional two-sided space fractional wave eqution. Some numerical results show that the second order convergence in both space and time directions.