Numerical Methods And Preconditioning Techniques For Several Classes Of Fractional Differential Equations

Fractional calculus are the extension of the traditional integral order calculus theory,which originated from some guesses of Leibniz and Euler and developed till now.Because fractional differential operators are nonlocal,they provide powerful tools for describing materials with memory functions and genetic properties in the real world.Therefore,they are widely used in the fields such as fluid mechanics,viscoelastic mechanics,anomalous diffusion,electromagnetism,signal processing and system recognition,modern control theory and so on.With the development of applications of fractional differential equations,to achieve the analytic solutions is still the primary goal.However,in general,obtained the analytic solution of fractional differential equations is very difficult.Even the analytic solutions of linear fractional differential equations contain some special functions,such as Mittag-Leffler function,Wright function,Hypergeometric function and so on,with slow convergence.Therefore,more and more scholars pay high attention to construct efficient numerical methods for fractional differential equations.In view of this,in the current thesis,we will focus on numerical methods and preconditioning techniques for several classes of fractional differential equations.In Chapter 1,we briefly introduce the historical background and recent research situation of the fractional differential equations.Then,the main work of this thesis is also presented.In Chapter 2,we consider the extended one-leg methods for a class of nonlinear stiff fractional differential equations with Caputo derivatives.It is proved under some suitable conditions that the extended one-leg methods are convergent and stable.Several numerical examples are given to illustrate the computational efficiency and accuracy of the methods.In Chapter 3,a type of extended boundary value methods for nonlinear stiff fractional differential equations with Caputo derivatives are derived.The local stability,unique solvability and convergence of the methods are studied.By performing several numerical examples,the computational efficiency,accuracy and comparability of the methods are further illustrated.In Chapter 4,a class of extended block boundary value methods for solving the nonlinear fractional differential equations with Caputo derivatives are obtained.It is proved under some appropriate conditions that the induced methods are convergent and globally stable.Several numerical examples are given to illustrate the theoretical results and the computational effectiveness and accuracy of the methods.In Chapter 5,a class of quasi-compact boundary value methods are constructed for solving space-fractional diffusion equations.In order to accelerate the convergence rate of this class of methods,the Kronecker product splitting(KPS)iteration method and the generalized minimal residual(GMRES)method with KPS preconditioner are proposed.Numerical experiment further illustrates the computational efficiency and accuracy of the proposed methods.Moreover,a numerical comparison with the GMRES method with Strang-type preconditioner is given,which shows that the GMRES method with KPS preconditioner is comparable in computational efficiency.In Chapter 6,an implicit difference scheme is derived for solving a class of twodimensional time-space fractional convection-diffusion equations.It is proved under some suitable conditions that the derived difference scheme is stable and convergent.By combining the KPS preconditioner with the GMRES method,a preconditioning strategy for implementing the difference scheme is introduced to accelerate the convergence rate.Finally,several numerical examples are presented to illustrate the computational accuracy and efficiency of the methods.In Chapter 7,an implicit finite difference method is presented to solve a class of twodimensional time-space fractional Fokker-Planck equation with a nonlinear source term.The convergence and stability of the numerical method are analyzed,and the convergence order is shown to be second-order accuracy in both time and space.Similar to Chapter 6,a preconditioning strategy is proposed by making use of the GMRES method with the KPS preconditioner to accelerate the convergence rate.At last,several numerical examples are given to illustrate the theoretical results.In Chapter 8,a brief conclusion is given and some future work is stated.