Spectral Methods For Fractional Partial Differential Equations And Its Application

In recent,decades,as a novel mathematical tool,fractional calculus theory has been widely applied to many fields such as physics,chemistry,biology,finance,engineering and so on.Fractional model provides a more profound and compre-hensive explanation of memory,heredity,non-locality and path dependence in complex environment.However,the complexity and non-locality of fractional operators bring many difficulties to the solution of fractional models.Numerical methods for solving fractional model are becoming more and more important.Many scholars have studied the numerical solutions of fractional models.Spec-tral method is a numerical method for solving partial differential equations and it has the characteristics of high efficiency and high accuracy.However,due to the particularity of spectral method's requirement for basis function and initial-boundary conditions,the research on solving fractional partial differential equa-tions by spectral method is relatively few.In addition,parameter estimation for integer-order model has been relatively mature,but.the parameter estimation of fractional models still lacks a relatively feasible method.In this paper,we mainly study the spectral methods and parameter estimation of several fractional partial differential equations and their related applications.In this paper,we first propose a space-time spectral method for solving the one-dimensional time-space fractional Fokker-Planck equation.Stability and convergence analysis are given.Moreover,we use Levenberg-Marquardt(L-M)method to estimate some relevant parameters.Secondly,for the two-dimensional Riesz space distributed-order advection-diffusion equation,the Gauss quadrature formula with higher accuracy than the midpoint formula is proposed to discretize the space distributed-order derivative.The numerical solution is obtained by Crank-Nicolson alternating direction implicit Legendre spectral method.The stability and convergence analysis of the semi-discrete scheme and the fully dis-crete scheme are proved.Thirdly.we study the one-dimensional nonlinear cou-pled space fractional Schrodinger equation.The numerical solution by Legendre spectral method is derived and the corresponding theoretical analysis is given.Based on the numerical solution of the direct problem,we propose the Bayesian method to estimate the releVant parameters.Fourthly,we give the Fourier spec-tral method for the one-dimensional time fractional Boussinesq equation.The stability and convergence analysis of the numerical method are proved.Fifthly,for the high-dimensional nonlinear partial differential equations,tit requires certain time-step restrictions in theoretical analysis.To solve this problem,we study the spectral method of two-dimensional nonlinear time fractional mobile/immobile adovection-diffusion equation.Based on the error splitting merhod,We obtain the error estimate without any time step size.A fast method is applied to decrease the memory requirement and computational cost.We introduce the correction method to deal with the non-smooth solutions case.Finally,we develop a stable second-order semi-implicit Fourier spectral method for for the two-dimensional nonlinear reaction-diffusion equation with space described by the fractional Lapla-cian.The optimal error estimate of the numerical scheme is obtained by using the time-space error splitting technique.The linear stability of the semi-implicit method is analyzed and a practical criterion for the time step is obtained.Con-cretely,it,concludes:In Chapter 1,we briefly introduce the development of fractional calculus.and give some definitions of fractional derivatives used in this paper.Then,a brief overview of the main research content of this paper is presented.In Chapter 2,we propose a space-time spectral method for the one-dimensional time-space fractional Fokker-Planck equation.The Jacobi polynomial is used for time discretization and the Legendre polynomial is used for space discretization.The stability and convergence analysis of the numerical scheme are proved.We give the detailed numerical implementation process.Moreover,the L-M method is presented to estimate the order ? of time fractional derivative and the order 2? of space fractional derivative.The numerical examples show the error and convergence order of time and space in the different norms.The numerical so-lution agrees well with the analytical solution.It implies that the space-time spectral method is effective for solving the one-dimensional time-space fractional Fokker-Planck equation.In order to verify the validity of L-M method,we derive the influence of the initial parameters and different level noise-contaminated data on the estimated results.It is found that different level noise-contaminated data and different initial parameters have little impact on the estimated results.which illustrates that L-M method is feasible for the parameter estimation.In Chapter 3,we study the two-dimensional Riesz space distributed-order advection-diffusion equation.The Gauss quadrature has a higher computation-al accuracy than the mid-point quadrature rule is proposed to approximate the distributed order Riesz space derivative so that the considered equation is trans-formed into a multi-term space fractional equation.The numerical solution is obtained by Crank-Nicolson alternating direction implicit Legendre spectral method.The Crank-Nicolson difference method is used to discretize the time and the Legendre spectral method is used to discretize the space.Stability and convergence analysis of the semi-discrete scheme and the fully discrete scheme are verified.Finally,we give two numerical examples.The first numerical example shows the convergence order of the numerical scheme.The comparison between the numerical solution and the analytical solution is also developed.The above results demonstrate the effectiveness of the numerical umethod.The accuracy and convergence order between the Gauss quadrature formula and the mid-point for-mula are compared,which proves that.the Gauss quadrature formula is superior to the mid-point formula.Based on the relevant research background,the second numerical example is provided.We discuss the influence of relevant coefficients on the solution.The difference and relationship between Riesz space distributed-order advection-diffusion equation and Riesz space fractional advection-diffusion equation are displayed.In Chapter 4,we develop the spectral method for the one-dimensional nonlin-ear coupled space fractional Schrodinger equation.The numerical implementation process is given.Crank-Nicolson difference method is used to discretize the time and Legendre spectral method is used to approximate the space.The convergence and conservation laws analysis of numerical scheme are proved.Based on the nu-merical solution,we use the Bayesian method to estimate the order a of space fractional derivative,the coefficients p and ? of nonlinear term.Finally,three numerical examples are derived.The convergence order of t.he numerical scheme is given by the first numerical example.The impact of different initial parameters and different maximum iterations on the estimated results is developed.It can also be shown that different initial parameters have little effect on the estimated results.As the maximum iterations become larger,the estimated results become more accurate and the errors become smaller.Therefore,the numerical method and the Bayesian method are valid.The second numerical example describes the properties of numerical solutions,and discusses the application of the model.The third numerical example further demonstrates the feasibility of the numerical scheme by adding a source term to the model.In Chapter 5,we consider the one-dimensional time fractional Boussinesq equation with periodic boundary conditions,which is usually used to describe the surface water waves whose horizontal scale is much larger than the depth of the water.L2 method is used to discretize the time fractional derivative and Fourier spectral method is given to approximate the space.The stability and convergence analysis are verified.Finally,two numerical examples are provided to confirm the theoretical analysis.The first numerical example presents the error.convergence order and CPU time of the numerical scheme.The numerical solution is also in good agreement with the analytical solution.The above results confirm the effectiveness of the proposed spectral method.The second numerical example shows the properties of the numerical solution and the influence of parameters on the solution.It further indicates that the proposed nuumerical method is effective.In Chapter 6,due to the existence of nonlinear term,the usual theoreti-cal analysis may cause certain time-step restrictions dependent on the spatial mesh size for high-dimensional nonlinear partial differential equations.The spec-tral method for the two-dimensional nonlinear time fractional mobile/immobile advection-diffusion equation is studied.We assume that the initial condition of the equation is zero(if the non-zero initial problem is encountered,we can make a transformation).So the time Caputo fractional derivative is equivalen-t to Riemann-Liouville fractional derivative.We use the weighted and shifted Griinwald-Letnikov difference method to discretize the time fractional derivative.This method can raise the convergence order in the time to the second order.The Legendre spectral method is used in the space,and the non-homogeneous boundary conditions are considered.In order to solve the high-dimensional equa-tions and long-time computation,we propose a novel fast method to reduce the storage space and computation time in the numerical implementation process.In addition,in terms of the temporal-spatial error splitting argument technique,an optimal error estimate of the numerical scheme is obtained unconditionally without any time-step size conditions.The time fractional partial differential equation often has singularity at t = 0,and the regularity of the solution is poor.We deal with this situation by the correction method.Finally,three numerical examples are presented.The first and the second numerical examples has homo-geneous and inhomogeneous boundary conditions,respectively.The analytical solutions are smooth.We give the convergence order and error.The difference in computing time between the fast method and the direct method is shown,as well as the error between the two methods.The results illustrate the validity of the numerical method and the fast method.The third numerical example develops the non-smooth solution.The accuracy and convergence order of adding different correction terms are displayed.It proves that the correction terms is feasible.In Chapter 7.we study the stabilized second-order semi-implicit Fourier spec-tral method for the two-dimensional nonlinear reaction-diffusion equation with space described by the fractional Laplacian.Semi-implicit second-order difference scheme is used in time direction and second-order stabilization term is added to improve stability.Fourier spectral method is applied in space direction.Based on time-space error splitting technique,an optimal error estimate of numerical schemes is obtained without any time-step size conditions.We also analyze the linear stability of the stabilized semi-implicit method and obtain a practical cri-terion to choose the time step size to guarantee the stability.Our approach is illustrated by solving several problems of practical interest,including the frac-tional Allen-Cahn,Gray-Scott and FitzHugh-Nagumo models.Finally,three nu-merical examples are presented.The first numerical example gives the error and convergence order of the numerical scheme,which verifies the effectiveness of the proposed spectral method.The second and third numerical examples consider the space fractional Gray-Scott model and the space fractional FitzHugh-Nagumo model,respectively.The properties of the numerical solution are given and the applications of the models are discussed.In Chapter 8,we give our conclusions and some possible research work in the future.