Spectral Method For Several Types Of Time Fractional Partial Differential Equations And Its Applications

As a new mat.hematical tool,fractional calculus has been widely used in many fields of engineering technology and real life,including physics,chemistry,biol-ogy,economics,etc.Fractional models can simulate some complex transmissions and diffusion mechanisms more accurately and effectively.However,due to the fact that fractional operators have memory in time and nonlocality in space,it also increases the difficulty of solving fractional models and the complexity of simulation.Many experts and scholars have accomplished a series of work in the numerical solutions of fractional partial differential equations.At present,the numerical simulation of fractional models has become mature.As one of the numerical methods,spectral method has the characteristics of high efficiency and high precision.However,due to the particularity of the spectral method for the basis functions and initial boundary conditions,there are relatively few studies on using spectral method to solve fractional partial differential equations.The main purpose of this paper is to combine fractional calculus theory,spectral method,finite difference method,Bayesian method and so on to solve several kinds of time-fractional models,provide theoretical analysis and esti-mate parameters,such as one-dimensional time-fractional Navier-Stokes equa-tion,one-dimensional time-fractional dual-phase-lag heat,conduction model,two-dimensional distributed-order time-fractional Cable equation,nonlinear two-dim-ensional time-fractional anomalous diffusion equation,two-dimensional nonlinear time-fractional advection-diffusion equation.In this paper,the finite difference method and spectral method are used to study the numerical solutions of the above-mentioned time-fractional partial differential equations,and the related theoretical analysis and applications are completed.The Bayesian method is firstly introduced into the parameter estimation of time-fractional dual-phase-lag model to estimate the parameters of the model and analyze the properties of them.For the high-dimensional model,the matrix diagonalization technology is employed into the numerical implementation of the algorithms.Compared with the directmethod,it improves the operation efficiency and reduces the storage requirement.In the theoretical analysis of high-dimensional nonlinear problems,the construction of spatial grids often limits the selection of time step,which is sometimes too strict,resulting in unnecessary small time step,increasing the de-mand of operation time and storage space.In view of this situation,we develop a new inequality,which weakens the limitation of space grids on time step in the process of theoretical analysis.The singularity of fractional operators lead to the singularity of solutions of fractional equations,and the accuracy is often reduced when numerical approximation is made to such nonlocal models.With the help of two different numerical methods,we analyze the two-dimensional nonlinear time-fractional advection diffusion equation with non-smooth solutions,and ob-tain high-precision numerical results.The organizational structure of this paper is as follows:In Chapter 1,we briefly introduce the generation and development of fractional calculus,and provide the relevant definitions and symbols to be used in the following chapters.Then,the main research contents of each chapter are briefly described.In Chapter 2,the numerical method for one-dimensional time-fractional Navier-Stokes equation satisfying periodic boundary conditions is considered.This model can be used in the study of turbulent and inhomogeneous medium flow,viscoelas-tic and electromagnetic theory.In time direction,L1 finite difference method is used to approximate Caputo fractional derivative with piecewise linear interpo-lation.In spatial direction,Fourier spectral method is employed for numerical approximation.The stability and convergence of the proposed method are proved.Finally,a numerical example is given to verify the theoretical analysis.When the parameters are different,the errors,convergence orders and CPU time of the time and space directions are given in the tables.Figures show that the numerical so-lution is in good agreement with the exact solution,which shows the proposed method is effective.In Chapter 3,the Galerkin-Legendre spectral method for one-dimensional time-fractional dual-phase-lag heat conduction model and its parameter estima-tion are introduced.After a brief description of the conduction of the model,we discuss the model in two aspects.Firstly,we consider the numerical method.The weighted and shifted Grunwald approximation is used in the time direction,which can achieve the second-order accuracy;in the space direction,the Legen-dre polynomials are used,which can achieve the spectral accuracy.We present the corresponding stability and convergence analysis.Then,we introduce the parameter estimation based on the full discrete scheme of the direct problem,and propose the Bayesian method to estimate the four parameters of the model,namely,orders of two time-fractional derivatives α and β,the delay time τT and the relaxation time τq.The main advantage of this method is that itcan esti-mate multiple parameters at the same time and can be used to deal with nonlinear problems.Finally,numerical examples are given to verify the correctness of theo-retical analysis and the effectiveness of the numerical methods.In the numerical experiments,the comparison between the exact solution and the numerical solu-tion is given,and it can be seen that they are in good agreement.The data in tables show the second order convergence in time direction.In addition,com-pared with other numerical schemes,we can see that the method in this chapter can achieve higher convergence accuracy through fewer grid points,and the op-eration time is relatively short.In the analysis of parameters,we verify that the results of parameter estimation are stable to the initial values and the times of iterations,and the estimation results are not affected by small disturbances.In Chapter 4,the distributed-order models have more general properties,so they are more suitable to describe complex dynamical systems than classical models and fractional models.Distributed-order time-fractional partial differen-tial equations are especially effective when considering local phenomena.In this chapter,for the two-dimensional distributed-order time-fractional Cable equa-tion,the midpoint formula is used to approximate the Riemann-Liouville time-fractional derivative of the distributed-order,so that the considered equation can be transformed into a multi-term time-fractional equation,and then the equa-tion can be solved numerically.In temporal direction,we use the weighted and shifted Griinwald approximation,in the spatial direction,we use Legendre spec-tral method.In the algorithm implementation,the detailed matrix operation form is given,and matrix diagonalization technology is proposed to solve the high-dimensional difficulty and algorithm complexity.The algorithm implemen-tation process of the two methods are given.Finally,the effectiveness of the proposed method is verified by numerical examples.From the results of numeri-cal examples,we can see that our method achieves the second-order convergence in the time direction,the second-order convergence in the approximate scheme for distributed-order fractional derivatives,and the spectral accuracy in the space direction.In Chapter 5,anomalous diffusion models can produce various kinds of models,which are widely used in science and engineering,and can be used to describe the heterogeneity in fractal porous media.In this chapter,the generalized two-dimensional nonlinear time-fractional anomalous diffusion model is analyzed.We use the L1/Fourier spectral method to solve the equation.Based on the frac-tional type Gronwall inequality,the stability and convergence of the numerical scheme are analyzed.In the implementation of the algorithm,based on the two-dimensional fast Fourier transform,we give a detailed algorithm,and use the implicit iterative method to approximate the nonlinear term.Finally,several numerical examples verify the effectiveness of the method.In Chapter 6,two numerical methods for solving nonlinear two-dimensional time-fractional advection diffusion equations with non-smooth solutions are in-troduced.In the first method,FCN method is used in time direction.In the case of non-smooth solution,correction terms are introduced.In the second method,we use L2-1θ finite difference method on graded meshes in temporal direc-tion.In spatial direction,Legendre spectral method is employed.Based on two different fractional type Gronwall inequalities,the stability and convergence of the two numerical methods are proved theoretically.In addition,in the analysis of high-dimensional nonlinear problems,the construction of spatial grids often limits the selection of time step,which is sometimes too strict.In view of this situation,we develop a new inequality to weaken the limit of time step.Finally,numerical experiments are carried out with two numerical schemes to verify the effectiveness of the theoretical analysis and numerical methods in this chapter.In Chapter 7,we give the summary of this thesis and the possible research directions for the future work.