Effcient Numerical Method For Fractional Partial Differential Equation And Its Parameter Estimation

Fractional calculus is an old and fresh topic.In the early stage,due to the lack of physical mechanism explanation,application background and other rea-sons,fractional calculus developed slowly,it is mainly studied by mathematicians to research the abstract concept.Then,with the development of science and tech-nology,the research has gradually shifted from theory to application,such as the application in high energy physics,anomalous diffusion,biomedical engineering,systems control and the mechanical constitutive relation of complex viscoelastic materials.Recently,researchers are devoted to the solution of fractional partial differential equations,but due to the existence of the differential-integral convo-lution operator in the definition of fractional derivative,it has the property of historically dependent and globally relevant,so it is always difficult to get the analytic solution of the fractional partial differential equations.Also,for the con-structed fractional model,it is of great significance to study the influence of the variation of parameters in the model to analyze the internal physical mechanism and mechanical mechanism of the problem.But,in the existence of parameter estimation of fractional model,the mainly method is 'trial-and-error method'.However,because of data collection and curve fitting process,the approach of parameter identification is usually time consuming.So,it is an interesting re-search to study the efficient numerical method for fractional partial differential equation and based on the numerical method of direct problem to get the param-eter estimation.In this paper,we focus on the efficient numerical method for fractional par-tial differential equation and its parameter estimation.Firstly,for the time-space fractional telegraph equation with periodic boundary conditions,we propose the numerical method which is based on the Crank-Nicolson finite difference scheme in temporal direction while Fourier spectral method in spatial direction,and the stability and convergence analysis are strictly proven.The fast Fourier transform(FFT)technique is applied to practical computation,which improves the comput-ing efficiency greatly.Secondly,we discuss the two dimensional fractional Stokes'first problem for a heated generalized second grade fluid with Dirichlet boundary condition,we propose the method based on the L1 finite difference scheme for the temporal direction while the Legendre spectral method for the spatial direction.Numerical stability and convergence of the method are rigorously established.At the mean time,the case of non-smooth solution is considered by adding some correction terms.Numerical experiments are included,which verify the theoreti-cal predictions.Thirdly,we study the electroosmotic flow of fractional Maxwell fluids,by means of the fractional Maxwell constitutive equation,and based on the experimental data,the nonlinear conjugate gradient method is proposed to get the viscoelastic parameters.Combined with the continuity equations and Cauchy momentum equations,the governing equations of velocity distribution are established.The fully discrete spectral method based on difference method in the temporal direction and Legendre spectral method in the spatial direction is introduced to solve the dimensionless governing equations.Fourthly,on account of the basic structural characteristics of porous coal matrix,by means of the gas adsorption and desorption experiments in coal samples and based on fractional Fick's law,the fractional fractal diffusion model is established to express the gas transport mechanism in heterogeneous coal matrix.The finite difference method in temporal direction while spectral collocation method in spatial direction is proposed to solve the model numerically.Attempts have been made by BFGS method,nonlinear conjugate gradient method,and Bayesian method to compare and contrast to obtain the physical parameters of the model.The results show that the fractional fractal diffusion model can better describe the gas transport process in coal matrix and all three methods can obtain effective parameter esti-mation of the model.Finally,a new numerical simulation technique of time frac-tional Cable equation is developed,the time-space Legendre spectral tau method based on the shifted Legendre polynomial and its operational matrices is used to solve the direct problem.Furthermore,we prove that the approximated solution of this method converges to the exact solution.In addition,the inverse problem is formulated by using the nonlinear conjugate gradient method,the stability and convergence for the inverse problem are provided,which improves the the-oretical framework of the inverse problem for fractional order model parameter estimation.Specifically:In chapter 1,we introduce the development of fractional calculus briefly,and give the definitions of fractional calculus used in this paper.Then,we summarize-the main content of this paper.In chapter 2,we research the numerical solution of the time-space fraction-al telegraph equation with periodic boundary conditions and give its parameter estimation.First,we construct the Crank-Nicolson(CN)finite difference scheme in temporal direction and Fourier spectral method in spatial direction to get the numerical solution.The fast Fourier transform(FFT)technique is applied to prac-tical computation.The stability and convergence analysis are strictly proven.For the inverse problem,we based on the finite difference Fourier spectral method,propose the Levenberg-Marquardt(LM)iterative method for the parameter esti-mation,by adding random perturbations or not to get the numerical simulation of experimental data,we get the estimated parameter of fractional order.Moreover,we give some numerical examples,the results show that the proposed method for direct problem is stable and convergent with 2-? order accuracy in time and spectral accuracy in space,which confirms the theoretical results.And for dif-ferent initial guess and random measurement error,the LM method can get the reasonable estimation,this fact demonstrates that the proposed LM method for parameter estimation is effective.In chapter 3,we consider the numerical solution of the two dimensional fractional Stokes' first problem for a heated generalized second grade fluid with Dirichlet boundary condition.The proposed method is based on the L1 finite difference scheme in the temporal direction while the Legendre spectral method in the spatial direction.Based on the properties of the coefficients which obtained from the approximation of the time fractional derivatives by L1 finite difference scheme,we give numerical stability and convergence analysis of the method rigor-ously.At the mean time,the case of non-smooth solution is considered by adding some correction terms,in which we make correction for low regularity terms to get reasonable numerical solution.Finally,we propose numerical experiments,we compare the errors between numerical solution and exact solution versus time step ? and space polynomial degree N,order and CPU time for different frac-tional order ?,the results demonstrate that the convergence order in temporal direction is 1+?,and it could obtain spectral order in spatial direction,which is in accordance with the theoretical analysis.In chapter 4,we study the electroosmotic flow of fractional Maxwell fluid in a rectangular microchannel.The applications related to non-Newtonian fluid flow through microchannels are mostly associated with the transport of biofluids,with blood as one of the most common example.In this chapter,we based on the viscoelastic moduli experimental data,construct the fractional Maxwell model.The nonlinear conjugate gradient method is proposed to get the estimation of fractional parameters ?,?,relaxation time ? and shear modulus E,in order to get more accurate fits,we propose the log residual error.we find that the fractional Maxwell model encounters a success in the description of the character of viscoelastic flow,and this indicates the nonlinear gradient method is a suitable method for the estimation of the unknown fractional model parameters.Then,we combine the continuity equations and Cauchy momentum equations,establish the governing equations of velocity distribution.And propose fully discrete spectral method based on finite difference method in the temporal direction and Legendre spectral method in the spatial direction to solve the dimensionless governing equations.The results have guiding significance for the design of microfluidic devices and the prediction of fluid viscoelastic behavior in microchannels.In chapter 5,we research the gas adsorption and desorption process in coal samples based on real experiment.The coal particles are adsorbed to methane,and due to the non-uniformity of coal matrix pores,which defer from hundreds of microns to tens of nanometers or even several angstroms in size.In such a porous medium,the diffusion path of gas molecules is distorted and molecules also collide with the wall surface.In addition,in the process of adsorption and desorption,the pressure of the gas changes with time,which causes the change of the mean free path.The free path is related to the diffusion path,so the diffusion process usually presents abnormal diffusion.Based on the basic structural characteristics of porous coal matrix,the fractional fractal diffusion model is established to ex-press the gas transport mechanism in heterogeneous coal matrix.The L1 finite difference method in temporal direction while spectral collocation method in s-patial direction is proposed to solve the model numerically.Then,based on the numerical solution,we get the parameter estimation of fractional order ?,fractal dimension df and structure parameter ? by BFGS method,nonlinear conjugate gradient method,and Bayesian method to compare and contrast to obtain the physical parameters of the model.Furthermore,advantages and limitations of different estimation methods are discussed.The results of this study fit the des-orption process of methane gas in coal particles well,and this research can provide theoretical support for the gas exploitation and safety production in underground coal mines.In chapter 6,we study the new numerical simulation technique for the di-rect and inverse problems of time fractional Cable equation.The time-space Legendre spectral tau method based on the shifted Legendre polynomial and its operational matrices is used to solve the direct problem.Furthermore,we prove that the approximated solution of this method converges to the exact solution.In addition,the inverse problem is formulated by using the nonlinear conjugate gradient method to get the fractional order in the equation,and based on the numerical solution of direct problem,the parameters are estimated by simulation test,and discuss the influence of the initial guess and measurement error for the estimation.The stability and convergence for the inverse problem are provided,which improves the theoretical framework of fractional order parameter estima-tion and give theoretical support for the application of parameter estimation in previous chapters.Finally,some numerical results are carried out to support the theoretical claims,and demonstrate the flexibly character of tau method to deal with the boundary condition.In chapter 7,we give the summary of this dissertation and the future research work prospects.