Analytical And Numerical Methods For Fractional Partial Differential Equation And Its Parameter Estimation Problems

As a novel mathematical tool,fractional calculus theory and the correspond-ing modelling technology have been widely used in physics,chemistry,biology,medicine,finance,control engineering,and so on.Benefiting from the memory and hereditary characters of the fractional calculus operator,fractional constitu-tive models have shown its great advantages in depicting the abnormal phenom-ena related to memory and hereditary,path dependence,holistic correlation,and self-similarity.With the rapid development of the fractional calculus,research papers on this fields have achieved growth spurt.More and more researchers pay their great attention to the analytical and numerical methods for solving the fractional partial differential equations.Based on its widely use in practical problems,fractional constitutive models' parameter estimation become to be an emerging research hot spot.The research on the parameter estimation for integer models has been relatively mature.But in the field of fractional models,param-eters are commonly determined by curve fitting,lacking of specific and feasible parameter inverse methods.Thus,in this paper,we mainly do the research on analytical and numerical methods for fractional partial differential equation and its parameter estimation problems.As for different kinds of fractional models,this paper studies the analyt-ical and numerical methods for the direct problem,as well as the parameter estimation methods for the corresponding inverse problem.Firstly,we propose a novel unstructured mesh finite element method for the time-space fractional wave equation on a two-dimensional irregular convex domain,the stability and conver-gence analysis of the numerical scheme are constructed.Secondly,we deduce a one-dimensional time fractional thermal wave equation with fractional heat flux conditions.The analytical solution is given by integral transformation.The least-squares method,which is given as a specific method for fractional inverse problem is used to estimate both the fractional order and the relaxation time,simultane-ously.Thirdly,based on the generalized fractional element(GFE)network,the Bayesian method is firstly proposed to deal with the fractional inverse problem of parameter estimation.It is demonstrated to be a validity and specific pa-rameter estimation method for fractional inverse problem with both the stability and convergence analysis proved.Fourthly,as for the fractional fractal diffusion model,the finite difference method is used to obtain the numerical solution,and then the Bayesian method is applied to deal with the practical problem.Based on the experimental data from the fast desorption process of methane in coal,the validity of the method is testified.Lastly,we generalize the time fractional derivative model to the multi-term case,the time efficient modified fractional predictor-corrector method is applied to obtain the numerical solution,and the modified hybrid Nelder-Mead simplex search and particle swarm optimization(MH-NMSS-PSO)algorithm is given as another feasible parameter estimation method for fractional parameter estimation inverse problem.Specifically:In chapter 1,we first give a brief introduction to the emersion and devel-opment of the fractional calculus,and present some kinds of definitions of the fractional derivative,which used in this dissertation.Then,we briefly overview the main contents of this dissertation.In chapter 2,as to the time-space fractional wave equation on a two-dimensional irregular convex domain,we develop a novel unstructured mesh finite element method.In time,the Crank-Nicolson scheme is used to discretize the Caputo time fractional derivative,while in space,the novel unstructured mesh Galerkin finite element method is used.The stability and convergence analysis of the unstructured mesh Crank-Nicolson Galerkin scheme are constructed,and the implementation of the scheme is detailed.Numerical results show that the un-structured mesh Crank-Nicolson Galerkin method works well in dealing with the two-dimensional time-space fractional wave equation on an irregular convex do-main.Besides,by the comparison of the unstructured mesh with the structured mesh in the implementation of the numerical scheme,the proposed unstructured mesh finite element method is shown to require a larger computational cost but leads to a better performance with smaller errors of the numerical scheme com-pared with the existing finite element method using the structured meshes.Given that many practical problems involve irregular convex domains,such as the hu-mane heart and brain,which are difficult to partition well with a structured mesh,research on the finite element method designed for an irregular convex domain using a completely unstructured mesh is of great significance.In chapter 3,we study a one-dimensional time fractional thermal wave e-quation with fractional heat flux conditions with Caputo fractional derivatives,and its corresponding parameter estimation problem.Based on the fractional Cattaneo theory,we first deduce the time fractional thermal wave model with fractional heat flux conditions.The analytical solution of the direct problem is obtained by using the fraction Laplace transform and Fourier cosine transform.Then the least-squares method is used to estimate both the fractional order ?and the relaxation time ? simultaneously,based on the measurable data of the inner temperature,which are simulated by the real temperature field and the random errors.Finally,two different heat flux distributions are given as different boundary conditions to perform the simulation experiments,respectively.Nu-merical results show that the least-squares method performs well in parameter estimation for this fractional thermal wave equation.In chapter 4,as for the fractional constitutive equations for viscoelastic ma-terials,we first propose the Bayesian method,which based on the principle of statistics,to conduct the parameter estimation of fractional constitutive models.Based on the analytical solution of the Zener model of viscoelasticity based on the generalized fractional element(GFE)network,the Bayesian method is first-ly proposed to obtain the optimal estimation of the four viscoelastic parameters(?,?,?,?)in the fractional model.Then the validity of the method is demonstrat-ed by three examples.In each example,results obtained by the Bayesian method display an excellent fitting between the calculative values and experimental data.It is shown that the Bayesian method is feasible in the inverse problem of pa-rameter estimation for the fractional constitutive equation,meanwhile,the GFE network Zener model is efficient to describe the stress relaxation behaviour of viscoelastic materials.This research provides a specific and feasible parameter inverse method for estimating parameters of fractional constitutive models.In chapter 5,we mainly consider a problem of parameters estimation for the fractional fractal diffusion model used to describe the anomalous diffusion in porous media.Firstly,the center Box difference method is employed to solve the fractional fractal diffusion equation subject to the initial-boundary conditions.Then based on the direct problem,we apply the Bayesian method to estimate three parameters for the model simultaneously,that is the fractional order ?,the fractal dimension df and the structure parameter ?.Finally,to testify the validity of the used methods,experimental data from the fast desorption process of methane in coal are used.It's shown that the numerical results modeled with parameters given by the Bayesian method coincide well with the desorption experimental data,indicating that the Bayesian method is efficient and valid in identifying multi-parameter for the fractional fractal diffusion model.Meanwhile,by comparison between the classic Fick's model and the fractional fractal diffusion model,the classic Fick's model is shown to be invalid and the fractional fractal diffusion model is demonstrated to be more capable than the classical Fick's model in describing the anomalous diffusion behaviour of gas in coal.Besides,to clarify the parameters' effect on the fractional fractal diffusion model,parameter sensitivity analysis of ??df?? is also carried out.Results show that all the three parameters have important influence on the fast desorption diffusion process of methane in coal,especially in the initial phase.This work provides a specific and convenient parameters estimation method for the fractional fractal diffusion model used to capture the anomalous diffusion behaviour in porous media.In chapter 6,we study the multi-term time fractional differential equation with the Caputo fractional derivative and its parameter estimation.Firstly,the time efficient modified fractional predictor-corrector method is applied to obtain the numerical solution for the direct problem.On this basis,the inverse prob-lem of estimating parameters is conducted by the modified hybrid Nelder-Mead simplex search and particle swarm optimization(MH-NMSS-PSO)algorithm.Fi-nally,to verify the efficiency of the proposed methods,three examples with exper-imental data of viscoelastic materials are simulated.Given the long experiment time taken in practical problem,in the inverse problem of estimating parameter,only the initial part of the experimental data are used in the inverse algorithm,then the estimated parameter values are testified whether suitable for the whole data during the whole experiment process or not.Results show that the esti-mated parameters are also appropriate for the whole experimentation,indicating that the fractional mathematical model is feasible in capturing and predicting the characteristics and behaviors of the real physical and mechanical phenome-na.Besides,the applied numerical method and the parameter inverse method are shown to be efficient in dealing with the parameter inverse method for the multi-term time fractional constitutive equations.In chapter 7,we give the summary of this dissertation and the future research work prospects.