Numerical Solution For Fractional Partial Differential Equations And Its Applications In Magnetohydrodynamics

As an extension of classical calculus,fractional calculus is more and more used in the fields of hydrodynamics,electromagnetism,system control,viscoelastic materials,electrochemistry,biological population and so on.The fractional operator can successfully describe various materials and processes with properties such as memory,non-locality and heredity,then it has become a powerful tool for mathematical modeling of complicated material science and physical processes.Considering the non-locality and complexity of fractional operators,the analytical solutions of fractional models are difficult to be given explicitly,so the numerical solution of fractional models is urgently important.Parameter estimation is an important inverse problem in the research of fractional models.At present,some methods have been developed to estimate unknown parameters in fractional models,but there are still relatively few.Efficient and feasible methods for parameter estimation also need to be further explored.In this paper,we mainly study the numerical solutions and parameter estimation of several types of fractional partial differential equations.Moreover,we introduce fractional calculus into magnetohydrodynamics(MHD),then we construct the fractional coupling model and study the numerical simulation of complex magnetic fluid.In this paper,firstly we establish a time fractional heat transfer model to describe the heat transfer during the gas adsorption of coalbed methane.We also give the numerical format based on the finite difference and spectral collocation.Moreover,the Cuckoo Search(CS)algorithm is applied to the parameter estimation of fractional models for the first time.Combined with the experimental heat flow data,the problem of parameter estimation on the time fractional heat transfer model is solved.Secondly,we develop the finite difference and Laguerre-Legendre spectral method for the generalized Oldroyd-B fluid in two-dimensional semi-infinite domain.This method can be solved the problem directly in the whole region without adding artificial boundary,which avoids the error caused by truncating the region.Moreover,we provide the stability and convergence of the numerical method.Thirdly,focusing on the unsteady space fractional MHD free convection and heat transfer of a viscous incompressible conductive fluid with Hall effects,we establish a space fractional MHD flow and heat transfer model,which is coupled by the incompressible space fractional Navier-Stokes equation and the heat conduction equation.Considering the incompressible constraints of the velocity and pressure,we propose a finite difference spectral decomposition method based on the pressure correction algorithm to solve the model.Then we successfully simulate the space fractional MHD flow and heat transfer in a two-dimensional closed square cavity.Finally,for the unsteady flow and heat transfer of the generalized second-order fluid in porous media,we establish a new time fractional MHD flow and heat transfer coupled model.We develop the finite difference Legendre spectral method to solve this model,then prove its stability and convergence.Furthermore,a fast algorithm for saving memory and computing time is proposed and the convergence of the fast method is proved.This paper goes as follows.In Chapter 1,we briefly introduce the research background of fractional calculus.Then,we give the specific definitions of several fractional operators used in this paper.Then,we introduce the main results of this paper.Finally,some definitions of common spaces and norms are given.In Chapter 2,we study the parameter estimation on the fractional heat conduction model in coalbed methane.Aiming at the heat transfer during the gas adsorption of coalbed methane,we establish a time fractional heat conduction model combined with the fractional Fourier law.We use the standard Grünwald-Letnikov formula in the time direction and use the Legendre spectral collocation method in the space direction.Then we give the fully discrete numerical scheme and the calculation formula of heat flow Q(t)on the surface of coal matrix.For the model parameters:fractional order α,adsorption parameter G and adsorption rate constant v,we use the CS algorithm to estimate them for the first time.By the experimental heat flow data,we successfully obtain the explicit values of three parameters.In order to verify the stability of the CS algorithm for solving the inverse problem of fractional models,we discuss the influence of the maximum number of iterations Niter,discovery probability p and nest number n on the results of parameter estimation.It turns out that the results vary little as the values of these parameters change,which indicates that the CS algorithm is feasible and effective for the parameter estimation on fractional models.Finally,we analyze the sensitivity of the parameters in the time fractional heat conduction model.It can be found that all of the three parameters have significant impacts on the time fractional heat conduction model.In Chapter 3,we study the numerical solution of the two-dimensional generalized Oldroyd-B model in a semi-infinite domain.For the time direction with multiple fractional derivatives,we use the second-order θ scheme combined with the weighted and shifted Grünwald-Letnikov formula to discretize it.For the case that the space is unbounded in one direction and bounded in the another direction,we propose a composite LaguerreLegendre spectral scheme in the whole region.In addition,we prove the stability and convergence theorems of the numerical scheme and we give a detailed numerical implementation process.Then we give two examples with different initial values to illustrate the effectiveness of the numerical method.Both examples show that the numerical method achieves second-order accuracy in the time direction and does not change with the values of the parameters θ,α,β.In the space direction,the error decays exponentially in both directions.According to the listed CPU time,the developed numerical method is very fast to solve the fractional model in the semi-infinite domain.The numerical method in this chapter has high accuracy and less calculation time,so it can be extended to solve other viscoelastic fluid models and time fractional partial differential equations in unbounded domain.In Chapter 4,we study the numerical solution of the unsteady space fractional MHD free convection and heat transfer for a viscous incompressible conductive fluid.Firstly,considering the Hall effects,viscous dissipation and Joule heat,we establish a space fractional MHD flow and heat transfer model in a two-dimensional closed square cavity.This model is coupled by the incompressible space fractional Navier-Stokes equation and the heat conduction equation.Due to the incompressible constraint between pressure and velocity in the model,it is difficult to be calculated directly.Therefore,we use the pressure correction algorithm to decouple the velocity and pressure in the momentum equation.Then,in the space direction with the fractional Laplace operator,we propose the spectral decomposition method for discretization.In the time direction,we use the second-order semi-implicit difference scheme to discretize and use extrapolation to deal with the nonlinear terms in the model,so as to give a fully discrete scheme.Using this numerical method,we successfully simulate the space fractional MHD flow and heat transfer in a two-dimensional closed square cavity.We obtain the fluid field and the temperature field changing with time.In addition,we give the profiles of velocity u,v and the trends of the average Nusselt number on the cold wall under different values of the Hartmann number Ha,the fractional order α,the Hall parameter m and the Reynolds number Re.According to results,the effects of relevant parameters in the model on fractional MHD flow and heat transfer are described in detail.In Chapter 5,we study the numerical solution for the fractional MHD flow and heat transfer coupled model of the generalized second-order fluid.For the unsteady flow and heat transfer of the generalized second-order fluid in porous media,a fractional MHD coupled model is established by combining with the fractional constitutive relation and the generalized Fourier law.We use the second order fractional backward difference formula(FBDF2)to discretize the time direction and Legendre spectral method to discretize the space direction,then the fully discrete numerical scheme is obtained.We prove the stability and convergence of the obtained fully discrete scheme.In order to reduce the computational time and memory requirements,we further develop a fast algorithm and strictly prove the convergence of the fast algorithm.In the numerical experiment,we give the specific numerical implementation process and verify the effectiveness of the numerical method through an example.The results show that the scheme has second-order accuracy in the time direction and spectral accuracy in the space direction.In addition,the developed fast algorithm can effectively save the calculation time and does not increase the extra error.Finally,we successfully simulate the unsteady fractional MHD flow and heat transfer of generalized second order fluid in porous media near a vertical infinite plate.According to the numerical results,the effects of relevant parameters in the model on velocity u,v,temperature T are discussed and the mechanism is analyzed in detail.In Chapter 6,we summarize the content of this paper and give the research proposal in the future.