Numerical Solution For Fractional Partial Differential Equations And Its Applications In Fractional Magnetohydrodynamic Coupled Models

In this paper,we mainly study the efficient numerical solutions of several kinds of fractional partial differential equations and their applications in the magnetohydrodynamic coupled fractional model.The finite difference method and spectral method are combined to establish numerical solutions for several kinds of fractional partial differential equations.The time-space error splitting technique is first applied to the high-dimensional complex magnetohydrodynamic coupled fractional model.It is proved that the numerical scheme is stable without applying the condition of Courant Friedrichs Lewy(CFL),and the optimal error estimate is given.In order to reduce the amount of storage and computation,a fast algorithm of time fractional derivative based on exponential sum rule is proposed for the magnetohydrodynamic coupled fractional model.For the singularity problem of the time fractional order complex magnetohydrodynamic model at t=0,a correction method and a graded mesh method are proposed,and Cuckoo Search(CS)algorithm is used to estimate the parameters in the equation.For the first time,the regularity compensation oscillation(RCO)technique is applied to the long-term dynamic model of space fractional complex magnetic fluid,which breaks through the convergence proof of improved uniform error.Specifically:In Chapter 1,first,we introduce the research background of fractional calculus,and give the specific definitions of several fractional calculus operators used in this paper.Secondly,the main research contents of this paper are introduced.In Chapter 2,spectral method is first used to solve the nonlinear coupled spatial fractional equations of complex magnetohydrodynamics.Based on the Crank-Nicolson scheme,the central difference scheme and the Fourier spectral method,an effective numerical method for the nonlinear coupled space fractional Klein-Gordon-Schr?dinger(NCSFKGS)equation in complex magnetohydrodynamics are studied.In Chapter 3,the linearized L1-Legendre-Galerkin spectral method for solving twodimensional nonlinear time fractional advection diffusion equation is studied.The error splitting technique is first combined with discrete fractional Gronwall inequality to prove that the numerical method is stable without the CFL condition.We also deal with the case of non-smooth solutions by adding some corrections.In Chapter 4,to reduce the amount of storage and computation,a fast time fractional derivative discrete algorithm based on the SOE rule is proposed for the two-dimensional nonlinear coupled time fractional Schr?dinger equation in complex magnetic fluid,and the CS algorithm is used to estimate the parameters in the equation.We derive an L1Legendre-Galerkin spectral method.Combining the error splitting technique and discrete fractional Gronwall inequality,we strictly prove that the numerical method is stable without CFL conditions.At the same time,we use the fully implicit method to deal with the nonlinear term.For the non-smooth solution,we adopt the graded mesh method.In Chapter 5,we study the stabilized second order scheme of the time fractional Allen-Cahn equation.The scheme uses the fractional backward difference formula(FBDF)to approximate the time fractional derivative,and uses the Legendre spectral method as the spatial approximation.The nonlinear term is implicitly treated by the secondorder stability term.Based on the fractional Gronwall inequality,we strictly prove that the proposed scheme has second-order convergence accuracy in time and spectral accuracy in space.In order to save computing time and storage,we developed a fast algorithm.Finally,some numerical examples are given to show the evolution of the phase field,and the effectiveness of the proposed method is verified.In Chapter 6,a fast algorithm to solve the the two-dimensional nonlinear coupled time-space fractional Klein-Gordon-Zakharov(KGZ)equations in magnetic fluid is developed.The L2-1σ method based on an SOE rule and a Fourier spectral method are used to approximate the time and space direction,respectively.And we use the previous time levels to deal with the nonlinear terms to obtain a linearized numerical scheme,which provides a powerful tool for solving the numerical simulation of practical problems.In Chapter 7,the RCO technique is first applied to the long time dynamic model of spatial fractional complex magnetic fluid.The improved uniform error convergence analysis of the exponential wave integral method for the spatial fractional nonlinear space fractional Klein-Gordon equation(NSFKGE)with weak cubic nonlinearity in complex magnetohydrodynamics is established,which breaks through the convergence proof of the improved uniform error.First,the NSFKGE is semi-discretized in time by trapezoidal quadrature,and then the complete discretization scheme is derived by the spatial Fourier spectral method.By using the RCO technique,the improved uniform error bounds for long time second order semi discretization and full discretization are obtained respectively.The complex valued NSFKGE case with general nonlinearity is also discussed.Finally,numerical results are given to prove the effectiveness of our numerical method.This provides a new method for the theoretical proof of long time dynamic models of other complex magnetic fluids.In Chapter 8,we summarize the content of this paper and give the research proposal in the future.