Conservative Numerical Methods For Some Fractional Partial Differential Equations

Fractional partial differential equation is an extension of integer partial differential equation.In recent two decades,with its unique properties in describing hereditary property and memory phenomenon,fractional partial differential equation has been widely utilized in anomalous diffusion,viscoelastic mechanics,quantum mechanics,plasma,system identification and so on.However,in general,the exact solution of fractional partial differential equation can not be obtained,or it may contain some special functions that are too complex to calculate,such as Mittag-Leffler function,Wright function,hypergeometric function and so on,which hinders greatly the practical application of fractional partial differential equation.Therefore,seeking numerical solutions of fractional partial differential equation has important significance for theoretical and practical application.The thesis mainly studies the initial boundary value problem in several kinds of fractional partial differential equation in physics,and provides new numerical methods and analyzes relative properties of numerical solutions.It is worth mentioning that these classical problems in physics often have conserved quantities.Keeping these conserved characteristics in the process of constructing numerical methods will greatly improve the accuracy and effectiveness of numerical methods.As a starting point,the main work of this thesis is as followsIn Chapter 1,the development of fractional partial differential equation and the significance of its study are introduced.The physics background and research status of equations the thesis studied is presented.At the end of this chapter,the main contents of the thesis are concluded.In Chapter 2,under the Dirichlet boundary condition,high-order conservation difference scheme of space fractional Klein-Gordon-Schr(?)inger equation is given.The scheme is firstly verified to satisfy the conservation laws of charge and energy.With the discrete energy method,the priori estimate and maximum norm convergence of the above numerical scheme are analyzed.Finally,the convergence and conservative of the scheme is verified with numerical examples.In Chapter 3,the Zakharov equation of double fractional order is considered.It is proved that the studied equation has two conserved quantities.The self-closed three layer linear difference scheme is established and its ability of conservation and accuracy is discussed,which is verified in efficiency of the scheme with the numerical example and the effect of two fractional order on some solitary wave solution behavior is studied.In Chapter 4,a three-point compact difference scheme is developed for solving the initial-boundary value problem for a class of fractional Schr(?)dinger-Boussinesq equations.The scheme is based on the use of “compact technique”to discretize the spatial derivatives in the low-order equations obtained by introducing new variables.A priori estimation of the numerical solution is obtained according to the discrete conservation law,then the existence and convergence of the numerical solution are proved by the discrete energy method.Numerical experiments verify the effectiveness and convergence of the proposed scheme.In Chapter 5,the damped fractional Schr(?)dinger equation with periodic boundary condition is investigated.The damped fractional Schr(?)dinger equation is transformed to an equivalent form through variable substitution,and the mass and energy conservation laws of the equivalent equation are presented.Based on the equivalent equation,the spatial semi-discretization scheme is obtained by the Fourier spectral method,then the convergence and conservation of the Fourier spatial semi-discretization scheme are proved.Further,the Crank-Nicolson scheme is employed to discrete the time variable,which leads to the Fourier spectral full-discretization scheme.It is proved that the mass and energy conservation laws are also preserved after the Fourier spectral full-discretization.The error analysis of the numerical solution shows that the proposed scheme has second-order accuracy in time and spectral accuracy in space.Moreover,numerical experiments are implemented to support the theoretical results and visually show the effect of fractional derivative and damping coefficient on the behavior of some solitary wave solutions.