An Averaged Vector Field Method For Fractional Partial Differential Equations

In the past few decades,fractional calculus has gained widespread use in science and engineering.Because the fractional derivative is inherently nonlocal,fractional differential equations have broad application prospects in describing physics,chemistry,biology and even economics.It is very necessary and important to establish a numerical scheme with high precision to solve the fractional differential equation.In recent years,many books have given some important analytical methods to solve some fractional partial differential equations,such as finite difference method,finite element method,finite volume method,spectral method and many other effective numerical methods.Its consistency,stability and convergence are proved.Spectral method has intuitionistic and convenient characteristics.Finite element method has higher order of convergence and is still in the first stage of development at present.This paper is divided into four parts,The main purpose of this paper is to construct energy preserving schemes for fractional partial differential equations by using spectral method and average vector field method.Moreover,the new schemes proposed in this paper are more effective than those previously proposed.Numerical experiments are carried out to verify the effectiveness of the new schemes,The results of numerical experiments are analyzed.In the first chapter,the relationship between fractional Laplacian and fractional derivative in Riesz Space is proved,and the spectral method is used to discretize the fractional derivative.In the spatial direction,the spectral method is used to discretize the fractional sine-Gordon equation.In the time direction,the fourth-order average vector field method and the Boole discrete line integral method are used to discretize the integral term.The new scheme is tested by MATLAB.The experimental results show that the new scheme is more novel and effective.The results of numerical simulation show that the new scheme can keep the energy conservation of the equation and accurately simulate the behavior of the solutions of the equations.In the second chapter,in the space direction,the spectral method is used to discretize the Riesz fractional nonlinear Klein-Gordon-Zakharov equation,and in the time direction,the average vector field method is used to discretize the Riesz fractional nonlinear Klein-Gordon Zakharov equation.Finally,the results of numerical simulation show that the new scheme can keep the energy conservation of the equations and accurately simulate the behavior of the solutions of the equations.In the third chapter,The spectral method and the fourth-order average vector field method are used to discretize the four coupled nonlinear Schrodinger equations in space and time respectively.The numerical experiments are carried out with MATLAB to verify the effectiveness of the new scheme.In the fourth chapter,the four coupled nonlinear Schrodinger equations are expressed as a multisensor structure.The spectral method and average vector field method are used to discretize the equations in the space and time directions respectively.The multi symplectic energy conservation scheme of the equations is constructed and the numerical simulation of the new scheme is carried out.The new scheme is simulated in MATLAB.The numerical results show that the scheme constructed in this paper is more effective than before.The results further verify that the new scheme can keep the energy conservation of the equation and simulate the solution of the equations accurately.