Numerical Methods Of Two Kinds Of Fractional Equations

Fractional differential equation has a very extensive application in the field of mathematics and physics,it can describe some anomalous diffusion phenomenon more accurately.However,there are no available analytical solutions for fractional problems in general because of the fact that the time fractional derivatives have singularity and memory at the origin and the spatial fractional derivatives have non-locality,which is different from integer-order derivatives.Therefore,the numerical algorithm of fractional order equations has become the focus of researchers.It is well known that the energy is a very important physical invariant,so it is of great theoretical and practical value to study the dissipation of the equation and the numerical method of energy conservation.In view of this,we focuses on the numerical methods of two kinds of fractional equations,that is,the numerical dissipation of time fractional order sub-diffusion equation and the energy conservation of space fractional order Schr（?）dinger equation.In the first part,we study the dissipation of semi-linear time fractional subdiffusion equations in L²（Ω）and prove that the decay rate is t^-α,0 <α< 1,which is essentially different from the exponential decay of integer order.Then the time derivative of Caputo and the classical spatial Laplace operator are discretized by the L1 method and the finite element method respectively and prove the numerical dissipation of the scheme.Finally,a numerical example is given to verify the correctness of the theoretical results.In the second part,we establish an energy conservation relaxation method satisfying the any nonlinear term of power for space fractional nonlinear Schr（?）dinger equation.Then construct the vector form of the discrete equation by introducing a new variable and prove the time convergence order of therelaxation scheme for the third power Schr（?）dinger equation in detail.The numerical results show that the numerical scheme established in this paper is not only energy conservation but also second order convergence with respect to time,which is consistent with the theoretical results demonstrated.In the third part,the space fractional logarithmic Schr（?）dinger equation is discussed.We eliminate the singularity of the logarithm terms at the origin by introducing a small parameter 0 <ε<< 1 and prove the fact that the solution of the approximate equation converges to the solution of the original equation.Then,we construct a regularized splitting spectrum method to obtain the convergence order of the numerical method adopted.Numerical results also demonstrate that the numerical method has the property of energy conservation,which proves the correctness of the theoretical analysis.