CONSIDER the following nonlinear Schrodinger equation: （SP）{iΔφ<sub>t</sub>+Δ<sup>2</sup>φ+β(|φ|<sup>2p</sup>φ)=0,x∈R<sup>2</sup>,t】0, （1,1） φ|<sub>i...CONSIDER the following nonlinear Schrodinger equation: （SP）{iΔφ<sub>t</sub>+Δ<sup>2</sup>φ+β(|φ|<sup>2p</sup>φ)=0,x∈R<sup>2</sup>,t】0, （1,1） φ|<sub>i=0</sub>=φ<sub>0</sub>（x）, （1,2）where p】0, β∈R are constants, φ<sub>0</sub>∈H<sup>3</sup> （R<sup>2</sup>）. For the problem arising from nonlinearplasma in nonhomogeneous media, see references [1,2].展开更多
In this paper, the initial value problem of nonlinear reaction-diffusion equation is considered. The Dufort-Frankel finite difference approximation for the long time scheme is given for the d-dimensional reaction-diff...In this paper, the initial value problem of nonlinear reaction-diffusion equation is considered. The Dufort-Frankel finite difference approximation for the long time scheme is given for the d-dimensional reaction-diffusion equation with the two different cases. The global solution and global attractor are discussed for the Dufort-Frankel scheme. Moreover properties of the solution are studied. The error estimate is presented in a finite time region and in the global time region for some special cases. Finally the numerical results for the equation are investigated for Allen-Cahn equation and some other equations and the homoclinic orbit is simulated numerically.展开更多
A conservative difference scheme is presented for the initial-boundary-value problem of a generalized Zakharov equations. On the basis of a prior estimates in L-2 norm, the convergence of the difference solution is pr...A conservative difference scheme is presented for the initial-boundary-value problem of a generalized Zakharov equations. On the basis of a prior estimates in L-2 norm, the convergence of the difference solution is proved in order O(h(2) + r(2)). In the proof, a new skill is used to deal with the term of difference quotient (e(j,k)(n))t. This is necessary, since there is no estimate of E(x, y, t) in L-infinity norm.展开更多
文摘CONSIDER the following nonlinear Schrodinger equation: （SP）{iΔφ<sub>t</sub>+Δ<sup>2</sup>φ+β(|φ|<sup>2p</sup>φ)=0,x∈R<sup>2</sup>,t】0, （1,1） φ|<sub>i=0</sub>=φ<sub>0</sub>（x）, （1,2）where p】0, β∈R are constants, φ<sub>0</sub>∈H<sup>3</sup> （R<sup>2</sup>）. For the problem arising from nonlinearplasma in nonhomogeneous media, see references [1,2].
基金This project is partly supported by the Natural Science Foundation of China and partly by StateEducation Committee.
文摘In this paper, the initial value problem of nonlinear reaction-diffusion equation is considered. The Dufort-Frankel finite difference approximation for the long time scheme is given for the d-dimensional reaction-diffusion equation with the two different cases. The global solution and global attractor are discussed for the Dufort-Frankel scheme. Moreover properties of the solution are studied. The error estimate is presented in a finite time region and in the global time region for some special cases. Finally the numerical results for the equation are investigated for Allen-Cahn equation and some other equations and the homoclinic orbit is simulated numerically.
文摘A conservative difference scheme is presented for the initial-boundary-value problem of a generalized Zakharov equations. On the basis of a prior estimates in L-2 norm, the convergence of the difference solution is proved in order O(h(2) + r(2)). In the proof, a new skill is used to deal with the term of difference quotient (e(j,k)(n))t. This is necessary, since there is no estimate of E(x, y, t) in L-infinity norm.
