Steklov eigenvalue problem of Helmholtz equation is considered in the present paper. Steklov eigenvalue problem is reduced to a new variational formula on the boundary of a given domain, in which the self-adjoint prop...Steklov eigenvalue problem of Helmholtz equation is considered in the present paper. Steklov eigenvalue problem is reduced to a new variational formula on the boundary of a given domain, in which the self-adjoint property of the original differential operator is kept and the calculating of hyper-singular integral is avoided. A numerical example showing the efficiency of this method and an optimal error estimate are given.展开更多
We construct and analyze a family of semi-discretized difference schemes with two parameters for the Korteweg-de Vries (KdV) equation. The scheme possesses the first four near-conserved quantities for periodic boundar...We construct and analyze a family of semi-discretized difference schemes with two parameters for the Korteweg-de Vries (KdV) equation. The scheme possesses the first four near-conserved quantities for periodic boundary conditions. The existence and the convergence of its global solution in Sobolev space L∞(0, T;H3)are proved and the scheme is also stable about initial values. Furthermore, the scheme conserves exactly the first two conserved quantities in the special case.展开更多
Consider solving the Dirichlet problem of Helmholtz equation on unbounded region R2\Г with Г a smooth open curve in the plane. We use simple-layer potential to construct a solution. This leads to the solution of a l...Consider solving the Dirichlet problem of Helmholtz equation on unbounded region R2\Г with Г a smooth open curve in the plane. We use simple-layer potential to construct a solution. This leads to the solution of a logarithmic integral equation of the first kind for the Helmholtz equation. This equation is reformulated using a special change of variable, leading to a new first kind equation with a smooth solution function. This new equation is split into three parts. Then a quadrature method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically. An error analysis in a Sobolev space setting is given. And numerical results show that fast convergence is clearly exhibited.展开更多
In this paper we use the spectral method to analyse the generalized KuramotoSiva-shinsky equations. We prove the ealstence and uniqueness of global smooth solution of the equations. Finally, we obtain the error estima...In this paper we use the spectral method to analyse the generalized KuramotoSiva-shinsky equations. We prove the ealstence and uniqueness of global smooth solution of the equations. Finally, we obtain the error estimation between spectral approkimate solution and exact solution on large time.展开更多
By using the concept of multigraphs, the difference graphs of a class of alternating block Crank-Nicolson methods are defined and described, which extends the results on difference graphs of parallel computing for the...By using the concept of multigraphs, the difference graphs of a class of alternating block Crank-Nicolson methods are defined and described, which extends the results on difference graphs of parallel computing for the finite difference method.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 Alien-Cahn equation and some other equations and the homoclinic orbit is simulated numerically.展开更多
In this paper we investigate the existence, uniqueness and regularity of thesolution of semilinear parabolic equations with coefficients that are discontinuousacross the interface, some prior estimates are obtained. A...In this paper we investigate the existence, uniqueness and regularity of thesolution of semilinear parabolic equations with coefficients that are discontinuousacross the interface, some prior estimates are obtained. A net shape of the finiteelements around the singular points was designed in [7] to solve the linear ellipticproblems, by means of that net, we prove that the approximate solution has thesame convergence rate as that without singularity.展开更多
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, a new method of boundary reduction is proposed, which reduces the biharmonic boundary value problem to a system of integro-differentialequations on the boundary and preserves the self-adjointness of the...In this paper, a new method of boundary reduction is proposed, which reduces the biharmonic boundary value problem to a system of integro-differentialequations on the boundary and preserves the self-adjointness of the original problem. Moreover, a boundary finite element method based on this integro-differentialequations is presented and the error estimates of the numerical approximations aregiven. The numerical examples show that this new method is effective.展开更多
Nonlinear Galerkin methods are numerical schemes adapted well to the long time integration of evolution partial differential equations. The aim of this paper is to discuss such schemes for reaction dchsion equations. ...Nonlinear Galerkin methods are numerical schemes adapted well to the long time integration of evolution partial differential equations. The aim of this paper is to discuss such schemes for reaction dchsion equations. The convergence results are proved.展开更多
A conservative difference scheme is presented for the init ial- b oundary- value problem of a generalized Zakharov equations. On the basis of a prior estimates in Li noring the convergence of the difference solution i...A conservative difference scheme is presented for the init ial- b oundary- value problem of a generalized Zakharov equations. On the basis of a prior estimates in Li noring the convergence of the difference solution is proved in order O(h2 + r2).In the proof, a new skill is used to deal with the term of difference quotient (enjh,k)t. This is necessary, since there is no estimate of E(x, y, t) in Lo∞ norm.展开更多
The generalized derivative Ginzburg-Landau equation in two spatial dimensions is discussed. The existence and uniqueness of global solution are obtained by Galerkin method and by a priori estimates on the solution in ...The generalized derivative Ginzburg-Landau equation in two spatial dimensions is discussed. The existence and uniqueness of global solution are obtained by Galerkin method and by a priori estimates on the solution in H+1 -norm and H +2-norm.展开更多
It has been observed numerically in [1] that, under certain conditions, all eigenvalues of the first-order Hermite cubic spline collocation differentiation matrices with unsymmetrical collocation points lie in one of ...It has been observed numerically in [1] that, under certain conditions, all eigenvalues of the first-order Hermite cubic spline collocation differentiation matrices with unsymmetrical collocation points lie in one of the half complex planes. In this paper, we provide a theoretical proof for this spectral result.展开更多
文摘Steklov eigenvalue problem of Helmholtz equation is considered in the present paper. Steklov eigenvalue problem is reduced to a new variational formula on the boundary of a given domain, in which the self-adjoint property of the original differential operator is kept and the calculating of hyper-singular integral is avoided. A numerical example showing the efficiency of this method and an optimal error estimate are given.
文摘We construct and analyze a family of semi-discretized difference schemes with two parameters for the Korteweg-de Vries (KdV) equation. The scheme possesses the first four near-conserved quantities for periodic boundary conditions. The existence and the convergence of its global solution in Sobolev space L∞(0, T;H3)are proved and the scheme is also stable about initial values. Furthermore, the scheme conserves exactly the first two conserved quantities in the special case.
基金the National Natural Science Foundation of China 19901004-1.
文摘Consider solving the Dirichlet problem of Helmholtz equation on unbounded region R2\Г with Г a smooth open curve in the plane. We use simple-layer potential to construct a solution. This leads to the solution of a logarithmic integral equation of the first kind for the Helmholtz equation. This equation is reformulated using a special change of variable, leading to a new first kind equation with a smooth solution function. This new equation is split into three parts. Then a quadrature method that takes special advantage of the splitting of the integral equation is used to solve the equation numerically. An error analysis in a Sobolev space setting is given. And numerical results show that fast convergence is clearly exhibited.
文摘In this paper we use the spectral method to analyse the generalized KuramotoSiva-shinsky equations. We prove the ealstence and uniqueness of global smooth solution of the equations. Finally, we obtain the error estimation between spectral approkimate solution and exact solution on large time.
文摘By using the concept of multigraphs, the difference graphs of a class of alternating block Crank-Nicolson methods are defined and described, which extends the results on difference graphs of parallel computing for the finite difference method.
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 Alien-Cahn equation and some other equations and the homoclinic orbit is simulated numerically.
文摘In this paper we investigate the existence, uniqueness and regularity of thesolution of semilinear parabolic equations with coefficients that are discontinuousacross the interface, some prior estimates are obtained. A net shape of the finiteelements around the singular points was designed in [7] to solve the linear ellipticproblems, by means of that net, we prove that the approximate solution has thesame convergence rate as that without singularity.
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].
文摘In this paper, a new method of boundary reduction is proposed, which reduces the biharmonic boundary value problem to a system of integro-differentialequations on the boundary and preserves the self-adjointness of the original problem. Moreover, a boundary finite element method based on this integro-differentialequations is presented and the error estimates of the numerical approximations aregiven. The numerical examples show that this new method is effective.
文摘Nonlinear Galerkin methods are numerical schemes adapted well to the long time integration of evolution partial differential equations. The aim of this paper is to discuss such schemes for reaction dchsion equations. The convergence results are proved.
文摘A conservative difference scheme is presented for the init ial- b oundary- value problem of a generalized Zakharov equations. On the basis of a prior estimates in Li noring the convergence of the difference solution is proved in order O(h2 + r2).In the proof, a new skill is used to deal with the term of difference quotient (enjh,k)t. This is necessary, since there is no estimate of E(x, y, t) in Lo∞ norm.
文摘The generalized derivative Ginzburg-Landau equation in two spatial dimensions is discussed. The existence and uniqueness of global solution are obtained by Galerkin method and by a priori estimates on the solution in H+1 -norm and H +2-norm.
基金The Project supported by A Grant from the Research Grants Council of the Hong Kong Spelial Administrative Region, China (Project No. CityU 1061/00p) the Foundation of Chinese Academy of Engineering Physics.
文摘It has been observed numerically in [1] that, under certain conditions, all eigenvalues of the first-order Hermite cubic spline collocation differentiation matrices with unsymmetrical collocation points lie in one of the half complex planes. In this paper, we provide a theoretical proof for this spectral result.
