谱问题、发展方程族及Lax表示的等价性

讨论了四对谱问题、相应的保谱方程族以及它们Ｌａｘ表示的等价住，并利用这种等价关系可导出一些方程的Ｌａｘ表示．     

On Spectral Method for Volterra Functional Integro-Differential Equations of Neutral Type

The main purpose of this work is to provide a numerical method for the solution of Volterra functional integro-differential equations of neutral type based on a spectral approach. We analyze the convergence properties of the spectral method to approximate smooth solutions of Volterra functional integro-differential equations of neutral type. It is shown that for the neutral integro-differential equations, the spectral methods yield an exponential order of convergence.     

INERTIAL MANIFOLDS FOR NONAUTONOMOUS SEMILINEAR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS WITH TIME DELAYS

The present paper deals with the long-time behavior of a class of nonautonomous retarded semilinear parabolic differential equations. When the time delays are small enough and the spectral gap conditions hold, the inertial manifolds of the nonautonomous retard parabolic equations are constructed by using the Lyapunov-Perron method.     

Extreme eigenvalues of real symmetric Toeplitz matrices

We exploit the even and odd spectrum of real symmetric Toeplitz matrices for the computation of their extreme eigenvalues, which are obtained as the solutions of spectral, or secular, equations. We also present a concise convergence analysis for a method to solve these spectral equations, along with an efficient stopping rule, an error analysis, and extensive numerical results.

     

Spectral discretization of Darcy equations coupled with Stokes equations by vorticity–velocity–pressure formulations

We propose to make the numerical analysis of a model coupling the Darcy equations in a porous medium with the Stokes equations in the cracks. The coupling is provided by a pressure continuity on the interface. We describe a discretization by spectral element methods. We derive a priori optimal error estimates and we present some numerical experiments which confirm the results of the analysis.© 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1628–1651, 2017     

Spectral methods for pantograph-type differential and integral equations with multiple delays

We analyze the convergence properties of the spectral method when used to approximate smooth solutions of delay differential or integral equations with two or more vanishing delays. It is shown that for the pantograph-type functional equations the spectral methods yield the familiar exponential order of convergence. Various numerical examples are used to illustrate these results.      

Efficient Implementation of a Spectral-Tau Method for a Class of Two-Point Boundary-Value and Initial-Boundary-Value Problems

A simple and efficient spectral method for solving the second, third order and fourth order elliptic equations with variable coefficients and nonlinear differential equations is presented. It is different from spectral-collocation method which leads to dense, ill-conditioned matrices. The spectral method in this paper solves for the coefficients of the solution in a Chebyshev series, leads to discrete systems with special structured matrices which can be factorized and solved efficiently. We also extend the method to boundary value problems in two space dimensions and solve 2-D separable equation with variable coefficients. As an application, we solve Cahn-Hilliard equation iteratively via first-order implicit time discretization scheme. Ample numerical results indicate that the proposed method is extremely accurate and efficient.     

Reciprocal Bäcklund transformations of autonomous evolution equations

We discuss the construction of reciprocal Bäcklund transformations for evolution equations using integrating factors of zeroth and higher orders with their corresponding conservation laws. As an example, we consider the Harry Dym equation and the Schwarzian KdV equation.     

Commutator identities on associative algebras and the integrability of nonlinear evolution equations

We show that commutator identities on associative algebras generate solutions of the linearized versions of integrable equations. In addition, we introduce a special dressing procedure in a class of integral operators that allows deriving both the nonlinear integrable equation itself and its Lax pair from such a commutator identity. The problem of constructing new integrable nonlinear evolution equations thus reduces to the problem of constructing commutator identities on associative algebras. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 3, pp. 477–491, March, 2008.     

非线性弱奇性Volterra积分方程的谱配置法

基于已有文献的研究成果及前期工作,我们考察了非线性弱奇性Volterra积分方程（VIE）的谱配置法,并对该方法进行了收敛性分析.得到的结论是数值误差呈谱收敛.误差收敛阶与配置点个数及方程解的正则性相关.数值实验也证实了这一结论.本文的方法解决了已有文献中类似数值方法（Allaei（2016）,Sohrabi（2017））存在的问题.     

Asymptotic behavior and stability of second order neutral delay differential equations

We study the asymptotic behavior of a class of second order neutral delay differential equations by both a spectral projection method and an ordinary differential equation method approach. We discuss the relation of these two methods and illustrate some features using examples. Furthermore, a fixed point method is introduced as a third approach to study the asymptotic behavior. We conclude the paper with an application to a mechanical model of turning processes.     

Mixed initial–boundary value problem for the Benjamin–Ono equation

We consider the mixed initial–boundary value problem for the Benjamin–Ono equation on a half-line. We study traditionally important problems of the theory of nonlinear partial differential equations, such as global in time existence of solutions to the initial–boundary value problem and the asymptotic behavior of solutions for large time.     

孤子族的生成及换位表示的一般结构

本文通过对谱问题ψｘ＝Ｕ（ｕ，λ）ψ的直接研究，利用谱梯度提供一条获得孤子方程族的途径，进一步，我们给出了孤子方程换位表示的一般结构，同时我们还将看到同一个谱问题可产生两族不同的孤子发展方程。     

Recovery of differential equations with nonlinear dependence on the spectral parameter

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line is studied. We give a formulation of the inverse problem, prove the uniqueness theorem and provided a procedure for constructing the solution of the inverse problem. We also establishe connections with inverse problems for partial differential equations.     

Numerical studies on nonlinear Schrödinger equations by spectral collocation method with preconditioning

In this study, we use the spectral collocation method with preconditioning to solve various nonlinear Schrödinger equations. To reduce round-off error in spectral collocation method we use preconditioning. We study the numerical accuracy of the method. The numerical results obtained by this way have been compared with the exact solution to show the efficiency of the method.     

Semiclassical spectral series of a Hartree-type operator corresponding to a rest point of the classical Hamilton-Ehrenfest system

We consider the classical equations of motion in quantum means, i.e., the Hamilton-Ehrenfest system. In the semiclassical approximation in the framework of the covariant approach based on these equations, we construct the spectral series of a nonlinear Hartree-type operator corresponding to a rest point. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 1, pp. 26–40, January, 2007.     

Zakharov–Shabat Spectral Transform on the Half-Line

The Zakharov–Shabat inverse spectral problem is constructed for a potential with support on the half-line and with a boundary value at the origin. This prescribed value is shown to produce a Jost solution with an essential singularity at large values of the spectral parameter; this requires particular attention when solving the related Hilbert boundary value problem. The method is then used to illustrate the sine-Gordon equation (in the light cone) and is discussed using a singular limit of the stimulated Raman scattering equations.     

二维不可压缩流的谱方法

本文以二维涡度方程为模型,介绍了谱方法和拟谱方法以及它们与差分方法和有限元法相结合的混合解法．这些方法可推广应用于其它一些类似的非线性问题．本文还给出了这些方法的某些数值例子和误差估计结果     

A method for solving nonlinear spectral problems for a class of systems of differential algebraic equations

A method for calculating eigenvalues of a nonlinear spectral problem for one class of linear differential algebraic equations is proposed under the assumption of an analytical dependence on spectral parameter of the matrices appearing in the system of equations and the matrices determining boundary conditions.     

广义Benjamin-Bona-Mahony方程Fourier谱方法的大时间误差估计

§ 1　ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅＢｅｎｊａｍｉｎ Ｂｏｎａ Ｍａｈｏｎｙｅｑｕａｔｉｏｎｕｔ+ｕｘ+ｕｕｘ -ｕｘｘ-ｕｘｘｔ =0 ( 1 .1 )ｉｎｃｏｒｐｏｒａｔｅｓｎｏｎｌｉｎｅａｒｄｉｓｐｅｒｓｉｖｅａｎｄｄｉｓｓｉｐａｔｉｖｅｅｆｆｅｃｔｓ ,ａｎｄｈａｓｂｅｅｎｐｒｏｐｏｓｅｄａｓａｍｏｄｅｌｆｏｒｂｏｔｈｔｈｅｂｏｒｅｐｒｏｐａｇａｔｉｏｎａｎｄｔｈｅｗａｔｅｒｗａｖｅｓ[1,2 ] .Ｔｈｅｅｘｉｓｔｅｎｃｅａｎｄｕｎｉｑｕｅｎｅｓｓｏｆｓｏｌｕｔｉｏｎｓｆｏｒｔｈｉｓｅｑｕａｔｉｏｎｈａｖｅｂｅｅ…