Initial-boundary value problem for a parabolic-hyperbolic equation with power-law degeneration on the type change line

We consider the first boundary value problem for equations of mixed type in a rectangular domain. A criterion for the solution uniqueness is proved by the spectral expansion method. The solution is constructed in the form of a series in the eigenfunctions of the corresponding one-dimensional spectral problem. The stability of the solution with respect to the initial function is proved.     

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…     

Some non‐local problems for the parabolic‐hyperbolic type equation with complex spectral parameter

In the present paper the unique solvability of two non‐local problems for the mixed parabolic‐hyperbolic type equation with complex spectral parameter is proved. Sectors for values of the spectral parameter where these problems have unique solutions are shown. Uniqueness of the solution is proved by the method of energy integral and existence is proved by the method of integral equations. In particular cases, eigenvalues and corresponding eigenfunctions of the studied problems are found. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fourier-Chebyshev spectral method for cavitation computation in nonlinear elasticity

A Fourier-Chebyshev spectral method is proposed in this paper for solving the cavitation problem in nonlinear elasticity. The interpolation error for the cavitation solution is analyzed, the elastic energy error estimate for the discrete cavitation solution is obtained, and the convergence of the method is proved. An algorithm combined a gradient type method with a damped quasi-Newton method is applied to solve the discretized nonlinear equilibrium equations. Numerical experiments show that the Fourier-Chebyshev spectral method is efficient and capable of producing accurate numerical cavitation solutions.     

THE LARGE TIME BEHAVIOR OF SPECTRAL APPROXIMATION FOR A CLASS OF PSEUDOPARABOLIC VISCOUS DIFFUSION EQUATION

The asymptotic behavior of the solutions to a class of pseudoparabolic viscous diffusion equation with periodic initial condition is studied by using the spectral method. The semidiscrete Fourier approximate solution of the problem is constructed and the error estiation between spectral approximate solution and exact solution on large time is also obtained. The existence of the approximate attractor AN and the upper semicontinuity d(AN,A)→0 are proved.     

On an inhomogeneous Tricomi problem for a parabolic-hyperbolic equation

We consider an inhomogeneous Tricomi problem for a parabolic-hyperbolic equation with noncharacteristic type change line and with Frankl type matching condition for the normal derivatives on the type change line. The auxiliary function method is used to establish an a priori estimate of the solution. The existence of the solution is proved by the spectral method.     

Mixed problem for a first-order partial differential equation with involution and periodic boundary conditions

The Fourier method is used to find a classical solution of the mixed problem for a first-order differential equation with involution and periodic boundary conditions. The application of the Fourier method is substantiated using refined asymptotic formulas obtained for the eigenvalues and eigenfunctions of the corresponding spectral problem. The Fourier series representing the formal solution is transformed using certain techniques, and the possibility of its term-by-term differentiation is proved. Minimal requirements are imposed on the initial data of the problem.     

Couette-Taylor流的谱Galerkin逼近

利用谱方法对轴对称的旋转圆柱间的Couette-Taulor流进行数值模拟．首先给出Navier-Stokes方程流函数形式,利用Couette流把边界条件齐次化．其次给出Stokes算子的特征函数的解析表达式,证明其正交性,并对特征值进行估计．最后利用Stokes算子的特征函数作为逼近子空间的基函数,给出谱Galerkin逼近方程的表达式．证明了Navier-Stokes方程非奇异解的谱Galerkin逼近的存在性、唯一性和收敛性,给出了解谱Galerkin逼近的误差估计,并展示了数值计算结果．     

About the Tricomi problem for the mixed parabolic–hyperbolic type equation with complex spectral parameter

The unique solvability of the Tricomi problem for the parabolic–hyperbolic equation with complex spectral parameter is proved. Uniqueness of the solution is shown by the method of energy integral and existence by the method of integral equations.     

Inverse problems for differential operators of any order on trees

Inverse spectral problems for ordinary differential operators of any order on compact trees are studied. As the main spectral characteristics, Weyl matrices, which generalize the Weyl m-function for the classical Sturm-Liouville operator are introduced and studied. A constructive solution procedure for the inverse problem based on Weyl matrices is suggested, and the uniqueness of the solution is proved. The reconstruction of differential equations from discrete spectral characteristics is also considered.     

一类非线性Schrodinger方程的差分／Legendre谱元法

考察一类带幂次非线性项的Schrodinger方程的Dirichlet初边值问题，提出了一个有效的计算格式，其中时间方向上应用了一种守恒的二阶差分隐格式，空间方向上采用Legendre谱元法．对于时间半离散格式，证职了该格式具有能量守恒性质，并给出了L^2误差估计，对于全离散格式，应用不动点原理证明了数值解的存在唯一性，并给出了L^2误差估计．最后，通过数值试验验证了结果的可信性．     

Boundary value problem for a parabolic-hyperbolic equation with a nonlocal integral condition

For a parabolic-hyperbolic equation with the heat and wave operators in a rectangular domain, we consider a problem with a nonlocal Samarksii-Ionkin condition. A criterion for the uniqueness of the solution is established by the spectral expansion method. The classical solution of the problem is constructed in the form of the sum of a biorthogonal series. The solution is proved to be stable with respect to the initial condition.     

The Genetic Sums' Representation for the Moments of a System of Stieltjes Functions and its Application

This paper deals with the spectral problems for high-order nonsymmetric difference operators. The method of investigation is based on the analysis of the genetic sums formulas for the moments of the operator. The parameters of these sums are shown to be connected with coefficients of the introduced vector Stieltjes continued fraction. The connections with vector orthogonality, Hermite—Padé approximation, and Hankel determinants are investigated. This gives a tool for the analysis of the solution of the direct and inverse spectral problem of the operator. It is applied to the integration of hierarchy of the discrete KdV equations. The existence of a global solution is proved. July 13, 1998. Date revised: July 12, 1999. Date accepted: July 26, 1999.     

一类具有备用部件的可修人机系统解的渐近稳定性

运用泛函分析的方法 ,通过分析系统主算子的谱特征 ,给出一类具有备用部件的可修人机系统解的渐近稳定性证明 .     

Neumann problem for the Lavrent’ev–Bitsadze equation with two type-change lines in a rectangular domain

The Neumann problem for an equation with two perpendicular internal type-change lines in a rectangular domain is investigated. Uniqueness and existence theorems are proved by applying the spectral method. The separation of variables yields an eigenvalue problem for an ordinary differential equation. This problem is not self-adjoint, and the system of its eigenfunctions is not orthogonal. A corresponding biorthogonal system of functions is constructed and proved to be complete. The completeness result is used to prove a necessary and sufficient uniqueness condition for the problem under study. Its solution is constructed in the form of the sum of a biorthogonal series.     

Inverse Problem for a Differential Operator with Nonseparated Boundary Conditions

Uniqueness theorems for the solution of an inverse problem for a fourth-order differential operator with nonseparated boundary conditions are proved. The spectral data for the problem is specified as the spectrum of the problem itself (or its three eigenvalues) and the spectral data of a system of three problems.     

The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions

In this paper a spectral method and a numerical continuation algorithm for solving eigenvalue problems for the rectangular von Kármán plate with different boundary conditions (simply supported, partially or totally clamped) and physical parameters are introduced. The solution of these problems has a postbuckling behaviour. The spectral method is based on a variational principle (Galerkin’s approach) with a choice of global basis functions which are combinations of trigonometric functions. Convergence results of this method are proved and the rate of convergence is estimated. The discretized nonlinear model is treated by Newton’s iterative scheme and numerical continuation. Branches of eigenfunctions found by the algorithm are traced. Numerical results of solving the problems for polygonal and ferroconcrete plates are presented. Communicated by A. Zhou.     

Spectral Method for Three-Dimensional Nonlinear Klein-Gordon Equation by Using Generalized Laguerre and Spherical Harmonic Functions

In this paper, a generalized Laguerre-spherical harmonic spectral method is proposed for the Cauchy problem of three-dimensional nonlinear Klein-Gordon equation. The goal is to make the numerical solutions to preserve the same conservation as that for the exact solution. The stability and convergence of the proposed scheme are proved. Numerical results demonstrate the efficiency of this approach. We also establish some basic results on the generalized Laguerre-spherical harmonic orthogonal approximation, which play an important role in spectral methods for various problems defined on the whole space and unbounded domains with spherical geometry.AMS subject classifications: 65M70, 41A30, 81Q05     

Inverse nodal and inverse spectral problems for discontinuous boundary value problems

Inverse nodal and inverse spectral problems are studied for second-order differential operators on a finite interval with discontinuity conditions inside the interval. Uniqueness theorems are proved, and a constructive procedure for the solution is provided.     

SOLVING COLORED YANG-BAXTER EQUATION BY WU＇S METHOD

In this article, we discuss nonsymmetric solutions of the colored Yang-Baxter equation dependent on spectral as well as colored parameters and give all seven-vertex solutions by Wu＇s method. It is also proved that the solutions are composed of six groups of basic solutions up to five solution transformations. Moreover, al l solutions can be classified into two categories called Baxter type and free-fermion type.