An unified numerical approach of steady convection between two parallel plates

This paper considers the problem of laminar forced convection between two parallel plates. We present an unified numerical approach for some problems related to this case: the problem of viscous dissipation with Dirichlet and Neumann boundary conditions and the Graetz problem. The solutions of these problems are obtained by a series expansion of the complete eigenfunctions system of some Sturm-Liouville problems. The eigenfunctions and eigenvalues of this Sturm-Liouville problem are obtained by using Galerkin’s method. Numerical examples are given for viscous fluids with various Brinkman numbers.     

On an analog of periodic boundary value problems for the Poisson equation in the disk

We consider a periodic and an antiperiodic problem for the Poisson equation in the unit disk and prove their well-posedness. The possibility of separation of variables is justified. We construct the Green functions of these problems in closed form and obtain an integral representation of the solution. The problems are self-adjoint, and we construct all eigenvalues and eigenfunctions in closed form.     

Higher order convergence results for the Rayleigh-Ritz method applied to eigenvalue problems: 2. Improved error bounds for eigenfunctions

The application of the Rayleigh-Ritz method for approximating the eigenvalues and eigenfunctions of linear eigenvalue problems in several dimensions is investigated. The object is to improve upon known error estimates for the approximate eigenfunctions. Results for the Galerkin approximation of the eigenfunctions are developed under varying assumptions on the boundary conditions and domain of definition of the eigenvalue problem. These results, coupled with a previous result relating Galerkin and Rayleigh-Ritz approximation of the eigenfunctions, are then used to obtain improved error estimates for the approximate eigenfunctions in theL ² and uniform norms.This research was supported in part by AEC Grant (11-1)-2075.     

Irregular multipoint eigenvalue problems

We prove asymptotic estimates for the Green's function of irregular multipoint eigenvalue problems, these estimates are fundamental for the expansion of functions into a series of eigenfunctions of irregular eigenvalue problems.     

Computable bounds for eigenvalues and eigenfunctions of elliptic differential operators

Summary We are concerned with bounds for the error between given approximations and the exact eigenvalues and eigenfunctions of self-adjoint operators in Hilbert spaces. The case is included where the approximations of the eigenfunctions don't belong to the domain of definition of the operator. For the eigenvalue problem with symmetric elliptic differential operators these bounds cover the case where the trial functions don't satisfy the boundary conditions of the problem. The error bounds suggest a certain defectminization method for solving the eigenvalue problems. The method is applied to the membrane problem.     

External finite-element approximations of eigenfunctions in the case of multiple eigenvalues

The paper deals with the finite-element analysis of second-order elliptic eigenvalue problems when the approximate domains Ω_h are not subdomains of the original domain . The considerations are restricted to piecewise linear approximations. Special attention is devoted to the convergence of approximate eigenfunctions in the case of multiple exact eigenvalues. As yet the approximate solutions have been compared with linear combinations of exact eigenfunctions with coefficients depending on the mesh parameter h. We avoid this disadvantage.     

与夹层梁稳定性相联系的一个本征值问题

本文指出了与夹层梁稳定性相联系的本征值问题的几个特点,并说明了本征函数的一些应用.     

A collocation/quadrature-based Sturm–Liouville problem solver

We present a computational method for solving a class of boundary-value problems in Sturm–Liouville form. The algorithms are based on global polynomial collocation methods and produce discrete representations of the eigenfunctions. Error control is performed by evaluating the eigenvalue problem residuals generated when the eigenfunctions are interpolated to a finer discretization grid; eigenfunctions that produce residuals exceeding an infinity-norm bound are discarded. Because the computational approach involves the generation of quadrature weights and arrays for discrete differentiation operations, our computational methods provide a convenient framework for solving boundary-value problems by eigenfunction expansion and other projection methods.     

Asymptotic Distribution of Eigenfunctions and Eigenvalues of the Basic Boundary-Contact Oscillation Problems of the Classical Theory of Elasticity

The basic boundary-contact oscillation problems are considered for a three-dimensional piecewise-homogeneous isotropic elastic medium bounded by several closed surfaces. Using Carleman's method, the asymptotic formulas for the distribution of eigenfunctions and eigenvalues are obtained.     

Oscillation properties for the equation of vibrating beam with irregular boundary conditions

In this paper we study the oscillatory properties for the eigenfunctions of some fourth-order eigenvalue problems, where the boundary conditions are irregular in the sense of the classification of [S. Janczewski, Oscillation theorems for the differential boundary value problems of the fourth order, Ann. of Math. 29 (1928) 521–542]. In this case, we show that these oscillatory properties are different from those of the Sturm–Liouville problem.     

The combined effect of numerical integration and approximation of the boundary in the finite element method for eigenvalue problems

Summary. The paper deals with the finite element analysis of second order elliptic eigenvalue problems when the approximate domains are not subdomains of the original domain and when at the same time numerical integration is used for computing the involved bilinear forms. The considerations are restricted to piecewise linear approximations. The optimum rate of convergence for approximate eigenvalues is obtained provided that a quadrature formula of first degree of precision is used. In the case of a simple exact eigenvalue the optimum rate of convergence for approximate eigenfunctions in the -norm is proved while in the -norm an almost optimum rate of convergence (i.e. near to is achieved. In both cases a quadrature formula of first degree of precision is used. Quadrature formulas with degree of precision equal to zero are also analyzed and in the case when the exact eigenfunctions belong only to the convergence without the rate of convergence is proved. In the case of a multiple exact eigenvalue the approximate eigenfunctions are compard (in contrast to standard considerations) with linear combinations of exact eigenfunctions with coefficients not depending on the mesh parameter . Received September 18, 1993 / Revised version received September 26, 1994     

Series solution of nonlinear eigenvalue problems by means of the homotopy analysis method

A general analytic approach for nonlinear eigenvalue problems is described. Two physical problems are used as examples to show the validity of this approach for eigenvalue problems with either periodic or non-periodic eigenfunctions. Unlike perturbation techniques, this approach is independent of any small physical parameters. Besides, different from all other analytic techniques, it provides a simple way to ensure the convergence of series of eigenvalues and eigenfunctions so that one can always get accurate enough approximations. Finally, unlike all other analytic techniques, this approach provides great freedom to choose an auxiliary linear operator so as to approximate the eigenfunction more effectively by means of better base functions. This approach provides us a new way to investigate eigenvalue problems with strong nonlinearity.     

Asymptotic Formulas for Precision of Discrete Spectral Problems

We find principal terms in the power expansion, with respect to the step of a square grid, of the eigenvalue error for a discrete analogue of spectral problems for elliptic operators of the second and fourth order. We use the compactness of a bounded set in a Hilbert space, which gives the mean convergence of piecewise-constant fillings of grid eigenfunctions and the weak convergence of these fillings for difference derivatives. This, in turn, allows one to prove that eigenfunctions of the initial problems belong to the corresponding Sobolev spaces.     

Green’s Function of Ordinary Differential Operators and an Integral Representation of Sums of Certain Power Series

The eigenvalues and eigenfunctions of certain operators generated by symmetric differential expressions with constant coefficients and self-adjoint boundary conditions in the space of Lebesgue squareintegrable functions on an interval are explicitly calculated, while the resolvents of these operators are integral operators with kernels for which the theorem on an eigenfunction expansion holds. In addition, each of these kernels is the Green’s function of a self-adjoint boundary value problem, and the procedure for its construction is well known. Thus, the Green’s functions of these problems can be expanded in series in terms of eigenfunctions. In this study, identities obtained by this method are used to calculate the sums of convergent number series and to represent the sums of certain power series in an intergral form.     

Reliable computation of eigenvalues of the magnetostatic integral operator

Some variational problems in magnetostatics can be reformulated as eigenvalue problems for vector surface integral operators in appropriate function spaces, e.g., the magnetostatic integral operator is of considerable interest in the theory of permanent magnetization of compact bodies. In the case that the underlying surface is either a sphere, a spheroid, or a triaxial ellipsoid, explicit expressions for eigenvalues and eigenfunctions are well known. For the ellipsoid, these quantities are given in terms of Lamé functions and surface ellipsoidal harmonics. Since there is an apparent lack in literature we provide an new effective scheme for the reliable computation of these functions and of the corresponding eigenvalues of the magnetostatic operator.     

Generalized solutions and generalized eigenfunctions of boundary-value problems on a geometric graph

We consider generalized solutions to boundary-value problems for elliptic equations on an arbitrary geometric graph and their corresponding eigenfunctions. We construct analogs of Sobolev spaces that are dense in L ₂. We obtain conditions for the Fredholm solvability of boundary-value problems of various types, describe their spectral properties and conditions for the expansion in generalized eigenfunctions. The results presented here are fundamental in studying boundary control problems of oscillations of multiplex jointed structures consisting of strings or rods, as well as in studying the cell metabolism.     

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)     

Spectral Solutions of Self-adjoint Elliptic Problems with Immersed Interfaces

This paper describes a spectral representation of solutions of self-adjoint elliptic problems with immersed interfaces. The interface is assumed to be a simple non-self-intersecting closed curve that obeys some weak regularity conditions. The problem is decomposed into two problems, one with zero interface data and the other with zero exterior boundary data. The problem with zero interface data is solved by standard spectral methods. The problem with non-zero interface data is solved by introducing an interface space H _Γ(Ω) and constructing an orthonormal basis of this space. This basis is constructed using a special class of orthogonal eigenfunctions analogously to the methods used for standard trace spaces by Auchmuty (SIAM J. Math. Anal. 38, 894–915, 2006). Analytical and numerical approximations of these eigenfunctions are described and some simulations are presented.     

Biorthogonal decompositions of the Radon transform

Summary. The concept of singular value decompositions is a valuable tool in the examination of ill-posed inverse problems such as the inversion of the Radon transform. A singular value decomposition depends on the determination of suitable orthogonal systems of eigenfunctions of the operators , . In this paper we consider a new approach which generalizes this concept. By application of biorthogonal instead of orthogonal functions we are able to apply a larger class of function sets in order to account for the structure of the eigenfunction spaces. Although it is preferable to use eigenfunctions it is still possible to consider biorthogonal function systems which are not eigenfunctions of the operator. With respect to the Radon transform for functions with support in the unit ball we apply the system of Appell polynomials which is a natural generalization of the univariate system of Gegenbauer (ultraspherical) polynomials to the multivariate case. The corresponding biorthogonal decompositions show some advantages in comparison with the known singular value decompositions. Vice versa by application of our decompositions we are able to prove new properties of the Appell polynomials. Received October 19, 1993     

Discontinuous Sturm-Liouville Problems Containing Eigenparameter in the Boundary Conditions

In this paper, discontinuous Sturm-Liouville problems, which contain eigenvalue parameters both in the equation and in one of the boundary conditions, are investigated. By using an operatortheoretic interpretation we extend some classic results for regular Sturm-Liouville problems and obtain asymptotic approximate formulae for eigenvalues and normalized eigenfunctions. We modify some techniques of [Fulton, C. T., Proc. Roy. Soc. Edin. 77 （A）, 293-308 （1977）], [Walter, J., Math. Z., 133, 301-312 （1973）] and [Titchmarsh, E. C., Eigenfunctions Expansion Associated with Second Order Differential Equations I, 2nd edn., Oxford Univ. Pres, London, 1962], then by using these techniques we obtain asymptotic formulae for eigenelement norms and normalized eigenfunctions.