Eigenfunction expansion method and its application to two-dimensional elasticity problems based on stress formulation

This paper proposes an eigenfunction expansion method to solve twodimensional (2D) elasticity problems based on stress formulation. By introducing appropriate state functions, the fundamental system of partial differential equations of the above 2D problems is rewritten as an upper triangular differential system. For the associated operator matrix, the existence and the completeness of two normed orthogonal eigenfunction systems in some space are obtained, which belong to the two block operators arising in the operator matrix. Moreover, the general solution to the above 2D problem is given by the eigenfunction expansion method.

Skeletons of 3D Surfaces Based on the Laplace-Beltrami Operator Eigenfunctions

In this work we describe the algorithms to construct the skeletons, simplified 1D representations for a 3D surface depicted by a mesh of points, given the respective eigenfunctions of the Discrete Laplace-Beltrami Operator (LBO). These functions are isometry invariant, so they are independent of the object’s representation including parameterization, spatial position and orientation. Several works have shown that these eigenfunctions provide topological and geometrical information of the surfaces of interest [1] [2]. We propose to make use of that information for the construction of a set of skeletons, associated to each eigenfunction, which can be used as a fingerprint for the surface of interest. The main goal is to develop a classification system based on these skeletons, instead of the surfaces, for the analysis of medical images, for instance.

New tomographic reconstruction technique based on Laplacian eigenfunction

This letter proposes a new tomographic reconstruction procedure based on the Laplacian eigenfunction(LEF) patterns, which are independent of the plasma cross-section and do not require the flux surface information. The process is benchmarked for the experimental data of Heliotron J plasma and the results are compared with the least-squares approximation by a Phillips–Tikhonov(PT)-type regularization, which is widely used as the standard technique for tomographic reconstruction. The reconstruction based on the LEF is found to be capable of determining the magnetic axis at different time locations efficiently in comparison with the PT-type regularization.

An Alternative Method of Eigenfunction Expansion Associated with Second Order Differential Equation in Infinite Domain and Its Application

In this paper a method of eigenfunction expansion associated with 2nd order differential equation is developed by using the concept of theory of distribution. An application of the method to the infinite long antenna is described in detail.

A complete symplectic eigenfunction expansion for the elastic thin plate with simply supported edges

The eigenvalue problem for the Hamiltonian operator associated with the mathematical model for the deflection of a thin elastic plate is investigated.First,the problem for a rectangular plate with simply supported edges is solved directly.Then,the completeness of the eigenfunctions is proved,thereby demonstrating the feasibility of using separation of variables to solve the problem. Finally,the general solution is obtained by using the proved expansion theorem.

Asymptotic Approximation of the Eigenvalues and the Eigenfunctions for the Orr-Sommerfeld Equation on Infinite Intervals

Asymptotic eigenvalues and eigenfunctions for the Orr-Sommerfeld equation in two-dimensional and three-dimensional incompressible flows on an infinite domain and on a semi-infinite domain are obtained. Two configurations are considered, one in which a short-wave limit approximation is used, and another in which a long-wave limit approximation is used. In the short-wave limit, Wentzel-Kramers-Brillouin (WKB) methods are utilized to estimate the eigenvalues, and the eigenfunctions are approximated in terms of Green’s functions. The procedure consists of transforming the Orr-Sommerfeld equation into a system of two second order ordinary differential equations for which the eigenvalues and the eigenfunctions can be approximated. In the long-wave limit approximation, solutions are expressed in terms of generalized hypergeometric functions. Our procedure works regardless of the values of the Reynolds number.

Characterization of Periodic Eigenfunctions of the Fourier Transform Operator

We generalize this result to p1,p2-periodic eigenfunctions of F?on R2 and to p1,p2,p3-periodic eigenfunctions of F?on R3.

A Remark on Eigenfunction Estimates by Heat Flow

In this paper, we consider L∞ estimates of eigenfunction, or more generally, the L∞ estimates of equation -Δu=fu. We use heat flow to give a new proof of the L∞ estimates for such type equations.

Fourier coefficients of restrictions of eigenfunctions

Let{e_(j)}be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold(M,g).Let H■M be a submanifold and{ψ_(k)}be an orthonormal basis of Laplace eigenfunctions of H with the induced metric.We obtain joint asymptotics for the Fourier coefficients<γHe_(j),ψ_(k)>L^(2)(H)=∫He_(j),ψ_(k)dV_(H)of restrictionsγHe_(j)of e_(j)to H.In particular,we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum{(μ_(k),λ_(j))}^(∞)_(j,k-0)of the(square roots of the)Laplacian△_(M)on M and the Laplacian△_(H)on H in a family of suitably‘thick'regions in R^(2).Thick regions include(1)the truncated coneμ_(k)/λ_(j)∈[a,b]■(0,1)andλ_(j)≤λ,and(2)the slowly thickening strip|μ_(k)-cλ_(j)|≤w(λ)andλ_(j)≤λ,where w(λ)is monotonic and 1■w(λ)≤λ^(1/2).Key tools for obtaining the asymptotics include the composition calculus of Fourier integral operators and a new multidimensional Tauberian theorem.

Completeness in the sense of Cauchy principal value of the eigenfunction systems of infinite dimensional Hamiltonian operator

The properties of eigenvalues and eigenfunctions of the infinite dimensional Hamiltonian operators are studied, and the suffcient conditions of the completeness in the sense of Cauchy principal value of the eigenfunction systems of the infinite dimensional Hamiltonian operators are given. In the end, concrete examples are constructed to justify the effectiveness of the criterion.

A Renormalized-Hamiltonian-Flow Approach to Eigenenergies and Eigenfunctions

We introduce a decimation scheme of constructing renormalized Hamiltonian flows,which is useful in the study of properties of energy eigenfunctions,such as localization,as well as in approximate calculation of eigenenergies.The method is based on a generalized Brillouin-Wigner perturbation theory.Each flow is specific for a given energy and,at each step of the flow,a finite subspace of the Hilbert space is decimated in order to obtain a renormalized Hamiltonian for the next step.Eigenenergies of the original Hamiltonian appear as unstable fixed points of renormalized flows.Numerical illustration of the method is given in the Wigner-band random-matrix model.

STABILITY EIGENFUNCTIONS OF A BOUNDARY-LAYER FLOW OVER VISCOELASTIC COMPLIANT WALL

The eigenfunctions in a stability problem of boundary-layer flow over a viscoelastic compliant wall were studied. Two categories of modes, TSI and CIFI, exist in the eigenvalue solutions. The eigenfunctions of flow-based TSI were investigated together with those in the flow over rigid wall, whereas the eigenfunctions of wall-based CIFI were compared with the wall functions in an individual wall without fluid constraint. The physical characteristics of the eigenmodes were discussed based on their eigenfunctions.

On Eigenvalue Intervals and Eigenfunctions of Nonresonance Singular Dirichlet Boundary Value Problems

In this paper we shall consider the nonresonance Dirichlet boundary value problemwhere λ】0 is a parameter, p】0 is a constant. Intervals of A are determined to ensure the existence of a nonnegative solution of the boundary value problem. For λ=1, we shall also offer criteria for the existence of eigenfunctions. The main results include and improve on those of [2,4,6,8].

THE SASA-SATSUMA EQUATION ON A NON-ZERO BACKGROUND: THE INVERSE SCATTERING TRANSFORM AND MULTI-SOLITON SOLUTIONS

We concentrate on the inverse scattering transformation for the Sasa-Satsuma equation with 3×3 matrix spectrum problem and a nonzero boundary condition. To circumvent the multi-value of eigenvalues, we introduce a suitable two-sheet Riemann surface to map the original spectral parameter k into a single-valued parameter z. The analyticity of the Jost eigenfunctions and scattering coefficients of the Lax pair for the Sasa-Satsuma equation are analyzed in detail. According to the analyticity of the eigenfunctions and the scattering coefficients, the z-complex plane is divided into four analytic regions of D_(j) : j = 1, 2, 3, 4. Since the second column of Jost eigenfunctions is analytic in D_(j), but in the upper-half or lowerhalf plane, we introduce certain auxiliary eigenfunctions which are necessary for deriving the analytic eigenfunctions in Dj. We find that the eigenfunctions, the scattering coefficients and the auxiliary eigenfunctions all possess three kinds of symmetries;these characterize the distribution of the discrete spectrum. The asymptotic behaviors of eigenfunctions, auxiliary eigenfunctions and scattering coefficients are also systematically derived. Then a matrix Riemann-Hilbert problem with four kinds of jump conditions associated with the problem of nonzero asymptotic boundary conditions is established, from this N-soliton solutions are obtained via the corresponding reconstruction formulae. The reflectionless soliton solutions are explicitly given. As an application of the N-soliton formula, we present three kinds of single-soliton solutions according to the distribution of discrete spectrum.

具有勢壘的Sturm-Liouville算子的特徵值和特徵函數

We aim to find the eigenvalues and eigenfunctions of the barrier potential case for Strum-Liouville operator on the finite interval [0,π] when λ > 0. Generally, the eigenvalue problem for the Sturm-Liouville operator is often solved by using integral equations, which are sometimes complex to solve, and difficulties may arise in computing the boundary values. Considering the said complexity, we have successfully developed a technique to give the asymptotic formulae of the eigenvalue and the eigenfunction for Sturm-Liouville operator with barrier potential. The results are of significant interest in the field of quantum mechanics and atomic systems to observe discrete energy levels.

Structure of Periodic Flows through a Channel with a Suddenly Expanded and Contracted Part

With respect to flows in a two-dimensional sudden expansion and contraction channel having a pair of cavities, numerical simulation was performed by imposing inlet/outlet boundary conditions giving a velocity distribution to the inlet. Periodic flows have been reproduced, which have a discrete spectrum about frequency. A fundamental wave occupies most part of the disturbance components, but higher harmonic waves are also included. The disturbance is excited by Kelvin-Helmholtz instability in a cavity section, where only the fundamental wave is generated. A wavenumber is regulated by a channel length under a periodic boundary condition, but there is no restriction in a main flow direction under the inlet/outlet boundary conditions, and therefore, some wavenumbers can occur. Therefore, an arbitrary frequency component of disturbance is a synthesized wave composed of various wave numbers. There are two kinds of components constituting this synthesized wave: a maximum of a velocity distribution is near a wall and in the center of the channel, which are called as wall mode and central mode in linear stability analysis of the plane Poiseuille flow. The synthesized wave composed of some modes shows a tendency to lower wavenumbers at the center of the channel.

Composite Hermite and Anti-Hermite Polynomials

The Weber-Hermite differential equation, obtained as the dimensionless form of the stationary Schroedinger equation for a linear harmonic oscillator in quantum mechanics, has been expressed in a generalized form through introduction of a constant conjugation parameter according to the transformation , where the conjugation parameter is set to unity () at the end of the evaluations. Factorization in normal order form yields composite eigenfunctions, Hermite polynomials and corresponding positive eigenvalues, while factorization in the anti-normal order form yields the partner composite anti-eigenfunctions, anti-Hermite polynomials and negative eigenvalues. The two sets of solutions are related by an reversal conjugation rule . Setting provides the standard Hermite polynomials and their partner anti-Hermite polynomials. The anti-Hermite polynomials satisfy a new differential equation, which is interpreted as the conjugate of the standard Hermite differential equation.

Analysis of Oblique Wave Interaction with a Comb-Type Caisson Breakwater

This study develops an analytical solution for oblique wave interaction with a comb-type caisson breakwater based on linear potential theory. The fluid domain is divided into inner and outer regions according to the geometrical shape of breakwater. By using periodic boundary condition and separation of variables, series solutions of velocity potentials in inner and outer regions are developed. Unknown expansion coefficients in series solutions are determined by matching velocity and pressure of continuous conditions on the interface between two regions. Then, hydrodynamic quantities involving reflection coefficients and wave forces acting on breakwater are estimated. Analytical solution is validated by a multi-domain boundary element method solution for the present problem. Diffusion reflection due to periodic variations in breakwater shape and corresponding surface elevations around the breakwater are analyzed. Numerical examples are also presented to examine effects of caisson parameters on total wave forces acting on caissons and total wave forces acting on side plates. Compared with a traditional vertical wall breakwater, the wave force acting on a suitably designed comb-type caisson breakwater can be significantly reduced. This study can give a better understanding of the hydrodynamic performance of comb-type caisson breakwaters.

BUECKNER'S WORK CONJUGATE INTEGRALS AND WEIGHT FUNCTIONS FOR A CRACK IN ANISOTROPIC SOLIDS

The Bueckner work conjugate integrals are studied for cracks inanisotropic clastic solids.The difficulties in separating Lekhnitskii’s two complexarguments involved in the integrals are overcome and explicit functional representa-tions of the integrals are given for several typical cases.It is found that the pseudo-orthogonal property of the eigenfunction expansion forms presented previously forisotropic cases,isotropic bimaterials,and orthotropic cases,are proved to be also validin the present case of anisotropic material.Finally,Some useful path-independent in-tegrals and weight functions are proposed.

Hydroelastic Response of A Circular Plate in Waves on A Two-Layer Fluid of Finite Depth

The hydroelastic response of a circular, very large floating structure(VLFS), idealized as a floating circular elastic thin plate, is investigated for the case of time-harmonic incident waves of the surface and interfacial wave modes, of a given wave frequency, on a two-layer fluid of finite and constant depth. In linear potential-flow theory, with the aid of angular eigenfunction expansions, the diffraction potentials can be expressed by the Bessel functions. A system of simultaneous equations is derived by matching the velocity and the pressure between the open-water and the platecovered regions, while incorporating the edge conditions of the plate. Then the complex nested series are simplified by utilizing the orthogonality of the vertical eigenfunctions in the open-water region. Numerical computations are presented to investigate the effects of different physical quantities, such as the thickness of the plate, Young's modulus, the ratios of the densities and of the layer depths, on the dispersion relations of the flexural-gravity waves for the two-layer fluid. Rapid convergence of the method is observed, but is slower at higher wave frequency. At high frequency, it is found that there is some energy transferred from the interfacial mode to the surface mode.