On the uniform convergence of the Fourier series for a spectral problem with squared spectral parameter in a boundary condition

We analyze the uniform convergence of the Fourier series expansions of Hölder functions in the system of eigenfunctions of a spectral problem with squared spectral parameter in a boundary condition. To this end, we first prove a theorem on the equiconvergence of such expansions with those in a well-known orthonormal basis.     

On the spectral problem arising in the solution of a mixed problem for the heat equation with a mixed derivative in the boundary condition

We study issues related to the uniform convergence of the Fourier series expansions of Hölder class functions in the system of eigenfunctions corresponding to a spectral problem obtained from a mixed problem for the heat equation. We prove a theorem on the equiconvergence of these expansions with expansions in a well-known orthonormal basis.     

On the uniform convergence in <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript> of Fourier series for a spectral problem with squared spectral parameter in a boundary condition

We study the uniform convergence in C ¹ of the Fourier series of a Hölder function in a system of eigenfunctions corresponding to a spectral problem with squared spectral parameter in a boundary condition. We preliminarily study one more spectral problem with a spectral parameter in a boundary condition.     

On the uniform convergence of spectral expansions in a problem for the Laplace operator on the square with spectral parameter in the boundary condition

We consider a classical spectral problem that arises when studying the natural vibrations of a loaded rectangular membrane. We establish conditions ensuring the uniform convergence of spectral expansions in the selected Riesz basis and in the entire system of eigen-functions.     

Asymptotic property and convergence estimation for the eigenelements of the Laplace operator

In this article, we provide a rigorous derivation of asymptotic expansions for eigenfunctions and we establish convergence estimation for both eigenvalues and eigenfunctions of the Laplacian. We address the integral equation method to investigate the interplay between the geometry, boundary conditions and spectral properties of the eigenelements of the Laplace operator under deformation of the domain. The asymptotic formula and convergence estimation are tested by numerical examples.     

Basis properties of a problem for the Laplace operator on the square with spectral parameter in a boundary condition

We consider a classical spectral problem that arises when studying the natural vibrations of a loaded rectangular membrane fixed on two sides, the load being distributed along one of the free sides. We study the completeness, minimality, and basis property of the system of eigenfunctions and establish conditions guaranteeing the equiconvergence of spectral expansions in this system and in a given basis.     

Asymptotics and Estimates of the Convergence Rate in a Three-Dimensional Boundary-Value Problem with Rapidly Alternating Boundary Conditions

We consider a singularly perturbed boundary-value eigenvalue problem for the Laplace operator in a cylinder with rapidly alternating type of the boundary condition on the lateral surface. The change of the boundary conditions is realized by splitting the lateral surface into many narrow strips on which the Dirichlet and Neumann conditions alternate. We study the case in which the averaged problem contains the Dirichlet boundary condition on the lateral surface. In the case of strips with slowly varying width we construct the first terms of the asymptotic expansions of eigenfunctions; moreover, in the case of strips with rapidly varying width we obtain estimates for the convergence rate.     

On the convergence in W 2 m of spectral expansions for a spectral problem with boundary conditions of the third kind one of which contains the spectral parameter

We consider a spectral problem with boundary conditions of the third kind one of which contains the spectral parameter occurring linearly. We study the basis property of the system of eigenfunctions of this spectral problem in W ₂ ^m . We obtain conditions under which the system becomes a basis in this space after the deletion of any single eigenfunction.     

Nonlinear spectral problem for a singular system of ordinary differential equations with a nonlocal condition

We consider a nonlinear spectral problem for a system of ordinary differential equations defined on an unbounded half-line and supplemented with a nonlocal condition specified by a Stieltjes integral. We suggest a numerically stable method for finding the number of eigenvalues lying in a given bounded domain of the complex plane and for the computation of these eigenvalues and the corresponding eigenfunctions. Our approach uses a simpler (with uncoupled boundary conditions) auxiliary boundary value problem for the same equation.     

On the multiple spectrum of a problem for the Bessel equation with spectral parameter in the boundary condition

A problem for the Bessel equation of zero order with complex physical and spectral parameters in the boundary condition is considered. We study whether the system of eigenfunctions has the basis property.     

Asymptotic Expansions of Eigenelements of the Laplace Operator in a Domain with Many “Light” Concentrated Masses Closely Located on the Boundary. Multi-Dimensional Case

A spectral problem for the Laplace operator is considered in a domain with singular density near the boundary (with concentrated masses). The boundary condition is of rapidly variable type. We construct asymptotic expansions for the eigenvalues and eigenfunctions of the problem with respect to a small parameter characterizing the diameter and density of masses, as well as distance between the masses. The expansions are justified. Bibliography: 59 titles. Illustrations: 3 figures.__________Translated from Problemy Matematicheskogo Analiza, No. 30, 2005, pp. 87–119.     

On the convergence of expansions in polyharmonic eigenfunctions

We consider expansions of smooth, nonperiodic functions defined on compact intervals in eigenfunctions of polyharmonic operators equipped with homogeneous Neumann boundary conditions. Having determined asymptotic expressions for both the eigenvalues and eigenfunctions of these operators, we demonstrate how these results can be used in the efficient computation of expansions. Next, we consider the convergence. We establish the key advantage of such expansions over classical Fourier series–namely, both faster and higher-order convergence–and provide a full asymptotic expansion for the error incurred by the truncated expansion. Finally, we derive conditions that completely determine the convergence rate.     

Dependence of estimates of the local convergence rate of spectral expansions on the distance from an interior compact set to the boundary

We consider the problem on the convergence rate of biorthogonal expansions of functions in systems of root functions of a wide class of ordinary second-order differential operators defined on a finite interval. These expansions are compared with expansions of the same functions in Fourier trigonometric series in an integral or uniform metric on any interior compact subset of the main interval. We find the dependence of the equiconvergence rate of resulting expansions on the distance from the compact set to the boundary of the interval, on the coefficients of the differential operation, and on the presence of infinitely many associated functions in the system of root functions.     

On the eigenvalues of a nonlinear spectral problem

We consider a nonlinear eigenvalue problem of the Sturm–Liouville type with conditions of the third kind, which describes the propagation of polarized electromagnetic waves in a plane dielectric waveguide. The equation is nonlinear in the unknown function, and the boundary conditions depend on the spectral parameter nonlinearly. We obtain an equation for the spectral parameter and formulas for the zeros of the eigenfunctions and show that the problem has at most finitely many isolated eigenvalues.     

Basis properties of a fourth order differential operator with spectral parameter in the boundary condition

We consider a fourth order eigenvalue problem containing a spectral parameter both in the equation and in the boundary condition. The oscillation properties of eigenfunctions are studied and asymptotic formulae for eigenvalues and eigenfunctions are deduced. The basis properties in L _p (0; l); p ∈ (1;∞); of the system of eigenfunctions are investigated.     

Spectral problem with Steklov condition on a thin perforated interface

A two-dimensional Steklov-type spectral problem for the Laplacian in a domain divided into two parts by a perforated interface with a periodic microstructure is considered. The Steklov boundary condition is set on the lateral sides of the channels, a Neumann condition is specified on the rest of the interface, and a Dirichlet and Neumann condition is set on the outer boundary of the domain. Two-term asymptotic expansions of the eigenvalues and the corresponding eigenfunctions of this spectral problem are constructed.     

On the convergence of solutions and eigenelements of a boundary value problem in a domain perforated along the boundary

We consider a boundary value problem in a model domain periodically perforated along the boundary. We assume that the homogeneous Neumann condition is posed on the external boundary and the homogeneous Dirichlet condition is posed on the boundary of the cavities. A limit (homogenized) problem is obtained. We prove the convergence of the solutions, eigenvalues, and eigenfunctions of the original problem to the solutions, eigenvalues, and eigenfunctions, respectively, of the limit problem.     

Nonlocal Dezin’s problem for Lavrent’ev–Bitsadze equation

Using the method of spectral analysis, for the mixed type equation u_xx + (sgny)u_yy = 0 in a rectangular domain we establish a criterion of uniqueness of its solution satisfying periodicity conditions by the variable x, a nonlocal condition, and a boundary condition. The solution is constructed as the sum of a series in eigenfunctions for the corresponding one-dimensional spectral problem. At the investigation of convergence of the series, the problem of small denominators occurs. Under certain restrictions on the parameters of the problem and the functions, included in the boundary conditions, we prove uniform convergence of the constructed series and stability of the solution under perturbations of these functions.     

EIGENFUNCTION EXPANSION FOR STURM-LIOUVILLE PROBLEMS WITH TRANSMISSION CONDITIONS AT ONE INTERIOR POINT

The purpose of this article is to extend some spectral properties of regular SturmLiouville problems to the special type discontinuous boundaxy-value problem,which consists of a Sturm-Liouville equation together with eigenparameter-dependent boundary conditions and two supplementary transmission conditions.We construct the resolvent operator and Green's function and prove theorems about expansions in terms of eigenfunctions in modified Hilbert space L_2[a,b].     

Estimates of the local convergence rate of spectral expansions for even-order differential operators

We study the convergence rate of biorthogonal series expansions of functions in systems of root functions of a wide class of even-order ordinary differential operators defined on a finite interval. These expansions are compared with the trigonometric Fourier series expansions of the same functions in the integral or uniform metric on an arbitrary interior compact set of the main interval as well as on the entire interval. We show the dependence of the equiconvergence rate of these expansions on the distance from the compact set to the boundary of the interval, on the coefficients of the differential operation, and on the existence of infinitely many associated functions in the system of root functions.