Eigenfunctions Of Boundary Value Problem Research Articles

On solvability of the inverse problem for a fourth-order parabolic equation with a complex-valued coefficient

In this paper, the inverse problem for a fourth-order parabolic equation with a variable complex-valued coefficient is studied by the method of separation of variables. The properties of the eigenvalues of the Dirichlet and Neumann boundary value problems for a non-self-conjugate fourth-order ordinary differential equation with a complex-valued coefficient are established. Known results on the Riesz basis property of eigenfunctions of boundary value problems for ordinary differential equations with strongly regular boundary conditions in the space L2(−1,1) are used. On the basis of the Riesz basis property of eigenfunctions, formal solutions of the problems under study are constructed and theorems on the existence and uniqueness of solutions are proved. When proving theorems on the existence and uniqueness of solutions, the Bessel inequality for the Fourier coefficients of expansions of functions from space L2(−1,1) into a Fourier series in the Riesz basis is widely used. The representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are derived. The convergence of the obtained solutions is discussed.

Inverse problems for the beam vibration equation with involution

This article considers inverse problems for a fourth-order hyperbolic equation with involution. The existence and uniqueness of a solution of the studied inverse problems is established by the method of separation of variables. To apply the method of separation of variables, we prove the Riesz basis property of the eigenfunctions for a fourth-order differential operator with involution in the space ${{L}_{2}}(-1,1)$. For proving theorems on the existence and uniqueness of a solution, we widely use the Bessel inequality for the coefficients of expansions into a Fourier series in the space ${{L}_{2}}(-1,1)$. A significant dependence of the existence of a solution on the equation coefficient $\alpha$ is shown. In each of the cases $\alpha <-1$, $\alpha >1$, $-1<\alpha<1$ representations of solutions in the form of Fourier series in terms of eigenfunctions of boundary value problems for a fourth-order equation with involution are written out.

High-order harmonic cut-off frequency in atomic silver irradiated by femtosecond laser pulses: theory and experiment

The results of theoretical and experimental study of high-order optical harmonic generation in the ensemble of silver atoms irradiated by intense femtosecond pulses of Ti:Sapphire laser are presented. It is shown that the photoemission spectra exhibit unusual behavior when the laser field strength approaches near-atomic values. In subatomic field strength the cut-off frequency increases linearly with laser pulse intensity. However, when the field strength approaches near-atomic region firstly cut-off frequency slows down and then saturates. To give new interpretation of such kind of photoemission spectrum behavior we have proposed the light-atom interaction theory based on the use of eigenfunctions of boundary value problem for “the atom in an external field” instead of the traditional basis of the “free atom” eigenfunctions. The results of computer simulations clearly demonstrate saturation of the cut-off frequency at near-atomic strength of a laser field. The problem of angular divergence of a harmonic emission is analyzed.

Spectral Properties of Non‐Self‐Adjoint Perturbations for a Spectral Problem with Involution

Full description of Riesz basis property for eigenfunctions of boundary value problems for first order differential equations with involutions is given.

Linear independence of boundary traces of eigenfunctions of elliptic and Stokes operators and applications

This paper is divided into two parts and focuses on the linear independence of boundary traces of eigenfunctions of boundary value problems. Part I deals with second-order elliptic operators, and Part II with Stokes (and Oseen) operators. Part I: Let λi be an eigenvalue of a second-order elliptic operator defined on an open, sufficiently smooth, bounded domain Ω in Rn, with Neumann homogeneous boundary conditions on Γ = ∂Ω. Let {φij} j=1 be the corresponding linearly independent (normalized) eigenfunctions in L2(Ω), so that `i is the geometric multiplicity of λi. We prove that the Dirichlet boundary traces {φij |Γ1} `i j=1 are linearly independent in L2(Γ1). Here Γ1 is an arbitrary open, connected portion of Γ , of positive surface measure. The same conclusion holds true if the setting {Neumann B.C., Dirichlet boundary traces} is replaced by the setting {Dirichlet B.C., Neumann boundary traces}. These results are motivated by boundary feedback stabilization problems for parabolic equations [L-T.2]. Part II: The same problem is posed for the Stokes operator with motivation coming from the boundary stabilization problems in [B-L-T.1]– [B-L-T.3] (with tangential boundary control), and [R] (with just boundary control), where we take Γ1 = Γ . The aforementioned property of boundary traces of eigenfunctions critically hinges on a unique continuation result from the boundary of corresponding over-determined problems. This is well known in the case of second-order elliptic operators of Part I; but needs to be established in the case of Stokes operators. A few proofs are given here. 2000 Mathematics Subject Classification: 35L20, 47, 49K20, 76N25, 76Q05, 93B29, 93C20.

A Boundary Value Problem for the Dirac System with a Spectral Parameter in the Boundary Conditions

Problems with a spectral parameter in equations and boundary conditions form an important part of spectral theory of linear differential operators. A bibliography of papers in which such problems were considered in connection with specific physical processes can be found in [1, 2]. Boundary value problems for ordinary differential operators with a spectral parameter in boundary conditions were considered in various settings in numerous papers [3–14]. The completeness and the basis property of eigenfunctions of boundary value problems with a spectral parameter in equations and boundary conditions were studied in more detail in [7, 8]. Consider the following boundary value problem for the Dirac system with the same spectral parameter in the equations and the boundary conditions:

Asymptotic expansions for the eigenvalues and the eigenfunctions of boundary value problems with rapidly oscillating coefficients in a layer

Asymptotic expansions for the eigenvalues and the eigenfunctions of boundary value problems with rapidly oscillating coefficients in a layer

Matrix methods for solving the boundary-value problem for a second-order differential equation with delayed argument

Zero-rank matrix numerical differentiation algorithms are applied to construct efficient numerical-analytical methods (so-called zero-rank matrix methods) to find the eigenvalues and eigenfunctions of boundary-value problems for second-order differential equations with a delayed argument. The proposed methods are analyzed.

Asymptotic solution of eigenvalue problems

A method is presented for the construction of asymptotic formulas for the large eigenvalues and the corresponding eigenfunctions of boundary value problems for partial differential equations. It is an adaptation to bounded domains of the method previously devised to deduce the corrected Bohr-Sommerfeld quantum conditions. When applied to the reduced wave equation in various domains for which the exact solutions are known, it yields precisely the asymptotic forms of those solutions. In addition it has been applied to an arbitrary convex plane domain for which the exact solutions are not known. Two types of solutions have been found, called the “whispering gallery” and “bouncing ball” modes. Applications have also been made to the Schrödinger equation.

On the Expansion of an Arbitrary Function in a Series of Eigenfunctions of Boundary Value Problems II

Let D be a bounded open domain, simply or multiply connected, whose boundary C is composed of a finite number of regular curves in the x,y plane, and let w n (x,y) be the eigenfunctions (normal and orthogonal) with the corresponding eigenvalues λ n > 0, n = 1, 2 , . . . of the boundary value problem.

On the expansion of an arbitrary function in a series of eigenfunctions of boundary value problems

On the expansion of an arbitrary function in a series of eigenfunctions of boundary value problems