Direct and Inverse Problems for the Heat Equation with a Dynamic type Boundary Condition

This paper considers the initial-boundary value problem for the heat equation with a dynamic type boundary condition. Under some regularity, consistency and orthogonality conditions, the existence, uniqueness and continuous dependence upon the data of the classical solution are shown by using the generalized Fourier method. This paper also investigates the inverse problem of finding a time-dependent coefficient of the heat equation from the data of integral overdetermination condition

Spectral properties of some regular boundary value problems for fourth order differential operators

In this paper we consider the problem y ıv + p2(x)y 00 + p1(x)y 0 + p0(x)y = λy, 0 < x < 1, y (s) (1) − (−1)σy (s) (0) +Xs−1 l=0 αs,ly (l) (0) = 0, s = 1, 2, 3, y(1) − (−1)σy(0) = 0, where λ is a spectral parameter; pj(x) ∈ L1(0, 1), j = 0, 1, 2, are complex-valued functions; αs,l, s = 1, 2, 3, l = 0, s − 1, are arbitrary complex constants; and σ = 0, 1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established in the case α3,2 + α1,0 =6 α2,1. It is proved that the system of root functions of this spectral problem forms a basis in the space Lp(0, 1), 1 < p < ∞, when α3,2 +α1,0 6= α2,1, pj(x) ∈ W j 1 (0, 1), j = 1, 2, and p0(x) ∈ L1(0, 1); moreover, this basis is unconditional for p = 2

Some problems of spectral theory of fourth-order differential operators with regular boundary conditions

In this paper, we consider the problem yıv + q (x) y = λy, 0 < x < 1, y (1) − (−1) σ y (0) + αy (0) + γ y (0) = 0, y (1) − (−1) σ y (0) + βy (0) = 0, y (1) − (−1) σ y (0) = 0, y (1) − (−1) σ y (0) = 0 where λ is a spectral parameter; q (x) ∈ L1 (0, 1) is a complex-valued function; α, β, γ are arbitrary complex constants and σ = 0, 1. The boundary conditions of this problem are regular, but not strongly regular. Asymptotic formulae for eigenvalues and eigenfunctions of the considered boundary value problem are established and it is proved that all the eigenvalues, except for a finite number, are simple in the case αβ = 0. It is shown that the system of root functions of this spectral problem forms a basis in the space L p (0, 1), 1 < p < ∞, when αβ = 0; moreover, this basis is unconditional for p = 2

On Basicity In Lp (0, 1) (1 < p < ∞) Of The System Of Eigenfunctions Of One Boundary Value Problem. I

The basis properties of the spectral problem is investigated for differential operator of the second order with the spectral parameter in both boundary conditions. In this part of the paper the oscillation properties of eigenfunctions are established and the asymptotic formulae are derived for eigen values and eigenfunctions

On oscillation properties of the eigenfunctions of a fourth order differential operator

The spectral problem for a fourth order ordinary differential operator is investigated. The oscillation properties of the eigenfunctions and their derivatives are established

Spectral Properties of the Differential Operators of the Fourth-Order with Eigenvalue Parameter Dependent Boundary Condition

We consider the fourth-order spectral problem y4 x−qxy x λyx, x ∈ 0, l with spectral parameter in the boundary condition. We associate this problem with a selfadjoint operator in Hilbert or Pontryagin space. Using this operator-theoretic formulation and analytic methods, we investigate locations in complex plane and multiplicities of the eigenvalues, the oscillation properties of the eigenfunctions, the basis properties in Lp0, l, p ∈ 1, ∞, of the system of root functions of this problem

On The Basis Properties And Convergence Of Expansions In Terms Of Eigenfunctions For A Spectral Problem With A Spectral Parameter In The Boundary Condition

In this paper, we consider the spectral problem − y 00 + q (x) y = λy, 0 < x < 1, y (0) = 0, y0 (0) − dλy (1) = 0, where λ is a spectral parameter, q (x) ∈ L1 (0, 1) is a complex-valued function and d is an arbitrary nonzero complex number. We study the spectral properties ( asymptotic formulae for eigenvalues and eigenfunctions, minimality and basicity of the system of eigenfunctions, the uniform convergence of expansions in terms of eigenfunctions ) of the considered boundary value problem

The Oscillation Properties Of The Boundary Value Problem With Spectral Parameter In The Boundary Condition

The spectral problem is investigated for the fourth order ordinary differential operator with spectral parameter in the boundary conditions. The oscillation properties of the eigenfunctions of this problem are established

An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions

This paper investigates the inverse problem of finding a time-dependent coefficient in a heat equation with nonlocal boundary and integral overdetermination conditions. Under some regularity and consistency conditions on the input data, the existence, uniqueness and continuous dependence upon the data of the solution are shown by using the generalized Fourier method

The Basis Property Of Sturm–Liouville Problems With Boundary Conditions Depending Quadratically On The Eigenparameter

We study basisness of root functions of Sturm–Liouville problems with a boundary condition depending quadratically on the spectral parameter. We determine the explicit form of the biorthogonal system. Using this we prove that the system of root functions, with arbitrary two functions removed, form a minimal system in L2, except some cases where this system is neither complete nor minimal. For the basisness in L2 we prove that the part of the root space is quadratically close to systems of sines and cosines. We also consider these basis properties in the context of general Lp