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 Property of the System of Eigenfunctions of a Spectral Problem with Spectral Parameter in the Boundary Condition

In the present paper, we study basis properties of the system of eigenfunctions of the boundary value problem (1.1),(1.2) in the spaces Lp (0, l)(1< p<∞). Boundary value problems for second-and fourth-order ordinary differential operators with a spectral parameter in boundary conditions were studied in a series of papers (eg, see [1–17]). A number of problems in mathematical physics can be reduced to such problems (eg, see [2–10]). Basis properties of the system of eigenfunctions of the Sturm–Liouville problem with a spectral parameter in the boundary condition were studied in [10–16] in various function spaces, and the existence of eigenvalues, estimates of eigenvalues and eigenfunctions, and expansion theorems were considered in [1, 6, 8, 9] for fourth-order ordinary differential operators with a spectral parameter in a boundary condition

The basis properties of eigenfunctions in the eigenvalue problem with a spectral parameter in the boundary condition

Boundary value problems for second- and fourth- order ordinary differential operators with a spectral parameter in the boundary conditions have been exten- sively studied (see, eg, [1–9]). In [3–5], such problems were associated with particular physical processes. The basis properties of the system of eigenfunctions in the Sturm–Liouville problem with a spectral param- eter in the boundary conditions were studied in various function spaces in [7–9]. The existence of eigenvalues, estimate for eigenvalues and eigenfunctions, and expansion theorems for fourth-order operators with a spectral parameter in the boundary condition were con- sidered in [1, 6] … This paper deals with the basis properties in Lp(0, l) (1 < p < ∞) of the system of eigenfunctions of boundary value problem (1), (2)

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