18 research outputs found
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
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
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
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