1 - 10 of 14 articles
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...
We consider a spectral problem generated by a Sturm-Liouville equation on the interval (0, π) with degenerate boundary conditions. We prove the existence of potentials q(x) ∈ L
2(0, π) such that the multiplicities of the eigenvalues λ
monotonically tend to infinity as n → ∞.
We study the Dirac operator with a complex-valued integrable potential in the space ℍ = L
2[0, π]. We obtain asymptotic formulas for a fundamental solution system of an operator. Remainders in each of the formulas are estimated.
We study the differential properties of the convolution of functions with a generalized Bessel-Macdonald kernel. The integral properties of a function are characterized in terms of its decreasing permutation. The differential properties of the convolution are described in terms of its modulus of...
We study sharp sufficient conditions on the growing lower coefficients of a parabolic equation guaranteeing the stabilization of the solution of the Cauchy problem to zero in some classes of growing initial functions.
For a system of Poisson equations in a three-dimensional domain, we consider two nonstandard problems with vector boundary conditions. On the basis of inequalities of the Friedrichs type, we show that these problems are well posed in the Hadamard-Petrovskii sense.
We study the asymptotics of solutions of partial differential equations with higher degenerations. Such equations arise, for example, when studying solutions of elliptic equations on manifolds with cuspidal singular points. We construct the asymptotics of a solution of the Laplace equation...
We consider general nonlinear evolution equations of arbitrary order. For these equations, we find conditions under which the Cauchy problem has no solutions global in t > 0. We also estimate the time beyond which the solution of the considered Cauchy problem necessarily does not exist.
We consider longitudinal elastic vibrations of a composite rod and find closedform expressions that describe optimal boundary controls bringing the rod from the quiescent state into a state with given displacement function φ(t) and velocity function ψ(t) in time T. We assume that the wave...
We consider mixed initial-boundary value problems for longitudinal vibrations described by the telegraph equation in the case of a system consisting of several parts with different densities and elasticities but with equal impedances. We consider the cases of control by displacements at both...
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