On the number of real eigenvalues of a certain boundary-value problem for a second-order equation with fractional derivative

The asymptotics as α → 0+ of the number of real eigenvalues λ _n (α) of the problem y″(x)+λD ₀ ^α (x) = 0, 0 < x < 1, y(0) = y(1) = 0, is obtained. The minimization of real eigenvalues is carried out. It is proved that . __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 137–155, 2006.     

On one boundary-value problem for a strongly degenerate second-order elliptic equation in an angular domain

We prove the existence and uniqueness of a classical solution of a singular elliptic boundary-value problem in an angular domain. We construct the corresponding Green function and obtain coercive estimates for the solution in the weighted Hölder classes.     

Cauchy problem for a second-order hyperbolic operator-differential equation with a singular coefficient

In a Hilbert space, we study the well-posedness of the Cauchy problem for a second-order operator-differential equation with a singular coefficient.     

Asymptotic behavior of the first boundary-value problem for elliptic equations in domains with a thin coating

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 6, pp. 74–84, November–December, 1988.     

Approximate solution of third boundary-value problem for second-order elliptic equation in domains of arbitrary shape

The fictitious domain method is applied to construct a difference scheme for solving the third boundary-value problem for a second-order elliptic equation in domains of arbitrary shape. A rate of convergence bound is derived.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 3–7, 1986.     

On the solvability of a boundary-value problem for an elliptic equation

Consider a boundary-value problem for a second-order linear elliptic equation in a bounded domain. The coefficient of the required function is nonpositive everywhere in the domain, except for a small neighborhood of an interior point. The following question arises: Under what constraints on this coefficient in the given small domain do the statements on the existence and uniqueness of the solution of the first boundary-value problem remain valid?     

A linear periodic boundary-value problem for a second-order hyperbolic equation

We study the boundary-value problemu _tt -u _xx =g(x, t),u(0,t) =u (π,t) = 0,u(x, t +T) =u(x, t), 0 ≤x ≤ π,t ∈ ℝ. We findexact classical solutions of this problem in three Vejvoda-Shtedry spaces, namely, in the classes of, and-periodic functions (q and s are natural numbers). We obtain the results only for sets of periods, and which characterize the classes of π-, 2π -, and 4π-periodic functions. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 2, pp. 281–284, February, 1999.     

Separate asymptotics of two series of eigenvalues for a single elliptic boundary-value problem

The spectral problem in a bounded domain Ω?Rⁿ is considered for the equation Δu= λu in Ω, ?u=λ?υ/?ν on the boundary of Ω (ν the interior normal to the boundary, Δ, the Laplace operator). It is proved that for the operator generated by this problem, the spectrum is discrete and consists of two series of eigenvalues {λ _j ⁰ } _j=1 ^∞ and {λ _j ^∞ } _j=1 ^∞ , converging respectively to 0 and +∞. It is also established that $$N^0 (\lambda ) = \sum\nolimits_{\operatorname{Re} \lambda _j^0 \geqslant 1/\lambda } {1 \approx const} \lambda ^{n - 1} , N^\infty (\lambda ) \equiv \sum\nolimits_{\operatorname{Re} \lambda _j^\infty \leqslant \lambda } {1 \approx const} \lambda ^{n/1} .$$ The constants are explicitly calculated.     

Solvability of the boundary-value problem for the second-order elliptic differential-operator equation with spectral parameter in the equation and boundary conditions

We study the solvability of a boundary-value problem for the second-order elliptic differential-operator equation with spectral parameter both in the equation and in boundary conditions. We also analyze the asymptotic behavior of the eigenvalues corresponding to the uniform boundary-value problem.     

Linear periodic boundary-value problem for a second-order hyperbolic equation. I

In three spaces, we obtain exact classical solutions of the boundary-value periodic problem u _tt−a ² u _xx=g(x,t), u(0,t)=u(π,t)=0, u(x,t+T)=u(x,t)=0, x,t∈ĝ Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 11, pp. 1537–1544, November, 1998.     

The asymptotic behavior of the Green function of the boundary-value problem for a second-order equation on a geometric graph

The boundary-value problem (where s is a complex parameter with Re s > 0) on an open connected geometric graph is considered. The asymptotic behavior as Im s → ∞ of the Green function is determined; it implies that the Green function has a preimage under the one-sided Laplace transform. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 38, Suzdal Conference-2004, Part 3, 2006.     

On the eigenvalues of second-order boundary-value problems

In this paper we investigate the properties of eigenvalues of some boundary-value problems generated by second-order Sturm-Liouville equation with distributional potentials and suitable boundary conditions. Moreover, we share a necessary condition for the problem to have an infinitely many eigenvalues. Finally, we introduce some ordinary and Frechet derivatives of the eigenvalues with respect to some elements of the data.     

Linear periodic boundary-value problem for a second-order hyperbolic equation. II. Quasilinear problem

In three spaces, we find exact classical solutions of the boundary-value periodic problem u_tt - a²u_xx = g(x, t) u(0, t) = u(π, t) = 0, u(x, t + T) = u(x, t), x ∈ ℝ, t ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator and whose right-hand side is a nonlinear operator. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1680–1685, December, 1998.