Vol 55, No 2 (2019)

The Sturm-Liouville operator with spectral parameter in the boundary conditions is considered, and sufficient conditions for the basis property of the system of eigenfunctions of this operator in the space L_p(0, 1), 1 < p < ∞, are obtained.

We consider the Sturm-Liouville operator generated in the space L²[0,+∞) by the expression −d²/dx² + x + aδ(x − b), where δ is the Dirac delta function, a < 0, and b > 0, and the boundary condition y(0) = 0. We prove that the eigenvalues λ_n of this operator satisfy the inequalities λ₁ < λ₁⁰ and λ_n−1⁰ < λ_n ≤ λ_n⁰, n = 2, 3,..., where {−λ_n⁰} is the sequence of zeros of the Airy function Ai (λ). The problem on the location of the first eigenvalue λ₁ depending on the parameters a and b is solved. In particular, we obtain conditions under which λ₁ is negative and provide a lower bound for λ₁.

The inverse Sturm-Liouville problem with nonseparated boundary conditions on a star-shaped geometric graph consisting of three edges with a common vertex is studied. It is shown that the Sturm-Liouville problem with general boundary conditions cannot be reconstructed uniquely from four spectra. A class of nonseparated boundary conditions is obtained for which two uniqueness theorems for the solution of the inverse Sturm-Liouville problem are proved. In the first theorem, the data used to reconstruct the Sturm-Liouville problem are the spectrum of the boundary value problem itself and the spectra of three auxiliary problems with separated boundary conditions. In the second theorem, instead of the spectrum of the problem itself, one only deals with five of its eigenvalues. It is shown that the Sturm-Liouville problem with these nonseparated boundary conditions can be reconstructed uniquely if three spectra of auxiliary problems and five eigenvalues of the problem itself are used as the reconstruction data. Examples of unique reconstruction of potentials and boundary conditions of the Sturm-Liouville problem posed on the graph under study are given.

A Carleman estimate is obtained for the solutions of a hyperbolic-parabolic system adjoint to the Navier–Stokes equations for a compressible medium.

We define a generalized Beltrami system, which is a broad generalization of the scalar Beltrami equation to vector equations. The Riemann-Hilbert boundary value problem is considered for such a system under the assumption that it is elliptic (i.e., the roots of the characteristic equation belong to the interior of the unit disk centered at zero). A Cordes type condition on the location of the roots of the characteristic equation of the system is obtained; this is a sufficient condition for the solution of this problem to have the Hölder property. The proof is based on the properties of singular integral operators in a domain.

Lomov’s regularization method is generalized to singularly perturbed integral equations with one-fold and multiple integral operators. We consider the case in which the kernel of the one-fold integral only depends on the time variable and is independent of the spatial variable. In this case, in contrast to Imanaliev’s works, we construct a regularized asymptotic solution of any order (with respect to the parameter). We also study the initialization problem, i.e., the problem of choosing a class of initial data of the problem for which it is possible to pass to the limit in its solution (as the small parameter tends to zero) to some limit operation mode on the whole prescribed set of independent variables, including the boundary layer region.

We consider a second-order differential equation that is a mathematical model of transverse vibrations of a string or longitudinal vibrations of an elastic rod. The coefficients of the second derivatives are piecewise constant functions. An existence and uniqueness theorem is proved and an explicit formula is given for the generalized solution of the Cauchy problem.

We consider linear inhomogeneous vector systems of higher-order ordinary differential equations in which the coefficient of the highest derivative of the unknown vector function is a matrix identically singular in the domain where the system is defined. We study how perturbations of the system by a Volterra operator, as well as perturbations of the initial data and the free term, affect the solutions. The corresponding estimates are obtained, which are then used to justify the application of the least squares method to the numerical solution of the corresponding initial value problems.