Рассматривается задача для уравнения Бесселя целого порядка с комплексным физическим и спектральным параметрами в граничном условии. Спектральный параметр в граничное условие входит квадратично. Изучается вопрос базисности системы собственных функций.

Асташова И.В., Барабанов Е.А., Боровских А.В., Бутузов В.Ф., Быков В.В., Ветохин А.Н., Глызин С.Д., Горицкий А.Ю., Денисова Н.В., Изобов Н.А., Ильин А.В., Ильяшенко Ю.С., Капустина Т.О., Кигурадзе И.Т., Козлов В.В., Колесов А.Ю., Коньков А.А., Ломов И.С., Моисеев Е.И., Палин В.В. и др.//Дифференциальные уравнения, 2021

We study two problems on planar steady-state weakly supersonic potential flows of anideal perfect gas with detached shock wave. In the first problem, we consider the flow past a finitewedge in an unbounded stream; the second problem deals with the flow past an infinite wedge in asteady jet. The free-stream velocity is close to the speed of sound, and therefore, the entropyincrements $$\DeltaS(\psi )=O(\varepsilon ^2)$$ on the shock waveand the derivative of the entropy with respect to the stream function are disregarded. In thevelocity hodograph plane, the cross-coupled sub- and supersonic flow behind a shock wave isdescribed by a solution of a Tricomi type problem. On part of the boundary depicting the shockwave, the directional derivative condition is set for the stream function. It is proved that itsdirection is not tangent to the domain boundary. The uniqueness of the solution for the problemsunder consideration follows from the “strong” Hopf maximum principle for uniformly ellipticequations. Replacing the Chaplygin equation with the Lavrent’ev–Bitsadze equation leads to twoHilbert problems for analytic functions with piecewise constant boundary conditions. Thesolutions of the Hilbert problems are expressed using the Schwarz operator.

We consider a problem for the zero-order Bessel equation with complex-valued physical and spectral parameters in the boundary condition. The spectral parameter is squared in the boundary condition. We study the basis property for the system of eigenfunctions.

We study the problem of boundary control of string vibrations on a subcritical time interval. The control is performed by displacements at one end of the string, while a homogeneous boundary condition with a noncharacteristic directional derivative is posed at the other end. The problem is studied in the classical sense. Necessary and sufficient conditions for the existence of a unique control are obtained, and the control itself is constructed in explicit analytical form.

The paper addresses the completeness of perturbed systems of sines and cosines with non-integer indices and, moreover, complex, in the space of Lebesgue-integrable functions. The criteria for the variation from the integer value have been found such that sine and cosine systems are complete.

Structure of the fundamental solutions of the Laplace equation in the elliptic and spheroidal coordinate systems is investigated. It was shown that among the solutions separated in these coordinate systems there are such solutions that have the form of harmonic polynomials in the original Cartesian coordinate system. These polynomials constitute basis for constructing with the help of radial multipliers the fundamental systems of functions that are consistent with the inhomogeneities of the elliptic and ellipsoidal shapes. These functions are used in problems of mechanics of inhomogeneous media for accurate modeling of physical processes near such inclusions. On the basis of these functions algorithms are constructed for defining effective thermophysical characteristics of inhomogeneous media with inclusions of elliptic and spheroidal shapes.

We study a boundary value problem with the oblique derivative on the semicircle and mixed conditions on the diameter for the Helmholtz equation in a semidisk, and also its relation to a 3D problem for the Laplace equation.

The Gellerstedt eigenvalue problem with homogeneous boundary conditions on interior characteristics is studied. We prove that the eigenfunction system of this problem is a Riesz basis in the L2 space in the elliptic domain.

The paper addresses the completeness of sines and cosines with non-integer indices in the space of Lebesgue-integrable functions. The criteria for the variation from the integer value have been found such that sine and cosine systems are complete.

We study the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with boundary conditions on parallel characteristics in the hyperbolic domain of the equation. Three distinct types of conditions on the type change lines are considered, and existence and uniqueness theorems for the corresponding problems are proved.