In this paper we consider the Gellerstedt problem for the Lavrentiev-Bitsadze equation with boundary conditions on parallel characteristics in the hyperbolic part of the equation.
We study the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with the oddness boundary condition on the boundary of the ellipticity domain. All eigenvalues and eigenfunctions are obtained in closed form. It is proved that the system of eigenfunctions is complete in the elliptic part of the domain and incomplete in the entire domain. The unique solvability of the problem is also proved; the solution is written in the form of a series if the spectral parameter is not equal to an eigenvalue. For the spectral parameter coinciding with an eigenvalue, solvability conditions are obtained under which the family of solutions is found in the form of a series. A condition for the solvability of the problem depending on the eigenvalues is obtained. The constructed analytical solutions can be used efficiently in numerical modeling of transonic gas dynamics problems.
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.
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.
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.
Two boundary value problems in which one of the conditions is nonlocal and contains a real parameter are studied for an equation of mixed type in a half-strip. Sufficient conditions for the unique solvability of these problems are obtained under some restrictions on the parameter.
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 influence exerted by the second time derivative with a small parameter added to the heat equation in the case of discontinuous periodic initial data is investigated. It is shown that, except for the initial instants of time, the error of hyperbolization vanishes as the square root of the addition.
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.
We study the solvability of the Gellerstedt problem for the Lavrent’ev–Bitsadze equation. An initial function is posed in the ellipticity domain of the equation on the boundary of the unit half-circle with center the origin. Zero conditions are posed on characteristics in the hyperbolicity domain of the equation. “Frankl-type conditions” are posed on the type change line of the equation. We show that the problem is either conditionally solvable or uniquely solvable. We obtain a closed-form solvability condition in the case of conditional solvability. We derive integral representations of the solution in all cases.
We study the solvability of the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with nonclassical matching conditions for the gradient of the solution (in the sense of Frankl) on the type change line of the equation. We prove that the inhomogeneous Gellerstedt problem with data on the external characteristics of the equation is solvable either uniquely or modulo a nontrivial solution of the homogeneous problem. We obtain integral representations of the solution of the problem in both the elliptic and the hyperbolic parts of the domain. The solution proves to be regular.
