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This article investigates the Cauchy problem for a third-order pseudo-parabolic equation with a Bessel operator. Using the Erdélyi–Kober operator transformation, the Riemann's function for this equation is constructed, which is expressed through the hypergeometric Kampé de Fériet function. In particular, we obtain the Riemann's function for a one-dimensional pseudo-parabolic equation. [ABSTRACT FROM AUTHOR]

In this work, a nonlocal boundary value problem for a degenerate equation of elliptic type with a singular coefficient and a spectral parameter is studied in a vertical half-strip. By applying the Hankel transformation and the method of separation of variables, a solution to the nonlocal boundary value problem is obtained in explicit form. [ABSTRACT FROM AUTHOR]

We study integrals containing gamma functions and derive a lot of new relations involving hypergeometric and other special functions. [ABSTRACT FROM AUTHOR]

The previously introduced Radon–Kipriyanov transformation is adapted for the study of singular differential Bessel operators of even order : , . In this paper, the odd Radon–Kipriyanov transform and full Radon–Kipriyanov transform are introduced to study more general equations containing odd B-derivatives , (in particular, gradients of functions). Formulas for the complete Radon–Kipriyanov transformation of the corresponding linear singular differential operators are obtained. Based on the Bessel transformations (direct and inverse) introduced by B.M. Levitan, and "odd" Bessel transformations (direct and inverse) introduced by I.A. Kipriyanov and V.V. Katrakhov, a connection is obtained between the complete Radon–Kipriyanov transform with the Fourier transform and the mixed Fourier–Levitan–Kipriyanov–Katrakhov transform. The basic elementary properties of the complete Radon–Kipriyanov transformation are given. [ABSTRACT FROM AUTHOR]

In this work, the third boundary value problem for a three-dimensional mixed-type equation with three singular coefficients in a domain consisting of a quarter cylinder and a rectangular triangular prism is studied. The existence and uniqueness of the formulated problem is proved by the method of spectral analysis. The solution of the considered problem is constructed as the sum of a double series. In substantiating the uniform convergence of the constructed series, asymptotic estimates for the Bessel functions of the real and imaginary arguments were used. Based on them, estimates were obtained for each member of the series, which made it possible to prove the convergence of the resulting series and its derivatives up to the second order inclusive, as well as the existence theorem in the class of regular solutions. [ABSTRACT FROM AUTHOR]

In this work, for the Lavrentiev–Bitsadze equation, the problem Gellerstedt with a nonlocal boundary condition in the form of a parity requirement on the domain boundary in the elliptic part. All eigenvalues and eigenfunctions are explicitly written out, it is proved that the system of eigenfunctions is complete in the elliptic part of the domain and incomplete in the entire domain. For a spectral parameter not equal to an eigenvalue, unique solvability is proved and the solution is written as a series. For a spectral parameter coinciding with an eigenvalue, solvability conditions are obtained. Under the solvability conditions, the family of solutions is written out in the form of a series. Specified for which situations of solvability arise, and for which they do not. Complete proofs of the theorems announced earlier are presented. The constructed analytical solutions can serve as an effective test in the numerical simulation of problems of transonic gas dynamics. [ABSTRACT FROM AUTHOR]

We consider a general statement of linear inverse problem for -th order abstract differential equation in a Banach space. A universal uniqueness criterion for a solution of this problem is given. It is expressed through zeros of special Mittag-Leffler functions. The main emphasis is placed on the cases when all zeros of the functions are found explicitly and the uniqueness criterion takes an elementary form. It is shown that the previous results for equations of the st and nd orders follow from the general criterion. Fundamentally new is the case of one inverse problem for the th order differential equation, where the result also takes an elementary form. This last case is discussed in detail in our paper. [ABSTRACT FROM AUTHOR]

This article is devoted to the problem of including a subject area in a semantic library. The subject area is considered, which was not previously presented in part of the ontology of the digital library. A method based on the technology of establishing semantic links with the already accumulated content of the library is proposed. Main idea of the inclusion method is to use dictionaries, thesauri and links to the mathematical encyclopedia. They are used to attach a new scientific publication metadata set to an existing library metadata set. In the course of preliminary processing of texts of scientific publications, their semantic analysis is carried out and a local description of the subject area of this array is formed within the content of the library. Such subject areas as ordinary differential equations, partial differential equations, equations of continuum mechanics, equations of composite mechanics and their solutions expressed in terms of special functions of mathematical physics are considered. The description of the mathematical encyclopedia in its classical form is the central resource. It allows us to identify the appropriate classifiers through the links of the equations of applied problems and their solutions and use them also to form a description of the subject area. This subject area is new and has not been described in our digital library before. Proposed procedure is made possible only by data integration in the library. This takes into account the expansion of the description of subject areas related to the applications of mathematics in interdisciplinary research and technology. The implementation of this procedure is one of the new directions for filling the semantic libraries. It uses the approach of data integration and saturation with connections that are revealed only in the process of analyzing new data. If new references are found in the library during the process of adding a new description, then this method will also make it possible to identify and fill in the gaps in the description of related areas of mathematics, such as applications of solutions to classical problems of mathematical physics. [ABSTRACT FROM AUTHOR]

Asymptotics of Hermite–Padé approximants of the first type for Weil functions of discrete Meixner measure and its derivative is studied. We describe a piecewise-analytic curve in the complex plane which attracts part of the zeros of the Meixner multiple orthogonal polynomials. This curve possesses a symmetry property in the equilibrium problem with an external field for logarithmic potential of vector measures with a constraint. Some connection between the considered Hermite–Padé approximants and the theory of Diophantine approximations is presented. [ABSTRACT FROM AUTHOR]

For an equation of mixed elliptic-hyperbolic type of the second kind, the boundary problem in a rectangular region is studied. The criterion of uniqueness is established. The solution is built as a sum row. When substantiating convergence, the problem of small denominators. In this connection, estimates of separation from zero small denominators with the corresponding asymptotics, which and allowed us to justify the existence of a solution in the class of regular solutions and solution stability from boundary data. [ABSTRACT FROM AUTHOR]

Linear ordinary differential equations of the second order of a general view are considered with In variable factors (input equations) reduced to the self-interfaced form. The common decision Each input equation, by means of the integral formula, it is represented through the common decision The accompanying equation of the same type, but with constant factors. It is considered that the general solution of the accompanying equation is known. In the integral formula, except an accompanying solution, the fundamental solution of an input equation enters. From the integral formula implies two modes deriving of the approached analytical solution of an input equation. The first mode is based on Search of a fundamental solution by a method of perturbations. In the second mode assumed smoothness solutions of the accompanying equation and a possibility of its representation in a point of area in the form of a series Taylor through value of an accompanying solution in other point. As a result an initial solution it is represented in the form of a series on derivative of an accompanying solution. Factors at derivatives Are called as structural functions. They, actually, are the weighed moments fundamental solutions. Structural functions are continuous and are completely defined by functional dependence Initial factors from co-ordinates. They are converted in zero at constant initial factors, coinciding with accompanying factors. For determination of structural functions the system is received the recurrent equations. The well-founded mode of a choice of constants implies from this system Factors of the accompanying equation. [ABSTRACT FROM AUTHOR]

We consider a three-dimensional equation of elliptic type with two singular coefficients, for which the Dirichlet problem is studied in a half-cylinder. The study of the problem is carried out using the method of spectral analysis. For the problem posed, using the Fourier method, one-dimensional spectral problems are obtained. Based on the completeness property of systems of eigenfunctions of these problems, the uniqueness theorem is proved. The solution of the problem under study is constructed in the form of the sum of the Fourier–Bessel series. In substantiating the uniform convergence of the constructed series, we used asymptotic estimates of the Bessel functions of the real and imaginary argument. Based on them, estimates are obtained for each member of the series, which made it possible to prove the convergence of the resulting series and its derivatives to the second order inclusive, as well as the existence theorem in the class of regular solutions. [ABSTRACT FROM AUTHOR]

In the work, parabolic equations with perpendicular time directions considered together and the Gevrey problem with generalized gluing conditions, and a problem with the Bitsadze–Samarskii conditions have been studied. The uniqueness and existence of the solution of the considered problems were proved. The Gevrey problem was equivalently reduced to the two-point problem for ordinary differential equation involving fractional differential operators. The uniqueness of the solution of the obtained problem was proved by the principle of extremum and the existence was proved by the method of integral equations. The theory of the second kind Volterra integral equations and properties of two-parameter Mittag-Leffler function are applied at studying a unique solvability of the problem with Bitsadze–Samarskii condition. Necessary conditions for given functions for solvability of considered problems have been found. [ABSTRACT FROM AUTHOR]

A linear second-order hyperbolic equation of two independent variables with a singular Bessel operator is considered. For particular types of such equations, a detailed literature review of known methods for constructing Riemann functions is given. It is shown that to construct the Riemann function for equations with a singular Bessel operator, we can use the Erdélyi–Kober transmutation operator. The Riemann function for the Euler–Poisson–Darboux differential equations is found in explicit form. In this paper, we give examples and an algorithm for constructing the Riemann function for second-order hyperbolic equations with the Bessel operator. [ABSTRACT FROM AUTHOR]

In the present work we study the Neumann and Dirichlet boundary value problems for a four-dimensional (4D) degenerate elliptic equation by means the fundamental solutions, which were constructed earlier. Fundamental solutions are expressed by Lauricella hypergeometric functions. To prove the uniqueness of the solutions to the problems under consideration, the energy-integral method is used. In the course of proving the existence of solutions of the problems, differentiation formulas, decomposition formulas, some adjacent relations formulas and the autotransformation formula of hypergeometric functions are used. To express problem's solution in explicit form the Gauss–Ostrogradsky formula is used. [ABSTRACT FROM AUTHOR]

In this paper, we study an analogue of the Gellerstedt problem for a loaded parabolic-hyperbolic equation of the third order in an infinite three-dimensional domain. The main method to study this Gellerstedt problem is the Fourier transform. Based on the Fourier transform, we reduce the considering problem to a planar analogue of the Gellerstedt spectral problem with a spectral parameter. The uniqueness of the solution of this problem is proved by the new extreme principle for loaded third-order equations of the mixed type. The existence of a regular solution of the Gellerstedt spectral problem is proved by the method of integral equations. In addition, the asymptotic behavior of the solution of the Gellerstedt spectral problem is studied for large values of the spectral parameter. Sufficient conditions are found such that all differentiation operations are legal in this work. [ABSTRACT FROM AUTHOR]

The direct Lyapunov method is used to prove the absolute linear instability of steadystate plane-parallel shear flows of an inviscid stratified incompressible fluid in the gravity field with respect to plane perturbations both in the Boussinesq and non-Boussinesq approximations. A strict description is given for the applicability of the known necessary condition for linear instability of steady-state plane-parallel shear flows of an ideal nonuniform (by density) incompressible fluid in the gravity field both in the Boussinesq and non-Boussinesq approximations (the Miles theorem). Analytical examples of illustrative character are constructed. [ABSTRACT FROM AUTHOR]