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We introduce integral transforms to study problems for the singular Bessel differential operators [B.sub.-[gamma]] with negative parameter -[gamma] < 0. A solution to the singular Bessel equation [B.sub.-[gamma]]u+u = 0 is a function u = [J.sub.[micro]] expressed in terms of the Bessel functions of the first kind with positive parameter [micro] = ([gamma] + 1)/2. Based on the notion of a T-pseudoshift, we formulate the Levitan addition theorem. We construct the Bessel J-transform and the corresponding class of singular J-pseudodifferential operators. We prove a theorem on the order of J-pseudodifferential operators in the class of Sobolev--Kipriyanov functions. Bibliography: 13 titles., In the early 1980s, I. A. Kipriyanov proposed to apply a special Bessel transform to study problems for equations containing singular Bessel differential operators with negative parameter. It was done [...]

A -hyperbolic equation with a Bessel operator in time and with an operator in space is considered. The operator includes singular differential Bessel operators with non-negative parameters and derivatives of these operators. The found fundamental solution is expressed in terms of the Bessel function of the first kind of negative order. [ABSTRACT FROM AUTHOR]

The paper introduces integral transformations for studying problems with a singular differential Bessel operator with a negative parameter . We construct the -Bessel transform and the corresponding class of singular -pseudodifferential operators. A theorem on the order of -pseudodifferential operators in a special class of Sobolev–Kipriyanov functions is proved. [ABSTRACT FROM AUTHOR]

Functions satisfying the Laplace equation for the Kipriyanov -operator are said to be -harmonic. The following properties of -harmonic functions are presented and proved: the Green-type integral representation of -functions, the spherical mean theorem, and the maximum principle. As a corollary, the uniqueness of the solution of the interior and exterior Dirichlet problems is proved. [ABSTRACT FROM AUTHOR]

We introduce integral transforms to study problems for the singular Bessel differential operators B−γ with negative parameter −γ < 0. A solution to the singular Bessel equation B−γu+u = 0 is a function u = 핁μ expressed in terms of the Bessel functions of the first kind with positive parameter μ = (γ + 1)/2. Based on the notion of a 핋-pseudoshift, we formulate the Levitan addition theorem. We construct the Bessel 핁-transform and the corresponding class of singular 핁-pseudodifferential operators. We prove a theorem on the order of 핁-pseudodifferential operators in the class of Sobolev–Kipriyanov functions. [ABSTRACT FROM AUTHOR]

Solutions of singular differential equations with the Bessel operator of negative order are studied. In this regard, of great interest are the solutions of the singular differential Bessel equation , which are presented in the paper as linearly independent functions and , . Some properties of the functions expressed in terms of the properties of the Bessel–Levitan -function are considered. The direct and inverse -Bessel transforms are introduced, and the -pseudoshift operator that commutes with the Bessel operator is defined. The fundamental solution of the ordinary singular differential operator is found. A representation of the fundamental solution of the Kipriyanov -operator with a singularity at the point and on the cone in the Euclidean -dimensional half-space is given. [ABSTRACT FROM AUTHOR]

A formula for the fundamental solutions to the heat equation with singular Bessel operator is obtained. Bibliography: 7 titles. 1 Introduction We consider functions defined in the Euclidean space of [...]