On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series Coefficients

On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series Coefficients

In this paper, we continue our study of the Abel equation with the right-hand side belonging to the Lebesgue weighted space. We have improved the previously known result— the existence and uniqueness theorem formulated in terms of the Jacoby series coefficients that gives us an opportunity to find a...

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Main Author: Maksim V. Kukushkin
Format: Article
Language:English
Published: MDPI AG 2020-07-01
Series:Axioms
Subjects:
Riemann-Liouville operator
Abel equation
Jacobi polinomials
weighted Lebesgue spaces
Online Access:https://www.mdpi.com/2075-1680/9/3/81
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spelling doaj-1bbfa3eeeb3d49ee86840d0fe9be36d12020-11-25T03:09:22ZengMDPI AGAxioms2075-16802020-07-019818110.3390/axioms9030081On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series CoefficientsMaksim V. Kukushkin0Department of Applied Mathematics, University of Civil Engineering, 129337 Moscow, RussiaIn this paper, we continue our study of the Abel equation with the right-hand side belonging to the Lebesgue weighted space. We have improved the previously known result— the existence and uniqueness theorem formulated in terms of the Jacoby series coefficients that gives us an opportunity to find and classify a solution by virtue of an asymptotic of some relation containing the Jacobi series coefficients of the right-hand side. The main results are the following—the conditions imposed on the parameters, under which the Abel equation has a unique solution represented by the series, are formulated; the relationship between the values of the parameters and the solution smoothness is established. The independence between one of the parameters and the smoothness of the solution is proved.https://www.mdpi.com/2075-1680/9/3/81Riemann-Liouville operatorAbel equationJacobi polinomialsweighted Lebesgue spaces
collection DOAJ
language English
format Article
sources DOAJ
author Maksim V. Kukushkin
spellingShingle Maksim V. Kukushkin
On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series Coefficients
Axioms
Riemann-Liouville operator
Abel equation
Jacobi polinomials
weighted Lebesgue spaces
author_facet Maksim V. Kukushkin
author_sort Maksim V. Kukushkin
title On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series Coefficients
title_short On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series Coefficients
title_full On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series Coefficients
title_fullStr On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series Coefficients
title_full_unstemmed On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series Coefficients
title_sort on smoothness of the solution to the abel equation in terms of the jacobi series coefficients
publisher MDPI AG
series Axioms
issn 2075-1680
publishDate 2020-07-01
description In this paper, we continue our study of the Abel equation with the right-hand side belonging to the Lebesgue weighted space. We have improved the previously known result— the existence and uniqueness theorem formulated in terms of the Jacoby series coefficients that gives us an opportunity to find and classify a solution by virtue of an asymptotic of some relation containing the Jacobi series coefficients of the right-hand side. The main results are the following—the conditions imposed on the parameters, under which the Abel equation has a unique solution represented by the series, are formulated; the relationship between the values of the parameters and the solution smoothness is established. The independence between one of the parameters and the smoothness of the solution is proved.
topic Riemann-Liouville operator
Abel equation
Jacobi polinomials
weighted Lebesgue spaces
url https://www.mdpi.com/2075-1680/9/3/81
work_keys_str_mv AT maksimvkukushkin onsmoothnessofthesolutiontotheabelequationintermsofthejacobiseriescoefficients
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