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Article

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

1
Moscow State University of Civil Engineering, 129337 Moscow, Russia
2
Kabardino-Balkarian Scientific Center, RAS, 360051 Nalchik, Russia
Axioms 2020, 9(3), 81; https://doi.org/10.3390/axioms9030081
Received: 22 June 2020 / Revised: 11 July 2020 / Accepted: 14 July 2020 / Published: 17 July 2020
(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)
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. View Full-Text
Keywords: Riemann-Liouville operator; Abel equation; Jacobi polinomials; weighted Lebesgue spaces Riemann-Liouville operator; Abel equation; Jacobi polinomials; weighted Lebesgue spaces
MDPI and ACS Style

Kukushkin, M.V. On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series Coefficients. Axioms 2020, 9, 81. https://doi.org/10.3390/axioms9030081

AMA Style

Kukushkin MV. On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series Coefficients. Axioms. 2020; 9(3):81. https://doi.org/10.3390/axioms9030081

Chicago/Turabian Style

Kukushkin, Maksim V. 2020. "On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series Coefficients" Axioms 9, no. 3: 81. https://doi.org/10.3390/axioms9030081

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