Babenko's approach to Abel's integral equations
type: Item
instance of: scholarly article
title: Babenko's approach to Abel's integral equations
zbMATH Open document ID: 1394.45005
author: Kyle Clarkson
author: Chenkuan Li
published in: Mathematics
publication date: 2018-07-05T00:00:00Z
review text: Summary: The goal of this paper is to investigate the following Abel's integral equation of the second kind: \(y(t) + \frac{\lambda}{\Gamma(\alpha)} \int_0^t(t - \tau)^{\alpha - 1} y(\tau) d \tau = f(t)\) \((t > 0)\), and its variants by fractional calculus. Applying Babenko's approach and fractional integrals, we provide a general method for solving Abel's integral equation and others with a demonstration of different types of examples by showing convergence of series. In particular, we extend this equation to a distributional space for any arbitrary \(\alpha \in \mathbb R\) by fractional operations of generalized functions for the first time and obtain several new and interesting results that cannot be realized in the classical sense or by the Laplace transform.
Mathematics Subject Classification ID: 45E10
Mathematics Subject Classification ID: 26A33
zbMATH DE Number: 6898724
zbMATH Keywords: fractional calculus
zbMATH Keywords: series convergence
zbMATH Keywords: Laplace transform
zbMATH Keywords: gamma function
zbMATH Keywords: Mittag-Leffler function
MaRDI profile type: Publication
full work available at URL: math6030032
OpenAlex ID: W2793165349
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DOI: 10.3390/MATH6030032
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description: scientific article; zbMATH DE number 6898724