Babenko’s Approach to Abel’s Integral Equations

Chenkuan Li and Kyle Clarkson
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Chenkuan Li: Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada
Kyle Clarkson: Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada

Mathematics, 2018, vol. 6, issue 3, 1-15

Abstract: The goal of this paper is to investigate the following Abel’s integral equation of the second kind: y ( t ) + ? ? ( ? ) ? 0 t ( t ? ? ) ? ? 1 y ( ? ) d ? = 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 ? ? 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.

Keywords: distribution; fractional calculus; convolution; series convergence; Laplace transform; Gamma function; Mittag–Leffler function (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2018
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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