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Axioms 2018, 7(3), 66;

Solutions to Abel’s Integral Equations in Distributions

Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada
Author to whom correspondence should be addressed.
Received: 10 August 2018 / Revised: 27 August 2018 / Accepted: 31 August 2018 / Published: 2 September 2018
(This article belongs to the Special Issue Mathematical Analysis and Applications)
Full-Text   |   PDF [281 KB, uploaded 3 September 2018]


The goal of this paper is to study fractional calculus of distributions, the generalized Abel’s integral equations, as well as fractional differential equations in the distributional space D ( R + ) based on inverse convolutional operators and Babenko’s approach. Furthermore, we provide interesting applications of Abel’s integral equations in viscoelastic systems, as well as solving other integral equations, such as θ π / 2 y ( φ ) cos β φ ( cos θ cos φ ) α d φ = f ( θ ) , and 0 x 1 / 2 g ( x ) y ( x + t ) d x = f ( t ) . View Full-Text
Keywords: distribution; fractional calculus; Mittag–Leffler function; Abel’s integral equation; convolution distribution; fractional calculus; Mittag–Leffler function; Abel’s integral equation; convolution
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Li, C.; Humphries, T.; Plowman, H. Solutions to Abel’s Integral Equations in Distributions. Axioms 2018, 7, 66.

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