Solutions to Abel’s Integral Equations in Distributions

• Published in: Axioms Vol. 7; no. 3; p. 66
• Main Authors: Li, Chenkuan, Humphries, Thomas, Plowman, Hunter
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01.09.2018
• Subjects: Abel’s integral equation; Calculus; convolution; Cybernetics; Differential equations; distribution; Fractional calculus; Generalized method of moments; Growth models; Inflation; Integral equations; Laplace transforms; Mathematical analysis; Mathematical functions; Mathematics; Mittag–Leffler function; Numerical analysis; Operators (mathematics); Partial differential equations; Quantum physics; Viscoelasticity; New York; United States > US
• ISSN: 2075-1680; 2075-1680
• DOI: 10.3390/axioms7030066

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 ) .