Solutions to Abel’s Integral Equations in Distributions

Abstract

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 ) .

Keywords

• distribution
• fractional calculus
• Mittag–Leffler function
• Abel’s integral equation
• convolution