Babenko’s Approach to Abel’s Integral Equations

Abstract

Abstract:The goal of this paper is to investigate the following Abel's integral equation of the second kind: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 α ∈ 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.

Uniqueness of the Hadamard-type integral equations

“…Lemma 1.1 If α > 0, μ ∈ R, and 0 < a < b < ∞, then the operator J α a+,μ is bounded in Xμ(a, b), and for u ∈ Xμ(a, b), There are a lot of studies on fractional differential and integral equations involving Riemann-Liouville or Caputo operators with boundary value problems or initial conditions [3][4][5][6][7][8][9][10][11]. Li and Sarwar [12] considered the existence of solutions for the following fractional-order initial value problems:…”

The half-order energy balance equation – Part 2: The inhomogeneous HEBE and 2D energy balance models

“…I therefore proceed to derive the inhomogeneous HEBE and GHEBE using Babenko's method. The more usual application is to find the surface heat flux given a solution to the conduction equation (see, for example, Magin et al, 2004;Chenkuan and Clarkson, 2018), although the following application appears to be original. In the inhomogeneous case with τ = τ (x), l h = l h (x), l v = l v (x), and α = α(x), there is no unique nondimensionalization.…”

Generalized Integral Transforms via the Series Expressions

“…In [16][17][18][19][20], the authors studied the series expansions of appropriate functionals which is called the Ito-Wiener-Chaos expansion. Our method introduced in Theorem 3 is also a kind of these expansions.…”