Babenko's approach to Abel's integral equations

Abstract

The goal of this paper is to investigate the following Abel's integral equation of the second kind: y(t) + λ /Γα ∫t0 (t - τ)α-1y(τ)dτ = 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 α ε 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.