Babenko's approach to Abel's integral equations
Summary: The goal of this paper is to investigate the following Abel's integral equation of the second kind: \(y(t) + \frac{\lambda}{\Gamma(\alpha)} \int_0^t(t - \tau)^{\alpha - 1} y(\tau) d \tau = f(t)\) \((t > 0)\), 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 \(\alpha \in \mathbb R\) 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.
scientific article; zbMATH DE number 6898724
scientific article