Babenko’s Approach to Abel’s Integral Equations

The goal of this paper is to investigate the following Abel’s integral equation of the second kind: y ( t ) + λ Γ ( α ) ∫ 0 t ( t − τ ) α − 1 y ( τ ) d τ = 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 ∈ 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.