The fractional Green's function by Babenko's approach
Chenkuan Li, Changpin Li
The goal of this paper is to derive the fractional Green's function for the first time in the distributional space for the fractional-order integro-differential equation
with constant coefficients. Our new technique is based on Babenko's approach, without using any integral transforms such as the Laplace transform along with
Mittag-Leffler function. The results obtained are not only much simpler, but also more generalized than the classical ones as they deal with distributions which
are undefined in the ordinary sense in general. Furthermore, several interesting applications to solving the fractional differential and integral equations,
as well as in the wave reaction-diffusion equation are provided, some of which cannot be achieved by integral transforms or numerical analysis.
Tbilisi Mathematical Journal, Vol. 13(3) (2020), pp. 19-42