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.
This work is partially supported by NSERC (Canada 2019-03907) and NSFC (China 11671251).
The authors are grateful to Dr. H.M. Srivastava for his careful reading of the paper with several productive suggestions and corrections.
"The fractional Green's function by Babenko's approach." Tbilisi Math. J. 13 (3) 19 - 42, September 2020. https://doi.org/10.32513/tbilisi/1601344896