Дифференциальные уравнения типа Эйлера дробного порядка

Natalia V. Zhukovskaya; Sergey M. Sitnik

doi:10.25587/SVFU.2018.98.14228

Zhukovskaya Natalia V., nata2105@tut.by Belarusian State University 4 Nezavisimost’ Avenue, Minsk 220030, Belarus
Sitnik Sergey M., sitnik@bsu.edu.ru Belgorod State National Research Institute, 85 Pobeda Avenue, Belgorod 308015, Russia

DOI: https://doi.org/10.25587/SVFU.2018.98.14228

Keywords: fractional Riemann–Liouville integrals, Riemann–Liouville fractional derivatives, direct and inverse Mellin transforms, fractional analogue of Green’s function

Abstract

Using the direct and inverse Mellin transforms, we find a solution to the nonhomogeneous Euler-type differential equation with Riemann–Liouville fractional derivatives on the half-axis in the class of functions represented by the fractional integral in terms of the fractional analogue of the Green’s function. Fractional analogues of the Green’s function are constructed in the case when all roots of the characteristic polynomial are different and in the case when there are multiple roots of the characteristic polynomial. Theorems of solvability of the nonhomogeneous fractional differential equations of Euler type on the half-axis are stated and proved. Special cases and examples are considered.

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Euler type differential equations of fractional order

Abstract

References