Special Functions of Fractional Calculus in the Form of Convolution Series and Their Applications

Yuri Luchko
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Yuri Luchko: Department of Mathematics, Physics, and Chemistry, Beuth Technical University of Applied Sciences Berlin, Luxemburger Str. 10, 13353 Berlin, Germany

Mathematics, 2021, vol. 9, issue 17, 1-15

Abstract: In this paper, we first discuss the convolution series that are generated by Sonine kernels from a class of functions continuous on a real positive semi-axis that have an integrable singularity of power function type at point zero. These convolution series are closely related to the general fractional integrals and derivatives with Sonine kernels and represent a new class of special functions of fractional calculus. The Mittag-Leffler functions as solutions to the fractional differential equations with the fractional derivatives of both Riemann-Liouville and Caputo types are particular cases of the convolution series generated by the Sonine kernel κ ( t ) = t α − 1 / Γ ( α ) , 0 < α < 1 . The main result of the paper is the derivation of analytic solutions to the single- and multi-term fractional differential equations with the general fractional derivatives of the Riemann-Liouville type that have not yet been studied in the fractional calculus literature.

Keywords: Sonine kernel; Sonine condition; general fractional derivative; general fractional integral; convolution series; fundamental theorems of fractional calculus; fractional differential equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (7)

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