Fractional Differential Equations with the General Fractional Derivatives of Arbitrary Order in the Riemann–Liouville Sense

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

Mathematics, 2022, vol. 10, issue 6, 1-24

Abstract: In this paper, we first consider the general fractional derivatives of arbitrary order defined in the Riemann–Liouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of fractional calculus that leads to a closed form formula for their projector operator. These results allow us to formulate the natural initial conditions for the fractional differential equations with the general fractional derivatives of arbitrary order in the Riemann–Liouville sense. In the second part of the paper, we develop an operational calculus of the Mikusiński type for the general fractional derivatives of arbitrary order in the Riemann–Liouville sense and apply it for derivation of an explicit form of solutions to the Cauchy problems for the single- and multi-term linear fractional differential equations with these derivatives. The solutions are provided in form of the convolution series generated by the kernels of the corresponding general fractional integrals.

Keywords: Sonine kernel; Sonine condition; general fractional integral; general fractional derivative of arbitrary order; fundamental theorems of fractional calculus; operational calculus; fractional differential equations; convolution series (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (7)

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