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

TL;DR

This paper develops a comprehensive theory for general fractional derivatives of arbitrary order in the Riemann-Liouville sense, including null space, initial conditions, and explicit solutions to fractional differential equations using operational calculus.

Contribution

It introduces explicit forms for null space, projector operators, and initial conditions, and develops an operational calculus for solving fractional differential equations with these derivatives.

Findings

Explicit null space and projector operator formulas

Natural initial conditions for fractional differential equations

Explicit solutions via convolution series

Abstract

In this paper, we first deal with 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 Mikusi\'nski 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.