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

Yuri Luchko

doi:10.3390/math10060849

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.

Highlights

In the framework of the abstract Volterra integral equations on the Banach spaces, the evolution equations with the integro-differential operators of the convolution type with different classes of kernels and on different spaces of functions were a subject of active research within the last few decades
We address an important topic that was not yet treated in the fractional calculus (FC) literature, namely, derivation of the closed form solutions to the Cauchy problems for the fractional differential equations with the Riemann–Liouville general fractional derivative (GFD) of arbitrary order
We develop an operational calculus of the Mikusiński type for the GFDs of arbitrary order in the Riemann–Liouville sense

Summary

Introduction

In the framework of the abstract Volterra integral equations on the Banach spaces, the evolution equations with the integro-differential operators of the convolution type with different classes of kernels and on different spaces of functions were a subject of active research within the last few decades (see [1,2] and references therein). This operation calculus is applied for derivation of an explicit form of solutions to the Cauchy problems for the single- and multi-term linear fractional differential equations with the GFDs of the Riemann–Liouville type.

Results

Conclusion