# Can we split fractional derivative while analyzing fractional differential equations?

## TL;DR

This paper investigates the possibility of decomposing fractional derivatives in differential equations, establishing conditions for splitting such equations into systems with lower order derivatives, and analyzing the sufficiency and necessity of existing conditions.

## Contribution

It proposes conditions under which fractional differential equations can be split into lower order systems and clarifies the limitations of existing results.

## Key findings

- Conditions for splitting fractional derivatives are established.
- Examples demonstrate that existing conditions are sufficient but not necessary.
- Analysis clarifies the composition rules for fractional derivatives.

## Abstract

Fractional derivatives are generalization to classical integer-order derivatives. The rules which are true for classical derivative need not hold for the fractional derivatives, for example, we cannot simply add the fractional orders $\alpha$ and $\beta$ in ${}_0^{C}\mathrm{D}_t^\alpha {}_0^{C}\mathrm{D}_t^\beta$ to produce the fractional derivative ${}_0^{C}\mathrm{D}_t^{\alpha+\beta}$ of order $\alpha+\beta$, in general. In this article we discuss the details of such compositions and propose the conditions to split a linear fractional differential equation into the systems involving lower order derivatives. Further, we provide some examples, which show that the related results in the literature are sufficient but not necessary conditions.