# Operational Calculus for the General Fractional Derivatives of Arbitrary Order

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. General Fractional Derivatives of Arbitrary Order

**Definition**

**1**

**.**Let the functions κ and k satisfy the condition

**Definition**

**2**

**.**Let $(\kappa ,\phantom{\rule{4pt}{0ex}}k)\in {\mathcal{L}}_{n}$. The GFI with the kernel κ and the GFD of arbitrary order with the kernel k are defined as follows:

**Theorem**

**1**

**.**The triple ${\mathcal{R}}_{-1}=({C}_{-1}(0,+\infty ),+,\ast )$ with the usual addition + and multiplication ∗ in form of the Laplace convolution is a commutative ring without unity with respect to multiplication and without divisors of zero.

**Theorem**

**2**

**Theorem**

**3**

**Definition**

**3.**

**Theorem**

**4**

**.**Let $(\kappa ,\phantom{\rule{0.166667em}{0ex}}k)\in {\mathcal{L}}_{n},\phantom{\rule{4pt}{0ex}}n\in \mathbb{N}$.

**Theorem**

**5**

**.**Let $(\kappa ,\phantom{\rule{4pt}{0ex}}k)\in {\mathcal{L}}_{n}$.

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

## 3. Operational Calculus for the GFD of Arbitrary Order

**Theorem**

**6**

**.**The triple ${\mathcal{F}}_{-1}=({C}_{-1}^{2}(0,+\infty )/\sim ,\phantom{\rule{4pt}{0ex}}+,\phantom{\rule{4pt}{0ex}}\xb7)$ is a field that is usually referred to as the field of convolution quotients.

**Definition**

**4**

**Definition**

**5.**

**Theorem**

**7.**

**Proof.**

**Remark**

**3.**

**Definition**

**6.**

**Theorem**

**8.**

**Remark**

**4.**

**Theorem**

**9**

**Theorem**

**10.**

**Proof.**

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Al-Kandari, M.; Hanna, L.A.-M.; Luchko, Y.
Operational Calculus for the General Fractional Derivatives of Arbitrary Order. *Mathematics* **2022**, *10*, 1590.
https://doi.org/10.3390/math10091590

**AMA Style**

Al-Kandari M, Hanna LA-M, Luchko Y.
Operational Calculus for the General Fractional Derivatives of Arbitrary Order. *Mathematics*. 2022; 10(9):1590.
https://doi.org/10.3390/math10091590

**Chicago/Turabian Style**

Al-Kandari, Maryam, Latif A-M. Hanna, and Yuri Luchko.
2022. "Operational Calculus for the General Fractional Derivatives of Arbitrary Order" *Mathematics* 10, no. 9: 1590.
https://doi.org/10.3390/math10091590