# Operator Kernel Functions in Operational Calculus and Applications in Fractals with Fractional Operators

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Function Space

**Definition**

**1.**

- (i)
- The function $f\left(t\right)$ has, at most, a finite number of discontinuity points in any finite interval.
- (ii)
- For arbitrary $t>a,$ the integration of $f\left(t\right)$ is bounded, i.e., ${\int}_{a}^{t}f\left(\tau \right)\mathrm{d}\tau <\infty $.

#### 2.2. Ring and Field

**Definition**

**2.**

**Definition**

**3.**

#### 2.3. Notation of Operators

**Definition**

**4.**

**Definition**

**5.**

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

## 3. Operator Kernel Function Method

#### 3.1. Uniqueness of Solutions of Operator Differential Equations

**Lemma**

**1.**

#### 3.2. Translation Operator

**Remark**

**4.**

**Lemma**

**2.**

**Proof.**

#### 3.3. The Equivalence of Operational Calculus and Integral Transformations

**Theorem**

**1.**

**Proof.**

#### 3.4. The Uniqueness of Operational Calculus and Integral Transformations

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

#### 3.5. Relationship with Carson-Laplace Transform

## 4. Applications of Operator Kernel Function Method

#### 4.1. The Heat Conduction Problem

#### 4.2. Fractional-Order Mechanics on a Fractal Tree

#### 4.3. Fractional-Order Mechanics on a Fractal Loop

#### 4.4. Hemodynamics on a Fractal Ladder

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

OKF | Operator kernal funtion |

OC | Operational Calculus |

LHS | Left hand side |

RHS | Right hand side |

## Appendix A

**Proof**

**of**

**Lemma**

**1.**

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**Figure 2.**Response curves for stress relaxation and strain creep of the fractal tree model, the Kelvin-Voigt, and the Maxwell models. (

**a**) The black line represented in the figure indicates the process of applying and releasing step stress; (

**b**) The black line shown in the figure denotes the application of step strain.

Operator | Kernel Function |
---|---|

$\frac{1}{{\mathit{p}}^{n}}$^{1} | $\frac{{t}^{n-1}}{(n-1)!}$ |

$\frac{1}{{\left(\mathit{p}-\alpha \right)}^{n}}$ | $\frac{{t}^{n-1}}{(n-1)!}{e}^{\alpha t}$ |

$\frac{1}{{\mathit{p}}^{2}+{\beta}^{2}}$ | $\frac{1}{\beta}sin\beta t$ |

$\frac{\mathit{p}}{{\mathit{p}}^{2}+{\beta}^{2}}$ | $cos\beta t$ |

${\left[{(\mathit{p}-\alpha )}^{2}+{\beta}^{2}\right]}^{-n}$ | $\frac{{e}^{at}}{{\left(2{\beta}^{2}\right)}^{n-1}}\left[{A}_{n}\left({\beta}^{2}{t}^{2}\right)\frac{1}{\beta}sin\beta t-{B}_{n}\left({\beta}^{2}{t}^{2}\right)tcos\beta t\right]$^{2} |

$\frac{{(\sqrt{{\mathit{p}}^{2}+{\alpha}^{2}}-\mathit{p})}^{n}}{\sqrt{{\mathit{p}}^{2}+{\alpha}^{2}}}$ | ${\alpha}^{n}{J}_{n}\left(\alpha t\right)$^{3} |

$cos\frac{1}{{\mathit{p}}^{2}}$ | $\sum _{i=0}^{\infty}}\frac{{(-1)}^{i}{t}^{({2}^{i+1}-1)}}{\left(2i\right)!({2}^{i+1}-1)!$ |

$\frac{1}{\mathit{p}}{e}^{-\frac{\lambda}{\mathit{p}}}$ | ${J}_{0}\left(2\sqrt{\lambda t}\right)$ |

$\frac{1}{\sqrt{\mathit{p}}}{e}^{-\frac{\lambda}{\mathit{p}}}$ | $\frac{1}{\sqrt{\pi t}}cosh2\sqrt{\lambda t}$ |

$\frac{1}{{\mathit{p}}^{2}}{e}^{-\frac{\lambda}{\mathit{p}}}$ | $\sqrt{\frac{t}{\lambda}}{J}_{2}\left(2\sqrt{\lambda t}\right)$ |

${e}^{\lambda (\mathit{p}-\sqrt{{\mathit{p}}^{2}+{\alpha}^{2}})}$ | $1-\frac{\lambda}{\sqrt{{t}^{2}+2\lambda t}}\alpha {J}_{1}\left(\alpha \sqrt{{t}^{2}+2\lambda t}\right)$ |

${e}^{-\lambda \sqrt{{\mathit{p}}^{2}+{\alpha}^{2}}}/\sqrt{{\mathit{p}}^{2}+{\alpha}^{2}}$ | ${J}_{0}\left(\alpha \sqrt{{t}^{2}-{\lambda}^{2}}\right)\phantom{\rule{1.em}{0ex}}(0\le \lambda <t)$ |

^{1}$n>0$ and $n\in Z$.

^{2}${A}_{n}$ and ${B}_{n}$ are parameters.

^{3}${J}_{n}$ represents the Bessel function of order n.

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**MDPI and ACS Style**

Yu, X.; Yin, Y.
Operator Kernel Functions in Operational Calculus and Applications in Fractals with Fractional Operators. *Fractal Fract.* **2023**, *7*, 755.
https://doi.org/10.3390/fractalfract7100755

**AMA Style**

Yu X, Yin Y.
Operator Kernel Functions in Operational Calculus and Applications in Fractals with Fractional Operators. *Fractal and Fractional*. 2023; 7(10):755.
https://doi.org/10.3390/fractalfract7100755

**Chicago/Turabian Style**

Yu, Xiaobin, and Yajun Yin.
2023. "Operator Kernel Functions in Operational Calculus and Applications in Fractals with Fractional Operators" *Fractal and Fractional* 7, no. 10: 755.
https://doi.org/10.3390/fractalfract7100755