# General Response Formula for CFD Pseudo-Fractional 2D Continuous Linear Systems Described by the Roesser Model

## Abstract

**:**

## 1. Introduction

## 2. Conformable Fractional Derivative Definition and Its Properties

**Definition**

**1**

**.**The Conformable Fractional Derivative (CFD) of a real continuous-time function $f\left(t\right)$, $t\in [0,\infty )$ of order $\alpha \in (0,1)$ is defined by

**Definition**

**2**

**.**Let $0<\alpha \le 1$ and $f:[0,\infty )\to \mathbb{R}$, then the fractional one-sided Laplace transform of α-order of $f\left(t\right)$ is defined by

**Lemma**

**1**

**.**Let ${F}^{\alpha}\left(s\right)$ be the one-sided fractional Laplace transform of a function $f\left(t\right)$. Then,

**Theorem**

**1**

**.**The fractional Laplace transform (3) of the Conformable Fractional Derivative operator (1) is given by

**Definition**

**3**

**.**Let $f\left(t\right)$ and $g\left(t\right)$ be two continuous-time functions. We define the α-convolution of $f\left(t\right)$ and $g\left(t\right)$ by

**Theorem**

**2**

**.**Fractional Laplace transform of the α-convolution of functions $f\left(t\right)$ and $g\left(t\right)$, where $\alpha >0$ is the order of the transform, is given by

**Definition**

**4.**

## 3. The CFD Pseudo-Fractional 2D System Described by the Roesser Model

## 4. General Response Formula

**Theorem**

**3.**

## 5. Extension of Cayley–Hamilton Theorem

**Theorem**

**4.**

**Remark**

**1.**

## 6. Step Response

**Example**

**1.**

## 7. Concluding Remarks

## Funding

## Conflicts of Interest

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Rogowski, K.
General Response Formula for CFD Pseudo-Fractional 2D Continuous Linear Systems Described by the Roesser Model. *Symmetry* **2020**, *12*, 1934.
https://doi.org/10.3390/sym12121934

**AMA Style**

Rogowski K.
General Response Formula for CFD Pseudo-Fractional 2D Continuous Linear Systems Described by the Roesser Model. *Symmetry*. 2020; 12(12):1934.
https://doi.org/10.3390/sym12121934

**Chicago/Turabian Style**

Rogowski, Krzysztof.
2020. "General Response Formula for CFD Pseudo-Fractional 2D Continuous Linear Systems Described by the Roesser Model" *Symmetry* 12, no. 12: 1934.
https://doi.org/10.3390/sym12121934