# Conformable Laplace Transform of Fractional Differential Equations

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## Abstract

**:**

## 1. Introduction

- Not all fractional derivatives obey the familiar product rule for two functions:$$L(fg)=fL(g)+gL(f).$$
- Not all fractional derivatives obey the chain rule:$$L(f\circ g)(t)=L(f)(g(t))\phantom{\rule{0.166667em}{0ex}}L(g)(t).$$

## 2. Brief on Conformable Fractional Calculus

**Definition**

**1**

**.**Let $f:[0,\infty )\to \mathbb{R}$ be a function. Then, the conformable fractional derivative of f of order α, $0<\alpha \le 1$, is defined by,

**Remark**

**1.**

**Notation**

**1.**

**Theorem**

**1.**

**Definition**

**2**(Conformable fractional integral)

**.**

**Lemma**

**1.**

**Definition**

**3**

**.**The conformable fractional exponential function is defined for every $t\ge 0$ by:

**Lemma**

**2**

**.**Let r be a continuous, nonnegative function on $0\le t<T$ (some $T\le \infty $) and a and b be nonnegative constants such that:

**Definition**

**4.**

**Theorem**

**2**

**.**Let $f:[0,\infty )\to \mathbb{R}$ be a function such that ${\mathfrak{L}}_{\alpha}\{f(t)\}(s)={F}_{\alpha}(s)$ exists. Then:

**Theorem**

**3**

- If c is a constant, then:$$\mathfrak{L}[c]=\frac{c}{s}.$$
- Let q be a constant:$${\mathfrak{L}}_{\alpha}[{t}^{q}](s)={\alpha}^{q/\alpha}\frac{\mathsf{\Gamma}(1+\frac{q}{\alpha})}{{s}^{1+q/\alpha}}$$
- If c, q are arbitrary constants:$${\mathfrak{L}}_{\alpha}[{t}^{q}{E}_{\alpha}(c,t)](s)={\alpha}^{q/\alpha}\frac{\mathsf{\Gamma}(1+\frac{q}{\alpha})}{{(s-c)}^{1+q/\alpha}}$$

**Remark**

**2.**

**Theorem**

**4**

**.**Let $f,g:[0,\infty ]\to \mathbb{R}$ be real valued functions and $0<\alpha \le 1$. Then, if ${F}_{\alpha}(s)={\mathfrak{L}}_{\alpha}[f({t}^{\alpha})](s)$ and ${G}_{\alpha}(s)={\mathfrak{L}}_{\alpha}[g(t)](s)$ both exist for $s\ge 0$, then:

**Definition**

**5.**

## 3. Validity of the Conformable Laplace Transform for Linear Fractional-Order Equations

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

## 4. Illustrative Examples

**Remark**

**3.**

**Example**

**1.**

**Example**

**2**(von Foerster model)

**.**

**Remark**

**4.**

**Example**

**3.**

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Silva, F.S.; Moreira, D.M.; Moret, M.A.
Conformable Laplace Transform of Fractional Differential Equations. *Axioms* **2018**, *7*, 55.
https://doi.org/10.3390/axioms7030055

**AMA Style**

Silva FS, Moreira DM, Moret MA.
Conformable Laplace Transform of Fractional Differential Equations. *Axioms*. 2018; 7(3):55.
https://doi.org/10.3390/axioms7030055

**Chicago/Turabian Style**

Silva, Fernando S., Davidson M. Moreira, and Marcelo A. Moret.
2018. "Conformable Laplace Transform of Fractional Differential Equations" *Axioms* 7, no. 3: 55.
https://doi.org/10.3390/axioms7030055