# Analytical and Numerical Simulations of a Delay Model: The Pantograph Delay Equation

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## Abstract

**:**

## 1. Introduction

## 2. Analytic Solution at $\mathit{c}\in \mathbb{R}$, $\mathit{c}\ne \pm \mathbf{1}$

#### The Solution in Simplest Form

## 3. Convergence Analysis

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Lemma**

**1.**

**Proof.**

**Remark**

**1.**

## 4. Exact Solution at $\mathit{c}=-\mathbf{1}$

## 5. Results

#### 5.1. $c\in (-1,1)$, $\left|\frac{b}{a}\right|<1$, $a\in \mathbb{R}-\left\{0\right\}$

#### 5.2. $c>1$, $a<0$, $b\in \mathbb{R}$

#### 5.3. $c=-1$, $a,b\in \mathbb{R}$

#### 5.4. $a=-1,\phantom{\rule{3.33333pt}{0ex}}b=c=\frac{1}{q},\phantom{\rule{3.33333pt}{0ex}}q>1$

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Plots of the approximate solutions ${\varphi}_{m}\left(t\right),\phantom{\rule{3.33333pt}{0ex}}m=3,5,7,9$ in Equation (38) vs. t at $\lambda =1$, $c=\frac{1}{2}$, $b=1$, and $a=-2$.

**Figure 2.**Plots of the approximate solutions ${\varphi}_{m}\left(t\right),\phantom{\rule{3.33333pt}{0ex}}m=3,5,7,9$ in Equation (38) vs. t at $\lambda =1$, $c=\frac{1}{2}$, $b=1$, and $a=2$.

**Figure 3.**Plots of the approximate solutions ${\varphi}_{m}\left(t\right),\phantom{\rule{3.33333pt}{0ex}}m=3,5,7,9$ in Equation (38) vs. t at $\lambda =1$, $c=-\frac{1}{2}$, $b=1$, and $a=-2$.

**Figure 4.**Plots of the approximate solutions ${\varphi}_{m}\left(t\right),\phantom{\rule{3.33333pt}{0ex}}m=3,5,7,9$ in Equation (38) vs. t at $\lambda =1$, $c=-\frac{1}{2}$, $b=1$, and $a=2$.

**Figure 5.**Plots of the approximate solutions ${\varphi}_{m}\left(t\right),\phantom{\rule{3.33333pt}{0ex}}m=3,4,5,6$ in Equation (39) vs. t at $\lambda =1$, $c=\frac{3}{2}$, $b=1$, and $a=-2$.

**Figure 6.**Plots of the approximate solutions ${\varphi}_{m}\left(t\right),\phantom{\rule{3.33333pt}{0ex}}m=5,6,7,8$ in Equation (39) vs. t at $\lambda =1$, $c=\frac{3}{2}$, $b=3$, and $a=-3$.

**Figure 7.**Plots of the approximate solutions ${\varphi}_{m}\left(t\right),\phantom{\rule{3.33333pt}{0ex}}m=3,4,5,6$ in Equation (39) vs. t at $\lambda =1$, $c=\frac{5}{2}$, $b=3$, and $a=-2$.

**Figure 8.**Plots of the approximate solutions ${\varphi}_{m}\left(t\right),\phantom{\rule{3.33333pt}{0ex}}m=1,2,3,4$ in Equation (39) vs. t at $\lambda =1$, $c=5$, $b=-1$, and $a=-5$.

**Figure 9.**Plots of the exact solution in Equation (35) vs. t at different values of $\lambda $ when $a=1$ and $b=0$.

**Figure 10.**Plots of the exact solution in Equation (36) vs. t at different values of $\lambda $ when $a=0$ and $b=1$.

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**MDPI and ACS Style**

El-Zahar, E.R.; Ebaid, A.
Analytical and Numerical Simulations of a Delay Model: The Pantograph Delay Equation. *Axioms* **2022**, *11*, 741.
https://doi.org/10.3390/axioms11120741

**AMA Style**

El-Zahar ER, Ebaid A.
Analytical and Numerical Simulations of a Delay Model: The Pantograph Delay Equation. *Axioms*. 2022; 11(12):741.
https://doi.org/10.3390/axioms11120741

**Chicago/Turabian Style**

El-Zahar, Essam Roshdy, and Abdelhalim Ebaid.
2022. "Analytical and Numerical Simulations of a Delay Model: The Pantograph Delay Equation" *Axioms* 11, no. 12: 741.
https://doi.org/10.3390/axioms11120741