# Comparison of Symbolic Computations for Solving Linear Delay Differential Equations Using the Laplace Transform Method

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Method of Steps (MoS)

#### 2.2. Lambert W Function

#### 2.3. Cauchy’s Residue Theorem

**Residues of Poles:**If $M=1$, then $f\left(z\right)$ has a simple pole at ${z}_{0}$. At a simple pole ${z}_{0}\in \mathbb{C}$, the residue of $f\left(z\right)$ is given by [54]:

**Residue Theorem:**Suppose C is a simple closed contour and let $f\left(z\right)$ be an analytic function inside and on C, excluding a finite number (K) of isolated singularities inside C. Then,

## 3. Laplace Transform (LT) Method

#### 3.1. Theory for Linear Non-Neutral DDE Case

#### 3.2. Theory for NDDE Case

## 4. Comparison of MoS and LT Methods Using Symbolic Computation

- ${e}_{MoS}\left(t\right)=|{y}_{Maple,MoS}\left(t\right)-{y}_{MATLAB,MoS}\left(t\right)|$: Absolute error in the solution $y\left(t\right)$ between Maple and MATLAB via the MoS.
- ${e}_{LT}\left(t\right)=|{y}_{Maple,LT}\left(t\right)-{y}_{MATLAB,LT}\left(t\right)|$: Absolute error in the solution $y\left(t\right)$ between Maple and MATLAB via the LT.
- ${e}_{Map}\left(t\right)=|{y}_{Maple,MoS}\left(t\right)-{y}_{Maple,LT}\left(t\right)|$: Absolute error in the solution $y\left(t\right)$ between the MoS and LT via Maple.
- ${e}_{MAT}\left(t\right)=|{y}_{MATLAB,MoS}\left(t\right)-{y}_{MATLAB,LT}\left(t\right)|$: Absolute error in the solution $y\left(t\right)$ between the MoS and LT via MATLAB.

#### 4.1. Linear Non-Neutral DDE Examples

**Example**

**1.**

**Example**

**2.**

#### 4.2. Linear NDDE Examples

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

- From Maple MoS solutions, the order of time computation is as follows:Example 5 > Example 6 > Example 7 > Example 4 > Example 3.
- From Maple LT solutions, the order of time computation is as follows:Example 7 > Example 3 > Example 6 > Example 5 > Example 4.
- From MATLAB MoS solutions, the order of time computation is as follows:Example 4 > Example 5 > Example 7 > Example 6 > Example 3.
- From MATLAB LT solutions, the order of time computation is as follows:Example 3 > Example 7 > Example 6 > Example 4 > Example 5.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Bromwich contour (interested readers are referred to [54]).

**Figure 2.**Solution results of Equation (19) over $t\in [0,9\tau ]$ ($n=5000$ terms): plot of (

**a**) the solution $y\left(t\right)$ via each method and software, (

**b**) ${e}_{MoS}\left(t\right)$, (

**c**) ${e}_{LT}\left(t\right)$, (

**d**) ${e}_{Map}\left(t\right)$ and ${e}_{MAT}\left(t\right)$.

**Figure 3.**Solution results of Equation (20) over $t\in [0,9\tau ]$ ($n=5000$ terms): plot of (

**a**) the solution $y\left(t\right)$ via each method and software, (

**b**) ${e}_{MoS}\left(t\right)$, (

**c**) ${e}_{LT}\left(t\right)$, (

**d**) ${e}_{Map}\left(t\right)$ and ${e}_{MAT}\left(t\right)$.

**Figure 4.**Solution results of Equation (21) over $t\in [0,5\tau ]$ ($n=2000$ terms): plot of (

**a**) the solution $y\left(t\right)$ via each method and software, (

**b**) ${e}_{MoS}\left(t\right)$, (

**c**) ${e}_{LT}\left(t\right)$, (

**d**) ${e}_{Map}\left(t\right)$ and ${e}_{MAT}\left(t\right)$.

**Figure 5.**Solution results of Equation (22) over $t\in [0,6\tau ]$ ($n=500$ terms): plot of (

**a**) the solution $y\left(t\right)$ via each method and software, (

**b**) ${e}_{MoS}\left(t\right)$, (

**c**) ${e}_{LT}\left(t\right)$, (

**d**) ${e}_{Map}\left(t\right)$ and ${e}_{MAT}\left(t\right)$.

**Figure 6.**Solution results of Equation (23) over $t\in [0,10\tau ]$ ($n=500$ terms): plot of (

**a**) the solution $y\left(t\right)$ via each method and software, (

**b**) ${e}_{MoS}\left(t\right)$, (

**c**) ${e}_{LT}\left(t\right)$, (

**d**) ${e}_{Map}\left(t\right)$ and ${e}_{MAT}\left(t\right)$.

**Figure 7.**Solution results of Equation (24) over $t\in [0,7\tau ]$ ($n=500$ terms): plot of (

**a**) the solution $y\left(t\right)$ via each method and software, (

**b**) ${e}_{MoS}\left(t\right)$, (

**c**) ${e}_{LT}\left(t\right)$, (

**d**) ${e}_{Map}\left(t\right)$ and ${e}_{MAT}\left(t\right)$.

**Figure 8.**Solution results of Equation (25) over $t\in [0,7\tau ]$ ($n=500$ terms): plot of (

**a**) the solution $y\left(t\right)$ via each method and software, (

**b**) ${e}_{MoS}\left(t\right)$, (

**c**) ${e}_{LT}\left(t\right)$, (

**d**) ${e}_{Map}\left(t\right)$ and ${e}_{MAT}\left(t\right)$.

Example # | Program | Method | Time (s) |
---|---|---|---|

1 | Maple | MoS | 0.407 |

LT | 246.031 | ||

MATLAB | MoS | 1.335 | |

LT | 64.914 | ||

2 | Maple | MoS | 0.485 |

LT | 269.828 | ||

MATLAB | MoS | 1.385 | |

LT | 64.914 |

Example # | Program | Method | Time (s) |
---|---|---|---|

3 | Maple | MoS | 0.203 |

LT | 84.219 | ||

MATLAB | MoS | 0.838 | |

LT | 870.978 | ||

4 | Maple | MoS | 0.296 |

LT | 15.470 | ||

MATLAB | MoS | 2.022 | |

LT | 32.663 | ||

5 | Maple | MoS | 0.453 |

LT | 17.110 | ||

MATLAB | MoS | 1.572 | |

LT | 29.052 | ||

6 | Maple | MoS | 0.438 |

LT | 42.344 | ||

MATLAB | MoS | 1.130 | |

LT | 284.252 | ||

7 | Maple | MoS | 0.297 |

LT | 89.093 | ||

MATLAB | MoS | 1.280 | |

LT | 349.196 |

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**MDPI and ACS Style**

Sherman, M.; Kerr, G.; González-Parra, G.
Comparison of Symbolic Computations for Solving Linear Delay Differential Equations Using the Laplace Transform Method. *Math. Comput. Appl.* **2022**, *27*, 81.
https://doi.org/10.3390/mca27050081

**AMA Style**

Sherman M, Kerr G, González-Parra G.
Comparison of Symbolic Computations for Solving Linear Delay Differential Equations Using the Laplace Transform Method. *Mathematical and Computational Applications*. 2022; 27(5):81.
https://doi.org/10.3390/mca27050081

**Chicago/Turabian Style**

Sherman, Michelle, Gilbert Kerr, and Gilberto González-Parra.
2022. "Comparison of Symbolic Computations for Solving Linear Delay Differential Equations Using the Laplace Transform Method" *Mathematical and Computational Applications* 27, no. 5: 81.
https://doi.org/10.3390/mca27050081