# Exact Solutions and Continuous Numerical Approximations of Coupled Systems of Diffusion Equations with Delay

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Separated Initial-Value Vector Delay Differential Problem

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

## 3. Exact Infinite Series Solution

**Theorem**

**2.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Proof of Theorem**

**2.**

## 4. Continuous Numerical Solutions

**Theorem**

**3.**

**Example**

**1.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Numerical solutions computed with $N=20$ for Example 1. (

**a**) First component. (

**b**) Second component.

**Figure 2.**Maximum error bounds (log-scale) for numerical solutions of Example 1 for $(t,x)\in \left[\right(m-1)\tau +\delta ,m\tau ]\times [0,l]$, with $\delta =0.1$, in terms of N. (

**a**) Error bounds for different m values. (

**b**) Individual contributions to total error bound, for $m=4$, of the first ($s1$), second ($s2$), and third term ($s3$) in (56), with different scales for the first two terms (left axis) and the third term (right axis).

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**MDPI and ACS Style**

Reyes, E.; Castro, M.Á.; Sirvent, A.; Rodríguez, F.
Exact Solutions and Continuous Numerical Approximations of Coupled Systems of Diffusion Equations with Delay. *Symmetry* **2020**, *12*, 1560.
https://doi.org/10.3390/sym12091560

**AMA Style**

Reyes E, Castro MÁ, Sirvent A, Rodríguez F.
Exact Solutions and Continuous Numerical Approximations of Coupled Systems of Diffusion Equations with Delay. *Symmetry*. 2020; 12(9):1560.
https://doi.org/10.3390/sym12091560

**Chicago/Turabian Style**

Reyes, Elia, M. Ángeles Castro, Antonio Sirvent, and Francisco Rodríguez.
2020. "Exact Solutions and Continuous Numerical Approximations of Coupled Systems of Diffusion Equations with Delay" *Symmetry* 12, no. 9: 1560.
https://doi.org/10.3390/sym12091560