# Hybrid Finite Element Method to Thermo-Elastic Interactions in a Piezo-Thermo-Elastic Medium under a Fractional Time Derivative Model

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model

## 3. Applications

## 4. Laplace Transforms

## 5. Finite Element Method

## 6. Numerical Results and Discussions

**Figure 1.**The temperature distributions via the distance $x$ for three values of fractional parameter.

**Figure 2.**The electric potential distribution via the distance $x$ for three values of fractional parameters.

**Figure 3.**The displacement distributions via the distance $x$ for three values of fractional parameters.

**Figure 4.**The electric field distributions via the distance $x$ for three values of fractional parameters.

**Figure 6.**The temperature distributions via the distance $x$ with and without thermal relaxation time.

**Figure 7.**The electric potential distribution via the distance $x$ with and without thermal relaxation time.

**Figure 8.**The displacement distribution via the distance $x$ with and without thermal relaxation time.

**Figure 9.**The electric field distribution via the distance $x$ with and without thermal relaxation time.

## 7. Conclusions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Saeed, T.
Hybrid Finite Element Method to Thermo-Elastic Interactions in a Piezo-Thermo-Elastic Medium under a Fractional Time Derivative Model. *Mathematics* **2022**, *10*, 650.
https://doi.org/10.3390/math10040650

**AMA Style**

Saeed T.
Hybrid Finite Element Method to Thermo-Elastic Interactions in a Piezo-Thermo-Elastic Medium under a Fractional Time Derivative Model. *Mathematics*. 2022; 10(4):650.
https://doi.org/10.3390/math10040650

**Chicago/Turabian Style**

Saeed, Tareq.
2022. "Hybrid Finite Element Method to Thermo-Elastic Interactions in a Piezo-Thermo-Elastic Medium under a Fractional Time Derivative Model" *Mathematics* 10, no. 4: 650.
https://doi.org/10.3390/math10040650