# Model Problems on Oscillations of Mechanical and Biological Membranes

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

- ${\alpha}^{2}\left(t\right){u}_{tt}+2\beta \left(t\right){u}_{t}+{\gamma}^{2}\left(t\right)u={u}_{xx}$—equations for voltage fluctuations in a limited telegraph line.
- ${\alpha}^{2}\left(t\right){u}_{tt}+2\beta \left(t\right){u}_{t}+{\gamma}^{2}\left(t\right)u={u}_{xx}+{u}_{yy}+{u}_{zz},\left(x,y,z\right):{x}^{2}+{y}^{2}+{z}^{2}\le {R}^{2}$—equation of damped gas oscillations in a spherical region.
- ${\alpha}^{2}\left(t\right){u}_{tt}+2\beta \left(t\right){u}_{t}+{\gamma}^{2}\left(t\right)u={u}_{xx}+{u}_{yy},\left(x,y\right):{x}^{2}+{y}^{2}\le {R}^{2}$—vibration equation of a circular membrane.

## 3. Results

#### 3.1. Spherically Symmetric Cases in Three-Dimensional Space

#### 3.2. Oscillations of Circular Membranes

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The graph of the solution at various points in time when $\alpha \equiv 1,\beta =4,\gamma =2$.

**Figure 2.**The graph of the solution at various points in time when $\alpha \equiv 1,\beta =16,\gamma =2$.

**Figure 3.**The graph of the solution at various points in time when $\alpha \equiv 1,\beta =0.4,\gamma =\sqrt{0.2}$.

**Figure 4.**The graph of the solution at various points in time when $\alpha \equiv 1,\beta =0.4,\gamma =\sqrt{2}$.

**Figure 5.**The graph of the solution at various points in time when $\alpha \equiv 1,\beta =0,\gamma =2$.

**Figure 6.**The graph of the solution at various points in time when $\alpha \equiv 1,\beta =16,\gamma =0$.

**Figure 7.**The graph of the solution at various points in time when $\alpha \equiv 1,\beta =0.1,\gamma =\sqrt{0.2}$.

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**MDPI and ACS Style**

Kostikov, Y.; Romanenkov, A.
Model Problems on Oscillations of Mechanical and Biological Membranes. *Inventions* **2023**, *8*, 139.
https://doi.org/10.3390/inventions8060139

**AMA Style**

Kostikov Y, Romanenkov A.
Model Problems on Oscillations of Mechanical and Biological Membranes. *Inventions*. 2023; 8(6):139.
https://doi.org/10.3390/inventions8060139

**Chicago/Turabian Style**

Kostikov, Yury, and Aleksandr Romanenkov.
2023. "Model Problems on Oscillations of Mechanical and Biological Membranes" *Inventions* 8, no. 6: 139.
https://doi.org/10.3390/inventions8060139