# On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term

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## Abstract

**:**

## 1. Introduction

## 2. The Existence and Uniqueness Theorem

**Theorem**

**1.**

- $b\left(t,-\mathsf{\pi}\right)=b\left(t,\mathsf{\pi}\right),b\left(t,\mathsf{\phi}\right)0$ and $\mathsf{\eta}\left(0\right)=0$.
- $b\left(t,\mathsf{\phi}\right)$ and $1/b\left(t,\mathsf{\phi}\right)$ are analytical with respect to $\mathsf{\phi}$ when $-\mathsf{\pi}\le \mathsf{\phi}\le \mathsf{\pi}$ and, in terms of $t$ in some neighborhood of the initial time $t=0$, $\mathsf{\eta}\left(u\right)$ is analytical with respect to $u$.

**Proof.**

**Stage**

**1.**

**Stage**

**2.**

- $g\left(\mathsf{\tau},\mathsf{\psi},s\right)$ is analytical in some neighborhood of $\mathsf{\tau}=0,s=0$ and at all $-\mathsf{\pi}\le \mathsf{\psi}\le \mathsf{\pi}$.
- ${G}_{i}$ and $i=1,2,3$ are quadratic polynomials with respect to the required function $v$ and its derivatives, whose coefficients are functions of the independent variables $\mathsf{\tau},\mathsf{\psi},s$, analytical in some neighborhood of $\mathsf{\tau}=0,s=0$ and at all $-\mathsf{\pi}\le \mathsf{\psi}\le \mathsf{\pi}$.

## 3. The BEM Solution Algorithm

- We assume ${u}^{\left(0\right)}\equiv 0$ as the initial iteration;
- Solving system (37), we determine the next, (n+1)-th, iteration of the nodal values of temperature and flux on the boundary ${S}^{\left(0\right)}$, ${u}_{m}^{\left(n+1\right)}$ and ${q}_{m}^{\left(n+1\right)}$;
- The determined nodal values give us the (n+1)-th iteration of the solution$$\begin{array}{ll}{u}^{\left(n+1\right)}\left(\xi \right)& ={\displaystyle \sum _{m=1}^{2N}\left({q}_{m}^{\left(n+1\right)}{\displaystyle \underset{{e}_{m}}{\int}{u}^{*}\left(\xi ,x\right)dS\left(x\right)}-{u}_{m}^{\left(n+1\right)}{\displaystyle \underset{{e}_{m}}{\int}{q}^{*}\left(\xi ,x\right)dS\left(x\right)}\right)}\\ & +{\displaystyle \sum _{i=1}^{K}{\mathsf{\alpha}}_{i}^{\left(n\right)}\left({\widehat{u}}_{i}\left(\xi \right)-{\displaystyle \sum _{m=1}^{2N}\left({\widehat{q}}_{i}\left({x}_{m}\right){\displaystyle \underset{{e}_{m}}{\int}{u}^{*}\left(\xi ,x\right)dS\left(x\right)}-{\widehat{u}}_{i}\left({x}_{m}\right){\displaystyle \underset{{e}_{m}}{\int}{q}^{*}\left(\xi ,x\right)dS\left(x\right)}\right)}\right)};\end{array}$$
- Solving system (35), we determine the coefficients ${\mathsf{\alpha}}_{i}^{(n+1)}$ for the next iteration;
- We go to the next iteration.

## 4. Exact Solutions

## 5. Test Examples

## 6. Analysis of Using Radial Basis Functions

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Calculation of Heat Flux on the Zero Front of a Heat Wave

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t | h | Relative Error | |||
---|---|---|---|---|---|

N = 150 | N = 250 | N = 300 | N = 350 | ||

0.3 | 0.1 | 0.00069 | 0.00047 | 0.00039 | 0.00035 |

0.6 | 0.1 | 0.00081 | 0.00058 | 0.00048 | 0.00043 |

1 | 0.1 | 0.00095 | 0.00071 | 0.00060 | 0.00052 |

0.3 | 0.05 | 0.00061 | 0.00041 | 0.00034 | 0.00029 |

0.6 | 0.05 | 0.00072 | 0.00051 | 0.00043 | 0.00037 |

1 | 0.05 | 0.00084 | 0.00062 | 0.00054 | 0.00045 |

t | h | Relative Error | ||||||
---|---|---|---|---|---|---|---|---|

${\mathit{f}}_{\mathit{i}}={\mathit{r}}_{\mathit{i}}$ | ${\mathit{f}}_{\mathit{i}}={\mathit{r}}_{\mathit{i}}^{3}$ | ${\mathit{f}}_{\mathit{i}}=1+{\mathit{r}}_{\mathit{i}}$ | ${\mathit{f}}_{\mathit{i}}=1+{\mathbf{\delta}}_{1}{\mathit{r}}_{\mathit{i}}$ | ${\mathit{f}}_{\mathit{i}}=1+{\mathbf{\delta}}_{2}{\mathit{r}}_{\mathit{i}}$ | ${\mathit{f}}_{\mathit{i}}=\sqrt{1+{\left({\mathbf{\delta}}_{1}{\mathit{r}}_{\mathit{i}}\right)}^{2}}$ | ${\mathit{f}}_{\mathit{i}}=\sqrt{1+{\left({\mathbf{\delta}}_{2}{\mathit{r}}_{\mathit{i}}\right)}^{2}}$ | ||

0.3 | 0.1 | 0.00064 | 0.00048 | 0.00035 | 0.00017 | 0.00015 | 0.00012 | 0.00011 |

0.6 | 0.1 | 0.00077 | 0.00060 | 0.00043 | 0.00020 | 0.00017 | 0.00016 | 0.00014 |

1 | 0.1 | 0.00094 | 0.00074 | 0.00052 | 0.00024 | 0.00021 | 0.00022 | 0.00019 |

0.3 | 0.05 | 0.00058 | 0.00040 | 0.00029 | 0.00016 | 0.00014 | 0.00010 | 0.00010 |

0.6 | 0.05 | 0.00072 | 0.00054 | 0.00037 | 0.00018 | 0.00015 | 0.00013 | 0.00012 |

1 | 0.05 | 0.00087 | 0.00067 | 0.00045 | 0.00023 | 0.00018 | 0.00019 | 0.00016 |

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**MDPI and ACS Style**

Kazakov, A.; Spevak, L.; Nefedova, O.; Lempert, A.
On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term. *Symmetry* **2020**, *12*, 921.
https://doi.org/10.3390/sym12060921

**AMA Style**

Kazakov A, Spevak L, Nefedova O, Lempert A.
On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term. *Symmetry*. 2020; 12(6):921.
https://doi.org/10.3390/sym12060921

**Chicago/Turabian Style**

Kazakov, Alexander, Lev Spevak, Olga Nefedova, and Anna Lempert.
2020. "On the Analytical and Numerical Study of a Two-Dimensional Nonlinear Heat Equation with a Source Term" *Symmetry* 12, no. 6: 921.
https://doi.org/10.3390/sym12060921