# A Collocation Method Using Radial Polynomials for Solving Partial Differential Equations

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## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Radial Polynomials

## 3. Accuracy and Convergence Analysis

## 4. Numerical Examples

#### 4.1. A Two-Dimensional Ameoba-Shaped Problem

#### 4.2. A Two-Dimensional Star-Shaped Problem

#### 4.3. A Three-Dimensional Modified Helmholtz Problem

#### 4.4. A Three-Dimensional Poisson Problem

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Results comparison between this study and the Kansa method with the optimal shape parameter.

${\mathit{M}}_{\mathit{b}}$ | ${\mathit{M}}_{\mathit{i}}$ | ${\mathit{M}}_{\mathit{c}}$ | RMSE | Condition Number | ||
---|---|---|---|---|---|---|

This Study | The Kansa Method (Optimal Shape Parameter) | This Study | The Kansa Method | |||

736 | 92 | 92 | $3.77\times 1{0}^{-12}$ | $2.96\times 1{0}^{-9}$$(c=0.70)$ | $8.33\times 1{0}^{22}$ | $5.32\times 1{0}^{20}$ |

1208 | 151 | 151 | $7.29\times 1{0}^{-12}$ | $2.44\times 1{0}^{-9}$$(c=0.95)$ | $1.09\times 1{0}^{23}$ | $2.63\times 1{0}^{20}$ |

1792 | 224 | 224 | $7.06\times 1{0}^{-12}$ | $7.53\times 1{0}^{-9}$$(c=1.05)$ | $4.52\times 1{0}^{23}$ | $7.30\times 1{0}^{21}$ |

2480 | 310 | 310 | $5.90\times 1{0}^{-12}$ | $5.70\times 1{0}^{-9}$$(c=1.05)$ | $3.86\times 1{0}^{24}$ | $4.21\times 1{0}^{20}$ |

3240 | 405 | 405 | $5.31\times 1{0}^{-12}$ | $9.42\times 1{0}^{-9}$$(c=1.30)$ | $1.81\times 1{0}^{24}$ | $1.52\times 1{0}^{21}$ |

4115 | 514 | 514 | $4.77\times 1{0}^{-12}$ | $7.52\times 1{0}^{-9}$$(c=1.25)$ | $1.18\times 1{0}^{24}$ | $2.87\times 1{0}^{21}$ |

${\mathit{M}}_{\mathit{b}}$ | ${\mathit{M}}_{\mathit{i}}$ | ${\mathit{M}}_{\mathit{c}}$ | RMSE | |
---|---|---|---|---|

This Study | The Kansa Method (Optimal Shape Parameter) | |||

1050 | 300 | 300 | $7.65\times 1{0}^{-8}$ | $5.46\times 1{0}^{-4}$$(c=1.00)$ |

1400 | 400 | 400 | $8.89\times 1{0}^{-8}$ | $4.72\times 1{0}^{-4}$$(c=0.95)$ |

1750 | 500 | 500 | $6.77\times 1{0}^{-8}$ | $4.14\times 1{0}^{-4}$$(c=1.05)$ |

2100 | 600 | 600 | $6.26\times 1{0}^{-8}$ | $3.79\times 1{0}^{-4}$$(c=1.05)$ |

2450 | 700 | 700 | $5.80\times 1{0}^{-8}$ | $3.56\times 1{0}^{-4}$$(c=1.30)$ |

2800 | 800 | 800 | $5.41\times 1{0}^{-8}$ | $3.45\times 1{0}^{-4}$$(c=1.25)$ |

${\mathit{M}}_{\mathit{b}}$ | ${\mathit{M}}_{\mathit{i}}$ | ${\mathit{M}}_{\mathit{c}}$ | RMSE | |
---|---|---|---|---|

This Study | The Kansa Method (Optimal Shape Parameter) | |||

6724 | 700 | 700 | $4.28\times 1{0}^{-11}$ | $1.10\times 1{0}^{-6}$$(c=1.30)$ |

7569 | 800 | 800 | $2.32\times 1{0}^{-11}$ | $8.48\times 1{0}^{-7}$$(c=1.30)$ |

8100 | 900 | 900 | $4.40\times 1{0}^{-11}$ | $1.01\times 1{0}^{-6}$$(c=1.90)$ |

9025 | 1000 | 1000 | $5.99\times 1{0}^{-11}$ | $1.08\times 1{0}^{-6}$$(c=1.10)$ |

10,000 | 1100 | 1100 | $5.03\times 1{0}^{-11}$ | $7.88\times 1{0}^{-7}$$(c=1.30)$ |

${\mathit{M}}_{\mathit{b}}$ | ${\mathit{M}}_{\mathit{i}}$ | ${\mathit{M}}_{\mathit{c}}$ | RMSE |
---|---|---|---|

This Study | |||

5706 | 630 | 630 | $1.75\times 1{0}^{-10}$ |

7357 | 756 | 756 | $1.82\times 1{0}^{-10}$ |

9208 | 882 | 882 | $1.87\times 1{0}^{-10}$ |

11,259 | 1008 | 1008 | $1.92\times 1{0}^{-10}$ |

13,510 | 1134 | 1134 | $1.94\times 1{0}^{-10}$ |

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## Share and Cite

**MDPI and ACS Style**

Ku, C.-Y.; Xiao, J.-E.
A Collocation Method Using Radial Polynomials for Solving Partial Differential Equations. *Symmetry* **2020**, *12*, 1419.
https://doi.org/10.3390/sym12091419

**AMA Style**

Ku C-Y, Xiao J-E.
A Collocation Method Using Radial Polynomials for Solving Partial Differential Equations. *Symmetry*. 2020; 12(9):1419.
https://doi.org/10.3390/sym12091419

**Chicago/Turabian Style**

Ku, Cheng-Yu, and Jing-En Xiao.
2020. "A Collocation Method Using Radial Polynomials for Solving Partial Differential Equations" *Symmetry* 12, no. 9: 1419.
https://doi.org/10.3390/sym12091419