# Symmetric Radial Basis Function Method for Simulation of Elliptic Partial Differential Equations

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

## 3. Numerical Scheme

## 4. Numerical Results

**Problem**

**1.**

**Problem**

**2.**

**Problem**

**3.**

**Problem**

**4.**

**Problem**

**5.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Numerical approximation (

**left**) and ${L}_{abs}$ error norm (

**right**) for Problem 1 for N = 256.

**Figure 2.**Numerical approximation (

**left**) and ${L}_{abs}$ error norm (

**right**) for Problem 2 for N = 400.

**Figure 3.**${L}_{\infty}$ error norm with different non-uniform nodes for Problem 2. (

**a**) Chebyshev nodes $N=100$; (

**b**) ${L}_{\infty}$ error norm using Chebyshev nodes; (

**c**) random nodes $N=100$; (

**d**) ${L}_{\infty}$ error norm using random nodes; (

**e**) Halton nodes $N=107$; (

**f**) ${L}_{\infty}$ error norm using Halton nodes.

**Figure 4.**Computational domain x-axis $[-1,1]$ and y-axis $[-1,1]$ (

**left**) and numerical solution (

**right**) for Problem 2.

**Figure 5.**Computational domain x-axis $[-2.5,2.5]$ and y-axis $[-2.5,2.5]$ (

**left**) and numerical solution (

**right**) for Problem 2.

**Figure 6.**Computational domain x-axis $[-2,2]$ and y-axis $[-2,2]$ (

**left**) and numerical solution (

**right**) for Problem 2.

**Figure 7.**Computational domain x-axis $[-3,3]$ and y-axis $[-3,3]$ (

**left**) and numerical solution (

**right**) for Problem 2.

**Figure 8.**Computational circle domain x-axis $[-3,3]$ and y-axis $[-3,3]$ (

**left**) and numerical solution (

**right**) for Problem 2.

**Figure 9.**Numerical approximation (

**left**) and ${L}_{abs}$ error norm (

**right**) for Problem 3 for N = 400.

**Figure 10.**Numerical approximation (

**left**) and ${L}_{abs}$ error norm (

**right**) for Problem 4 for N = 1600.

**Figure 12.**Computational domain (

**left**) and ${L}_{abs}$ error norm on the line $x=y=z$ (

**right**) for Problem 5.

**Figure 13.**Computational domain (

**left**) and ${L}_{abs}$ error norm on the line $x=y=z$ (

**right**) for Problem 5.

N | c | ${\mathit{L}}_{\mathbf{\infty}}$ | RMS |
---|---|---|---|

81 | 0.124 | $1.6296\times {10}^{-11}$ | $4.4250\times {10}^{-11}$ |

256 | 0.125 | $6.1207\times {10}^{-11}$ | $2.3986\times {10}^{-11}$ |

400 | 0.125 | $4.6252\times {10}^{-11}$ | $2.8510\times {10}^{-10}$ |

900 | 0.129 | $1.6221\times {10}^{-10}$ | $1.3108\times {10}^{-9}$ |

1600 | 0.126 | $8.8824\times {10}^{-11}$ | $1.4803\times {10}^{-9}$ |

2500 | 0.126 | $2.5943\times {10}^{-10}$ | $2.9648\times {10}^{-9}$ |

N | c | ${\mathit{L}}_{\mathbf{\infty}}$ | RMS |
---|---|---|---|

81 | 0.406 | $5.3081\times {10}^{-7}$ | $1.5419\times {10}^{-6}$ |

256 | 0.52 | $3.1206\times {10}^{-7}$ | $1.1868\times {10}^{-6}$ |

400 | 0.96 | $5.3085\times {10}^{-8}$ | $3.0250\times {10}^{-7}$ |

900 | 1.67 | $2.3015\times {10}^{-8}$ | $2.6587\times {10}^{-7}$ |

1600 | 2.15 | $2.1188\times {10}^{-8}$ | $3.7525\times {10}^{-7}$ |

2500 | 2.52 | $1.4109\times {10}^{-8}$ | $4.6924\times {10}^{-7}$ |

t | N | c | ${\mathit{L}}_{\mathbf{\infty}}$ | RMS |
---|---|---|---|---|

1 | 81 | 0.3 | $5.0926\times {10}^{-8}$ | $1.3014\times {10}^{-7}$ |

400 | 0.8 | $1.5166\times {10}^{-8}$ | $8.7744\times {10}^{-8}$ | |

900 | 1.59 | $4.9779\times {10}^{-9}$ | $3.4086\times {10}^{-8}$ | |

5 | 81 | 0.43 | $7.7247\times {10}^{-5}$ | $1.8988\times {10}^{-4}$ |

400 | 1.1 | $7.5979\times {10}^{-7}$ | $3.6823\times {10}^{-6}$ | |

900 | 1.69 | $1.7120\times {10}^{-7}$ | $9.8686\times {10}^{-7}$ | |

10 | 81 | 1.15 | $1.8200\times {10}^{-2}$ | $3.7200\times {10}^{-2}$ |

400 | 1.22 | $1.5067\times {10}^{-5}$ | $6.9115\times {10}^{-5}$ | |

900 | 1.93 | $4.1883\times {10}^{-8}$ | $2.5267\times {10}^{-5}$ | |

20 | 81 | 4.13 | $2.4950\times {10}^{-1}$ | $4.4920\times {10}^{-1}$ |

400 | 1.97 | $2.9000\times {10}^{-3}$ | $8.2000\times {10}^{-3}$ | |

900 | 2.22 | $1.2416\times {10}^{-4}$ | $5.4952\times {10}^{-4}$ |

N | c | ${\mathit{L}}_{\mathbf{\infty}}$ | RMS |
---|---|---|---|

81 | 0.73 | $8.5033\times {10}^{-6}$ | $2.2291\times {10}^{-5}$ |

256 | 0.9 | $3.5236\times {10}^{-7}$ | $1.8206\times {10}^{-6}$ |

400 | 1.31 | $7.2561\times {10}^{-8}$ | $4.5238\times {10}^{-7}$ |

900 | 2.02 | $1.8554\times {10}^{-8}$ | $1.4499\times {10}^{-7}$ |

1600 | 2.42 | $4.5357\times {10}^{-9}$ | $4.9253\times {10}^{-8}$ |

2500 | 2.92 | $1.9209\times {10}^{-9}$ | $1.5399\times {10}^{-8}$ |

N | c | ${\mathit{L}}_{\mathbf{\infty}}$ | RMS |
---|---|---|---|

343 | 0.06 | $3.6458\times {10}^{-6}$ | $1.7514\times {10}^{-5}$ |

1331 | 0.06 | $3.7429\times {10}^{-6}$ | $2.9703\times {10}^{-5}$ |

3375 | 0.06 | $5.1290\times {10}^{-6}$ | $6.9616\times {10}^{-5}$ |

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## Share and Cite

**MDPI and ACS Style**

Thounthong, P.; Khan, M.N.; Hussain, I.; Ahmad, I.; Kumam, P.
Symmetric Radial Basis Function Method for Simulation of Elliptic Partial Differential Equations. *Mathematics* **2018**, *6*, 327.
https://doi.org/10.3390/math6120327

**AMA Style**

Thounthong P, Khan MN, Hussain I, Ahmad I, Kumam P.
Symmetric Radial Basis Function Method for Simulation of Elliptic Partial Differential Equations. *Mathematics*. 2018; 6(12):327.
https://doi.org/10.3390/math6120327

**Chicago/Turabian Style**

Thounthong, Phatiphat, Muhammad Nawaz Khan, Iltaf Hussain, Imtiaz Ahmad, and Poom Kumam.
2018. "Symmetric Radial Basis Function Method for Simulation of Elliptic Partial Differential Equations" *Mathematics* 6, no. 12: 327.
https://doi.org/10.3390/math6120327