# A Simplified Radial Basis Function Method with Exterior Fictitious Sources for Elliptic Boundary Value Problems

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Elliptic Boundary Value Problems

**x**is the Cartesian coordinate, defined as $x=(x,y,z)$;

**A**is the velocity, defined as $A=({A}_{x},{A}_{y},{A}_{z})$; $B(x)$ is the given function; $f(x)$ is the given function value; $g(x)$ defines the given boundary conditions; and $\Omega $ is the domain with the boundary $\partial \Omega $.

#### 2.2. Simplified Radial Basis Functions

**x**denotes the interior point; ${x}^{s}$ denotes the source point, defined as ${x}^{s}=({x}^{s},{y}^{s},{z}^{s})$; and R denotes the characteristic length, which is the maximum radial distance. We can easily obtain the simplified MQ RBF as follows:

#### 2.3. Discretization

#### 2.3.1. Discretization in Two Dimensions

#### 2.3.2. Discretization in Three Dimensions

**f**is an ${N}_{i}\times 1$ vector of the function values for the interior points, written as $f=[{f}_{1},\text{}{f}_{2},\text{}\dots ,\text{}{f}_{{N}_{i}}]$;

**g**is an ${N}_{b}\times 1$ vector of boundary data, written as $g=[{g}_{1},\text{}{g}_{2},\text{}\dots ,\text{}{g}_{{N}_{b}}]$; ${N}_{i}$ is the number of interior points; and ${N}_{b}$ is the number of boundary points. Once the unknown coefficients are determined, we can collocate the validation points uniformly placed inside the domain to obtain the computed results.

#### 2.4. Location of Fictitious Sources

#### 2.4.1. Type A: Uniform Centers

#### 2.4.2. Type B: Randomly Fictitious Centers

#### 2.4.3. Type C: Exterior Fictitious Sources

## 3. Validation of the Methodology

#### 3.1. Example 1

#### 3.1.1. The Gaussian RBF

^{−12}. It seems that the simplified Gaussian RBFs utilizing the exterior fictitious sources of type C have the best accuracy among those Gaussian RBFs for type A and type B even when different values of the shape parameter are considered.

#### 3.1.2. The MQ RBF

^{−2}to 10

^{−7}as the shape parameter ranges from 0.2 to 5. However, the RMSE of the simplified MQ RBF in type C is 10

^{–}

^{13}. It was found that the RMSE of the simplified MQ RBF without a shape parameter in type C has the best accuracy among the MQ RBFs for type A and type B for different values of the shape parameter.

#### 3.1.3. The IMQ RBF

^{−12}, 10

^{−13}, and 10

^{−12}, respectively. From the results, we also demonstrated that the above simplified RBFs with exterior fictitious sources can be used to solve this two-dimensional Laplace problem with very high accuracy. From Table 2, the comparison of the computing time also illustrates the efficiency of the proposed method.

^{–11}and 10

^{–13}while the dilation factor ranges from 2.5 to 5. The results obtained show that the dilation factor has low sensitivity regarding the numerical accuracy while the dilation factor is greater than 2.5. Accordingly, the following numerical implementations of type C were solved using $\eta =3$.

#### 3.2. Example 2

^{−14}, 10

^{−15}, and 10

^{−13}, respectively. It is significant that excellent agreement was achieved, and highly accurate results were acquired using the simplified RBFs. From these results, it is demonstrated that the simplified RBFs with exterior fictitious sources can be used to solve the three-dimensional stationary Laplace equation with very high accuracy.

## 4. Application Examples

#### 4.1. Application Example 1

^{–}

^{13}. Figure 7b demonstrates the RMSE of the MQ RBF for the three collocation types. According to Figure 7b, the RMSE of the MQ RBF in type A and type B was in the order of 10

^{−2}to 10

^{−7}as the shape parameter ranged from 0 to 5. The RMSE of the MQ RBF without a shape parameter in type C was in the order of 10

^{–}

^{10}. The IMQ RBFs was analyzed by adopting the same perspective. The RMSE values of the IMQ for the three collocation types are illustrated in Figure 7c. Similar to the results obtained in Figure 7b, we also found the IMQ without the shape parameter for type C reached the order of 10

^{–}

^{8}. From the results, it is significant that the Gaussian RBF without the shape parameter for type C showed a high-accuracy performance.

#### 4.2. Application Example 2

^{−1}to 10

^{−6}as the shape parameter ranged from 0.5 to 5. However, the RMSE of the Gaussian RBF without a shape parameter in type C reached the order of 10

^{–11}. The MQ and IMQ RBFs were analyzed by adopting the same perspective. The RMSE of the MQ and IMQ RBFs for the three collocation types are illustrated in Figure 9b,c, respectively. Similar to the results shown in Figure 9a, we also found that the MQ and IMQ RBFs without the shape parameter for type C reached the order of 10

^{–}

^{9}and 10

^{–}

^{8}, respectively.

#### 4.3. Application Example 3

^{−2}to 10

^{−6}as the shape parameter ranged from 0.5 to 5. However, the RMSE of the simplified Gaussian, MQ, and IMQ RBFs (type C) without a shape parameter reached the order of 10

^{−8}, 10

^{−8}, and 10

^{−10}, respectively.

## 5. Conclusions

- (1)
- In this study, we demonstrated that the simplified RBFs, which consider many exterior fictitious sources outside the domain, can achieve accurate results to solve elliptic boundary value problems. The obtained results demonstrate that the simplified RBFs obtain a better accuracy than the original RBFs with the optimum shape parameter when solving elliptic boundary value problems.
- (2)
- Identification of the shape parameter is often very challenging and tedious in the original RBFs when solving partial differential equations. In this study, we proposed three simplified Gaussian, MQ, and IMQ RBFs without the shape parameter. The simplified RBFs have the advantages of a simple mathematical expression, high precision, and easy implementation.
- (3)
- With the consideration of many exterior fictitious sources outside the domain, we found that the radial distance is always greater than zero. The simplified Gaussian, MQ, and IMQ RBFs and their derivatives in the governing equation are always smooth and nonsingular.
- (4)
- Comparative analysis was conducted on the three different collocation types considering conventional uniform centers, randomly fictitious centers, and exterior fictitious sources. It was found that the exterior fictitious sources proposed in this study significantly improved the accuracy when solving problems.
- (5)
- Numerical examples, including elliptic BVPs in two and three dimensions, were carried out. The simplified radial basis function method with exterior fictitious sources can be applied to three-dimensional problems with ease and high accuracy.
- (6)
- In this study, we attempted to remove the shape parameter in conventional RBFs to solve partial differential equations. We achieved a promising result for three simplified Gaussian, MQ, and IMQ RBFs, especially for solving Laplace-type equations in two and three dimensions. Further studies to investigate the characteristics of the proposed method to solve different kinds of PDEs are suggested.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Location of the fictitious sources for the two-dimensional and three-dimensional domain. (

**a**) A two-dimensional domain: Type A. (

**b**) A two-dimensional domain: Type B. (

**c**) A two-dimensional domain: Type C. (

**d**) A three-dimensional domain: Type A. (

**e**) A three-dimensional domain: Type B. (

**f**) A three-dimensional domain: Type C.

**Figure 2.**The RMSE of the three RBFs with three different collocation types: (

**a**) Gaussian RBF, (

**b**) MQ RBF, and (

**c**) IMQ RBF.

**Figure 4.**Problem domain and location of the fictitious sources for example 2. (

**a**) Problem domain. (

**b**) Type A. (

**c**) Type B. (

**d**) Type C (blue and red circles denote the source and interior points, respectively).

**Figure 5.**RMSEs of three RBFs using three different collocation types: (

**a**) Gaussian RBF, (

**b**) MQ RBF, and (

**c**) IMQ RBF.

**Figure 7.**RMSEs of three RBFs with three different collocation types: (

**a**) Gaussian RBF, (

**b**) MQ RBF, and (

**c**) IMQ RBF.

**Figure 8.**Collocation points of the three types in the application example 2. (

**a**) Type A. (

**b**) Type B. (

**c**) Type C.

**Figure 9.**RMSEs of RBFs with three different collocation types: (

**a**) Gaussian RBF, (

**b**) MQ RBF, and (

**c**) IMQ RBF.

**Figure 10.**Problem domain and RMSEs of RBFs using three different collocation types: (

**a**) problem domain, (

**b**) Gaussian RBF, (

**c**) MQ RBF, and (

**d**) IMQ RBF.

Type of RBFs | Original RBFs | Simplified RBFs |
---|---|---|

Gaussian | ${\varphi}_{Gaussian}(r)={e}^{-{(\frac{r}{c})}^{2}}$ | ${\varphi}_{Gaussian\_S}(r)={e}^{-{(\frac{r}{R})}^{2}}$ |

Multiquadric (MQ) | ${\varphi}_{MQ}(r)=\sqrt{{r}^{2}+{c}^{2}}$ | ${\varphi}_{MQ\_S}\left(r\right)=r$ |

Inverse multiquadric (IMQ) | ${\varphi}_{IMQ}(r)=\frac{1}{\sqrt{{r}^{2}+{c}^{2}}}$ | ${\varphi}_{IMQ\_S}\left(r\right)=\frac{1}{r}$ |

RBF | RMSE | ||
---|---|---|---|

Type A | Type B | Type C ($\mathit{\eta}=3$) | |

Gaussian | 1.24 × 10^{–7} | 9.73 × 10^{–8} | 7.87 × 10^{–12} |

($c=1.75$) | ($c=2.0$) | ($c=1$) | |

(t = 5.84 s) | (t = 4.62 s) | (t = 8.11 s) | |

MQ | 1.42 × 10^{–7} | 1.46 × 10^{–7} | 4.35 × 10^{–13} |

($c=1.5$) | ($c=1.75$) | ($c=0$) | |

(t = 5.78 s) | (t = 5.75 s) | (t = 7.96 s) | |

IMQ | 1.47 × 10^{–7} | 8.46 × 10^{–8} | 6.37 × 10^{–12} |

($c=1.5$) | ($c=1.5$) | ($c=0$) | |

(t = 6.12 s) | (t = 6.28 s) | (t = 8.51 s) |

RBF | RMSE | ||
---|---|---|---|

Type A | Type B | Type C ($\mathit{\eta}=3$) | |

Gaussian | 2.45 × 10^{–8} | 1.33 × 10^{–8} | 9.50 × 10^{–13} |

($c=1.75$) | ($c=1.25$) | ($c=1$) | |

(t = 3.82 s) | (t = 7.02 s) | (t = 8.81 s) | |

MQ | 4.61 × 10^{–8} | 4.41 × 10^{–8} | 1.39 × 10^{–10} |

($c=2.25$) | ($c=1.75$) | ($c=0$) | |

(t = 3.80 s) | (t = 6.90 s) | (t = 8.77 s) | |

IMQ | 3.38 × 10^{–8} | 2.85 × 10^{–8} | 1.37 × 10^{–9} |

($c=2.5$) | ($c=2.0$) | ($c=0$) | |

(t = 3.80 s) | (t = 6.99 s) | (t = 8.83 s) |

RBF | RMSE | ||
---|---|---|---|

Type A | Type B | Type C ($\mathit{\eta}=3)$ | |

Gaussian | 1.18 × 10^{–6} | 1.61 × 10^{–6} | 2.76 × 10^{–11} |

($c=2.50$) | ($c=2.25$) | ($c=1$) | |

(t = 7.24 s) | (t = 9.47 s) | (t = 12.57 s) | |

MQ | 6.28 × 10^{–6} | 3.70 × 10^{–6} | 5.04 × 10^{–9} |

($c=1$) | ($c=1$) | ($c=0$) | |

(t = 7.28 s) | (t = 10.34 s) | (t = 13.01 s) | |

IMQ | 4.54 × 10^{–6} | 4.32 × 10^{–6} | 4.59 × 10^{–8} |

($c=1.25$) | ($c=1.25$) | ($c=0$) | |

(t = 7.24 s) | (t = 11.67 s) | (t = 12.67 s) |

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**MDPI and ACS Style**

Liu, C.-Y.; Ku, C.-Y.
A Simplified Radial Basis Function Method with Exterior Fictitious Sources for Elliptic Boundary Value Problems. *Mathematics* **2022**, *10*, 1622.
https://doi.org/10.3390/math10101622

**AMA Style**

Liu C-Y, Ku C-Y.
A Simplified Radial Basis Function Method with Exterior Fictitious Sources for Elliptic Boundary Value Problems. *Mathematics*. 2022; 10(10):1622.
https://doi.org/10.3390/math10101622

**Chicago/Turabian Style**

Liu, Chih-Yu, and Cheng-Yu Ku.
2022. "A Simplified Radial Basis Function Method with Exterior Fictitious Sources for Elliptic Boundary Value Problems" *Mathematics* 10, no. 10: 1622.
https://doi.org/10.3390/math10101622