# Multiquadrics without the Shape Parameter for Solving Partial Differential Equations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Collocation Method of the Multiquadric Radial Basis Function

**x**is a point inside the domain, ${s}_{j}$ is the center point, M is the total number of center points, and ${\lambda}_{j}$ is the unknown coefficient to be determined.

**A**is an $M\times N$ matrix, $\alpha $ refers to the coefficients with a size of $N\times 1$ to be evaluated, $b$ refers to the $M\times 1$ given functions. If $M={M}_{i}+{M}_{b}$, where ${M}_{b}$ is the number of boundary points and ${M}_{i}$ is the number of interior points, then Equation (10) is expressed as:

## 3. Accuracy and Convergence Analysis

^{−7}and 10

^{−5}, respectively. The RMSE and the MAE, using the proposed approach, are in the order of 10

^{−8}and 10

^{−7}, respectively. Our approach acquires more accurate results than the Kansa method, even when the optimum shape parameter is adopted. The proposed method utilizing the exterior source collocation of case D provides a promising solution for the two-dimensional Laplace problem. Results illustrate that the locations of fictitious sources are not sensitive to the accuracy if the fictitious irregular circular boundary is adopted.

^{−5}and 10

^{−7}, respectively. The MAE and the RMSE, utilizing our method, are in the order of 10

^{−7}and 10

^{−9}, respectively. The proposed IMQ RBF adopting the exterior source collocation yields more accurate solutions than the conventional IMQ RBF.

## 4. Numerical Examples

#### 4.1. A Two-Dimensional Wave Problem

^{0}to 10

^{−4}, while the shape parameter ranged from 0 to 5. The MAE of the proposed collocation method fluctuated between 10

^{−5}and 10

^{−7}, while the dilation parameter ranged from 0 to 5.

#### 4.2. A Two-Dimensional Groundwater Flow Problem

^{−1}to 10

^{−4}, while the shape parameter ranged from 0 to 5. However, the MAE of our approach remained 10

^{−6}, while the dilation parameter ranged from 0 to 5.

#### 4.3. An Unsaturated Flow Problem

## 5. Conclusions

- We considered the center point as the fictitious source collocated outside the domain. The radial distance between the interior point and the source point was, therefore, always greater than zero. As a result, the MQ and IMQ RBFs and their derivatives in the governing equation were smooth and globally infinitely differentiable. Boundary value problems are solved by the proposed collocation method using MQ and IMQ RBFs without the shape parameter.
- The fictitious boundary shape surrounded by the source points and the locations of the source points may affect the accuracy. We conducted a sensitivity analysis to examine the accuracy. In the sensitivity analysis, we investigated the source points placed at different positions outside the domain. Moreover, four irregular fictitious boundary shapes surrounded by the source points were studied. Results illustrate that the locations of fictitious sources are not sensitive to the accuracy if the suggested fictitious irregular circular boundary is adopted.
- Several examples were conducted to verify the robustness and accuracy of our method. The results demonstrated that the proposed method using MQ and IMQ RBFs acquires more accurate results than the RBFCM, even with the optimum shape parameter. Additionally, it was found that the locations of fictitious sources were not sensitive to the accuracy.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Hon, Y.C.; Sarler, B.; Yun, D.F. Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface. Eng. Anal. Bound. Elem.
**2015**, 57, 2–8. [Google Scholar] [CrossRef] - Kansa, E.J.; Hon, Y.C. Circumventing the ill-conditioning problem with multi- quadric radial basis functions: Applications to elliptic partial differential equations. Comput. Math. Appl.
**2000**, 39, 123–137. [Google Scholar] [CrossRef] [Green Version] - Kansa, E.J. Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics. I Surface approximations and partial derivative estimates. Comput. Math. Appl.
**1990**, 19, 127–145. [Google Scholar] [CrossRef] [Green Version] - Mahdavi, A.; Chi, S.W.; Zhu, H. A gradient reproducing kernel collocation method for high order differential equations. Comput. Mech.
**2019**, 64, 1421–1454. [Google Scholar] [CrossRef] - Ku, C.Y.; Liu, C.Y.; Xiao, J.E.; Huang, W.P.; Su, Y. A spacetime collocation Trefftz method for solving the inverse heat conduction problem. Adv. Mech. Eng.
**2019**, 11, 1–11. [Google Scholar] [CrossRef] - Mahdavi, A.; Chi, S.W.; Kamali, N. Harmonic-Enriched Reproducing Kernel Approximation for Highly Oscillatory Differential Equations. J. Eng. Mech.
**2020**, 146, 04020014. [Google Scholar] [CrossRef] - Ku, C.Y.; Xiao, J.E. A Collocation Method Using Radial Polynomials for Solving Partial Differential Equations. Symmetry
**2020**, 12, 1419. [Google Scholar] [CrossRef] - Cheng, A.H.-D. Multiquadric and its shape parameter—A numerical investigation of error estimate, condition number, andround-off error by arbitrary precision computation. Eng. Anal. Bound. Elem.
**2012**, 36, 220–239. [Google Scholar] [CrossRef] - Gutmann, H.M. A radial basis function method for global optimization. J. Glob. Optim.
**2001**, 19, 201–227. [Google Scholar] [CrossRef] - Hardy, R.L. Multiquadric equations of topography and other irregular surfaces. J. Geophys. Res.
**1971**, 76, 1905–1915. [Google Scholar] [CrossRef] - Kansa, E.J. Multiquadrics—A scattered data approximation scheme with applications to computational fluid-dynamics. II Solutions to parabolic, hyperbolic and elliptic partial-differential equations. Comput. Math. Appl.
**1990**, 19, 147–161. [Google Scholar] [CrossRef] [Green Version] - Chen, W.; Tanaka, M. A meshless, integration-free, and boundary-only RBF technique. Comput. Math. Appl.
**2002**, 43, 379–391. [Google Scholar] [CrossRef] [Green Version] - Ring, M.; Eskofier, B.M. An approximation of the Gaussian RBF kernel for efficient classification with SVMs. Pattern Recognit. Lett.
**2016**, 84, 107–113. [Google Scholar] [CrossRef] - Chen, C.S.; Karageorghis, A.; Dou, F. A novel RBF collocation method using fictitious centres. Appl. Math. Lett.
**2020**, 101, 106069. [Google Scholar] [CrossRef] - Soleymani, F.; Barfeie, M.; Haghani, F.K. Inverse multi-quadric RBF for computing the weights of FD method: Application to American options. Commun. Nonlinear Sci. Numer. Simul.
**2018**, 64, 74–88. [Google Scholar] [CrossRef] - Keller, W.; Borkowski, A. Thin plate spline interpolation. J. Geod.
**2019**, 93, 1251–1269. [Google Scholar] [CrossRef] - Fasshauer, G.E.; Zhang, J.G. On choosing “optimal” shape parameters for RBF approximation. Numer. Algorithms
**2007**, 45, 345–368. [Google Scholar] [CrossRef] - Biazar, J.; Hosami, M. Selection of an interval for variable shape parameter in approximation by radial basis functions. Adv. Numer. Anal.
**2016**, 2016, 1–11. [Google Scholar] [CrossRef] [Green Version] - Chen, W.; Hong, Y.; Lin, J. The sample solution approach for determination of the optimal shape parameter in the Multiquadric function of the Kansa method. Comput. Math. Appl.
**2018**, 75, 2942–2954. [Google Scholar] [CrossRef] - Ng, Y.L.; Ng, K.C.; Sheu, T.W.H. A new higher-order RBF-FD scheme with optimal variable shape parameter for partial differential equation. Numer. Heat Transf. Part B Fundam.
**2019**, 75, 289–311. [Google Scholar] [CrossRef] - Franke, R. Recent advances in the approximation of surfaces from scattered data. In Topics in Multivariate Approximation; Academic Press: Cambridge, MA, USA, 1987; pp. 79–98. [Google Scholar]
- Chen, C.-S.; Karageorghis, A.; Li, Y. On choosing the location of the sources in the MFS. Numer. Algorithms
**2016**, 72, 107–130. [Google Scholar] [CrossRef] - Li, X. On solving boundary value problems of modified Helmholtz equations by plane wave functions. J. Comput. Appl. Math.
**2006**, 195, 66–82. [Google Scholar] [CrossRef] [Green Version] - Strack, O.D.L.; Ausk, B.K. A formulation for vertically integrated groundwater flow in a stratified coastal aquifer. Water Resour. Res.
**2015**, 51, 6756–6775. [Google Scholar] [CrossRef] - Ku, C.Y.; Liu, C.Y.; Su, Y.; Xiao, J.E. Modeling of transient flow in unsaturated geomaterials for rainfall-induced landslides using a novel spacetime collocation method. Geofluids
**2018**, 2018, 1–16. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**The collocation scheme for the conventional radial basis function collocation method (RBFCM).

**Figure 3.**The exterior source collocation scheme with different positions of the source points: (

**a**) case A; (

**b**) case B; (

**c**) case C; (

**d**) case D.

**Figure 4.**The computed maximum absolute error (MAE) for the Kansa method and the proposed multiquadric radial basis function (MQ RBF) without the shape parameter: (

**a**) case A; (

**b**) case B; (

**c**) case C; (

**d**) case D.

**Figure 5.**The computed MAE for the conventional inverse multiquadric (IMQ) RBF and the proposed IMQ RBF without the shape parameter: (

**a**) case C; (

**b**) case D.

**Figure 8.**The computed MAE of the MQ RBF for the conventional RBFCM and the proposed method without the shape parameter.

**Figure 9.**The computed MAE of the IMQ RBF for the conventional RBFCM and the proposed method without the shape parameter.

**Figure 12.**The computed MAE of the MQ RBF for the conventional RBFCM and the proposed method without the shape parameter. (

**a**) ${D}_{p}=0.5$. (

**b**) ${D}_{p}=10$.

**Figure 13.**Convergence analysis of the IMQ RBF for the conventional RBFCM and the proposed method without the shape parameter. (

**a**) ${D}_{p}=0.5$. (

**b**) ${D}_{p}=10$.

**Figure 16.**The computed MAE of the MQ RBF for the conventional RBFCM and the proposed method without the shape parameter.

**Figure 17.**Convergence analysis of the IMQ RBF for the conventional RBFCM and the proposed method without the shape parameter.

**Table 1.**A comparison of the results between the Kansa method and our approach without the shape parameter.

The Kansa Method | The Proposed Method (MQ RBF) | ||||
---|---|---|---|---|---|

With the Optimum Shape Parameter | Without the Shape Parameter | ||||

Case A | Case B | Case C | Case D | ||

(c = 1.65) | $(\mathit{\eta}=4.70)$ | $(\mathit{\eta}=4.75)$ | $(\mathit{\eta}=0.15)$ | $(\mathit{\eta}=1.20)$ | |

MAE | 2.06 × 10^{−5} | 8.58 × 10^{−6} | 2.98 × 10^{−6} | 8.11 × 10^{−7} | 5.15 × 10^{−7} |

RMSE | 6.53 × 10^{−7} | 4.20 × 10^{−7} | 1.12 × 10^{−7} | 5.14 × 10^{−8} | 2.48 × 10^{−8} |

**Table 2.**A comparisons of the results between the conventional inverse multiquadric (IMQ) RBF and the proposed IMQ RBF without the shape parameter.

The Conventional IMQ RBF | The Proposed Method (IMQ RBF) | ||
---|---|---|---|

With the Optimum Shape Parameter | Without the Shape Parameter | ||

Case C | Case D | ||

(c = 2.40) | $(\mathit{\eta}=1.70)$ | $(\mathit{\eta}=4.70)$ | |

MAE | 1.82 × 10^{−5} | 3.26 × 10^{−7} | 4.35 × 10^{−7} |

RMSE | 3.26 × 10^{−7} | 9.63 × 10^{−9} | 2.19 × 10^{−8} |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Ku, C.-Y.; Liu, C.-Y.; Xiao, J.-E.; Hsu, S.-M.
Multiquadrics without the Shape Parameter for Solving Partial Differential Equations. *Symmetry* **2020**, *12*, 1813.
https://doi.org/10.3390/sym12111813

**AMA Style**

Ku C-Y, Liu C-Y, Xiao J-E, Hsu S-M.
Multiquadrics without the Shape Parameter for Solving Partial Differential Equations. *Symmetry*. 2020; 12(11):1813.
https://doi.org/10.3390/sym12111813

**Chicago/Turabian Style**

Ku, Cheng-Yu, Chih-Yu Liu, Jing-En Xiao, and Shih-Meng Hsu.
2020. "Multiquadrics without the Shape Parameter for Solving Partial Differential Equations" *Symmetry* 12, no. 11: 1813.
https://doi.org/10.3390/sym12111813