## 1. Introduction

Recently, the meshless approach has raised extensive attention due to its computational efficiency as well as simple collocation scheme. Many varieties of the radial basis functions (RBFs) have been developed for dealing with partial differential equations (PDEs) [

1,

2,

3]. Most popular RBFs, such as the Gaussian [

4,

5,

6], multiquadric (MQ) [

7,

8], and inverse multiquadric (IMQ) [

9,

10,

11], require the shape parameter. Among them, the Kansa method [

12] is recognized as one of the most popular domain-type meshfree approaches for solving PDEs. The MQ RBF adopted by the Kansa method becomes the well-known RBF, which has been successfully adopted for solving numerous engineering problems [

13,

14]. Despite the success of the Kansa method, limitations regarding to the accuracy affecting by the shape parameter still remain. The MQ RBF depends on the shape parameter that plays an important role for remaining the RBF as a smooth and non-singular function for solving PDEs. Attempts regarding for identifying proper value for the shape parameter of the MQ RBF have been widely studied, such as the LOOCV optimization technique [

15,

16,

17]. The question of finding the optimal shape parameter in the MQ RBF, however, is still very challenging.

In this study, we propose radial polynomials (RPs) rooted in the MQ RBF for solving PDEs. Formulated from the binomial series using the Taylor series expansion of the MQ RBF, the new global RPs include only even order radial terms. The proposed RPs may be regarded as the equivalent expression of the MQ RBF in series form. Not only are the RPs infinitely smooth and differentiable in nature, but the proposed RPs do not require any extra shape parameters. Therefore, the challenging task for finding the optimal shape parameter in the Kansa method is avoided. Several numerical implementations, including two- and three-dimensional problems, are conducted to verify the accuracy and robustness of the proposed RPs. The structure of this article is organized as follows: In

Section 2, formulation of the radial polynomial basis function is presented. To verify the proposed RPs, we conduct a convergence analysis in

Section 3.

Section 4 is devoted to present several numerical examples in two and three dimensions. The discussion of this paper is addressed in

Section 5. Conclusions are given finally.

## 2. Formulation of the Radial Polynomials

Considering a region,

$\Omega $, with the boundary,

$\partial \Omega $, the governing equation for the three-dimensional PDE can be expressed as follows.

in which

$\Delta $ represents Laplace operator,

$\mathbf{x}=(x,y,z)$,

$u(\mathbf{x})$ is the unknown,

$D$,

$E$,

$F$,

$G$ and

$H$ are given functions.

$\Omega $ is a bounded domain with boundary

$\partial {\Omega}^{D}$ and

$\partial {\Omega}^{N}$.

$\partial {\Omega}^{D}$ denotes boundary subjected to Dirichlet data,

$\partial {\Omega}^{N}$ denotes boundary subjected to Neumann data,

$g(\mathbf{x})$ and

$f(\mathbf{x})$ represent given boundary data. The meshless method using the MQ RBF is often named the Kansa method, where the RBFs are directly implemented for the approximation of the solution of partial differential equations. We may express the unknown by the RBF as follows.

where

${r}_{j}$ is the radial distance,

${r}_{j}=\left|\mathbf{x}-{\mathbf{s}}_{j}\right|$,

$\phi ({r}_{j})$ represents the RBF which is the distance of

$\mathbf{x}$ and

${\mathbf{s}}_{j}$,

${\mathbf{s}}_{j}$ is the center,

$\mathbf{x}$ denotes an arbitrary point inside the domain,

${\lambda}_{j}$ is the coefficient to be solved and

${M}_{c}$ is the number of the center points. The MQ RBF may be expressed as follows.

With the introduction of the shape parameter, the MQ RBF becomes a smooth and non-singular function. Because the Kansa method is a domain-type method, it has to discretize the governing equation inside the domain using the MQ RBF. We may insert the above equation into Equation (1). After obtaining the MQ RBF derivatives, we may obtain the following equation in two-dimensions.

The above equation demonstrates that the derivatives of the MQ basis function may become singular at the center point (

${r}_{j}=0$) if the shape parameter is zero. It is obvious that the MQ RBF is infinitely differentiable depending on the shape parameter. To avoid the singularity, the shape parameter must not be equal to zero. In this study, we propose RPs based on the MQ RBF without the shape parameter. For the mathematical formulation of the RPs, we may start from the MQ RBF. Equation (5) can be rewritten as follows.

Using the binomial series from the Taylor series of Equation (7), we have

where

$\left(\begin{array}{c}\alpha \\ k\end{array}\right):=\frac{\alpha (\alpha -1)(\alpha -2)\cdots (\alpha -k+1)}{k!}$ and

$\alpha =1/2$.

Using the finite terms,

${M}_{n}$, to approximate the solution, we may express the MQ RBF in series form as follows.

where

${M}_{n}$ is the order of the radial polynomials. In this study, we propose a novel meshless method to approximate the solution in terms of the RPs as follows.

where

${M}_{c}$ represents the center point number. The above equation proves that the MQ RBF can be expressed as a radial polynomial with only even order terms. Equation (8) can be regarded as the equivalent series form of the MQ RBF. Inserting Equation (8) into Equation (10), we have

Combining the constants in the above equation, we obtain

in which

${b}_{j,k}$ are the coefficients to be solved. Using Equation (12) for the discretization of Equation (1), we may obtain the following equation:

where

${L}_{1c}=4{k}^{2}$,

${L}_{2c}=(D(x-{x}_{j})+E(y-{y}_{j}))$ and

${L}_{1c}=4{k}^{2}+2k$,

${L}_{2c}=(D(x-{x}_{j})+E(y-{y}_{j})+F(z-{z}_{j}))$ are in two and three dimensions, respectively. To determine the unknown coefficients, we apply the approximate solution with the boundary data at collocation points to satisfy the governing equation. We may get the system of simultaneous equations.

where

$\mathbf{b}$ is the unknown coefficient with the size of

$N\times 1$ to be evaluated,

$\mathbf{R}$ is the known function with the size of

$M\times 1$,

$\mathbf{A}$ is an

$M\times N$ matrix where

$M={M}_{i}+{M}_{b}$ and

$N={M}_{c}\times {M}_{n}$. The above equation can be written as follows:

In the preceding equations,

${\mathbf{A}}_{\mathrm{I}}$ represents the

${M}_{i}\times N$ submatrix from the inner collocation points,

${\mathbf{A}}_{\mathrm{B}}$ represents the

${M}_{b}\times N$ submatrix from the boundary collocation points,

${\mathbf{R}}_{\mathrm{I}}$ is the vector of function values at the inner points which is a

${M}_{i}\times 1$ vector,

${\mathbf{R}}_{\mathrm{B}}$ is the data at the boundary points which is an

${M}_{b}\times 1$ vector,

${M}_{b}$ is the boundary point number,

${M}_{i}$ is the inner point number. The root mean square error (RMSE) is adopted to evaluate the accuracy which is defined by

in which

${M}_{m}$ represents the number of the measuring points with uniform distribution;

$u({x}_{i})$ and

$\widehat{u}({x}_{i})$ are the exact and approximate solutions at the

${i}^{th}$ collocation point, respectively.

## 3. Accuracy and Convergence Analysis

We first investigate a Laplacian problem in two dimensions enclosed by an irregular domain. The governing equation is

The star–like object boundary in two dimensions can be expressed in the following form:

The exact solution of Equation (17) is designated as

To verify the accuracy and convergence, we conduct a series of testing cases for the radial polynomial terms in which all cases adopt the same configurations of the boundary, center and inner points as shown in

Figure 1. In the analysis,

${M}_{b}$,

${M}_{i}$ and

${M}_{c}$ are 1208, 151, and 151, respectively. The number of the RPs terms,

${M}_{n}$, needs to be given for the proposed method. As shown in

Figure 2, for the RPs, it is found that the RMSE decreases with the increase in the number of RPs terms in which solutions with high accuracy may be found with the radial polynomial terms from 6 to 12.

On the other hand, other testing cases using the Kansa method for considering different shape parameters are conducted.

Figure 2 shows different shape parameters versus the RMSE. The optimal shape parameter is found within a narrow range of 0.5 to 1. We may also observe that the shape parameter is very sensitive to the accuracy in the Kansa method, such that attempts regarding for identifying proper values for the shape parameter of the Kansa method may be important. It is apparent that the minimums of the RMSE for the Kansa method and the RPs are in the order of

$1{0}^{-9}$ and

$1{0}^{-12}$, respectively.

In addition, to investigate the accuracy, another convergence analysis for investigating the boundary and inner point number is carried out.

Table 1 shows the comparison of this study with the Kansa method. We find that very high accurate results may be obtained using the proposed RPs.

## 5. Discussion

This study presents a collocation method using RPs which is regarded as the equivalent expression of the MQ RBF in series form for PDEs. The conception of the new global RPs includes only even order radial terms formulated from the binomial series using the Taylor series expansion of the MQ RBF. The discussions for this study are as follows.

The MQ RBF adopted by the Kansa method becomes one of the most successful RBFs for solving numerous problems. With the introduction of the shape parameter, the MQ RBF becomes a smooth and non-singular function. Even though the MQ RBF and its derivatives are smooth and global infinitely differentiable, the discretization of the governing equation may become singular while $c=0$ at ${r}_{j}=0$. It is obvious the shape parameter plays a role for shifting from the singularity while the center point is coincided with the inner point. However, the near singular effects still remain. This may explain that the accuracy in the Kansa method is strongly affected by the shape parameter.

To deal with the issue, we adopt the RPs as the basis function in which the proposed RPs are the equivalent expression of the MQ RBF in series form. It is advantageous that the proposed RPs and their derivatives are infinitely smooth and differentiable in nature without using the shape parameter. Because RPs are a non-singular series function, there are no near singular or singular effects at all. Accordingly, the method may obtain more accurate solutions than the Kansa method in our numerical implementations. In addition, accurate results can be directly obtained without using the tedious procedure for finding the optimal shape parameter.

Even though the shape parameter is not required in the proposed method, the radial polynomial terms have to be decided in advance. From the numerical implementations, solutions with high accuracy in the order of $1{0}^{-8}$ may be found with radial polynomial terms from 6 to 12. The radial polynomial terms are selected to be 9 in our numerical examples. It demonstrates that the radial polynomial terms are considerably less significant to the accuracy than the shape parameter. In the numerical examples, it is found that satisfactory solutions could be obtained while the terms of the RPs are within the range of 6 to 12.