0 Introduction

It is well-known that most of the problems in biological, physical, electrical science and other scientific areas can be described by partial differential equations (PDEs). Analytical solutions for PDEs are almost inaccessible for relatively complex problems. Numerical approximate solutions of PDEs are an alternative. The traditional mesh-based finite element method^[1-3] is stable in early scientific and engineering branches. However, the mesh-generation is a drawback for a large scope of practical demanding problems.

In this study, the emphasis is on the domain-type meshless methods with RBF collocation technique involved. The basic theory of the RBF was derived from the work of Hardy^[4], and it was connected with a topology on quadric surface. This type of functions was later applied to simulate PDEs. Edward Kansa in 1990 successfully used a globally supported interpolation, the multiquadric (MQ) RBF, to obtain the approximate solution^[5-6]. At present, the Gaussian (GS) RBF is another commonly-used choice in computation^[7-10]. It should be pointed out that shape parameters for the MQ and GS should be carefully investigated.

Recently, Biazar and Hosami^[11] investigated the shape parameter encountered in RBF approximation. Urleb and Vrankar^[12] present a procedure for searching for an optimal shape parameter for the RBF. Besides, the conical type (CT) radial basis function is a parameter-free type^[13]. Recently, Reutskiy^[14] used the CT RBF to simulate the heat conduction problems. Since the CT RBF is parameter-free, it was compared with MQ and GS in this work.

Because of its good performance, the above-mentioned three radial basis function has been triumphantly applied to solve different PDEs. The numerical solution of three-dimensional (3D) PDEs were considered by Song et al.^[15] A finite RBF-based subdomain collocation method was proposed by Chu et al.^[16] Numerical and theoretical investigation on the collocation based RBFs have been formulated in Ref. [17]. The RBF was used for solving economic problems^[18].

In this work, the above-mentioned three radial basis function, MQ, GS, and CT, were applied to handle PDEs with variable coefficients without iterations. Numerical examples were provided to verify the validity and stability of the three radial basis functions. The parameters in MQ and GS as well as the collocation point numbers were also considered. In Section 1, a description is given for the RBFs. In Section 2, the RBF algorithm is applied to PDEs with variable coefficients. In Section 3, comparative numerical results are illustrated for solving 2D PDEs with variable coefficients. Some conclusions are presented in Section 4.

1 RBFs

RBF is a function Φ: R^t→R in case Φ(X)=φ(‖X‖) with φ: [0, ∞)→R and X∈R^t, and t is the space dimensionality.

Interpolation with RBFs takes the following form:

$ S(X)=\sum\limits_{i=1}^{N} \alpha_{i} \varphi\left(\left\|X-X_{i}\right\|\right) $

(1)

The aim is to find the unknown coefficients α_i from the system

$ s\left(X_{i}\right)=f\left(X_{i}\right) $

(2)

where X={X_i}_i=1^N is a given points set in R^t and f(X_i) is a prescribed function. Eq.(1) and (2) have matrix form

$ \boldsymbol{Q} \alpha=\boldsymbol{f} $

(3)

$ Q_{i j}=\varphi\left\|X_{i}-X_{j}\right\| $

(4)

with

$ \boldsymbol{f}=\left[f\left(X_{1}\right), \ldots, f\left(X_{N}\right)\right]^{\mathrm{T}} $

(5)

$ \boldsymbol{\alpha}=\left[\alpha_{1}, \alpha_{2}, \ldots, \alpha_{N}\right]^{\mathrm{T}} $

(6)

where T represents the transpose of vector or matrix. Here, the emphasis is on the three commonly-used RBFs, i.e.,

$ \varphi(r)=\left(r^{2}+c^{2}\right)^{n / 2} $

(7)

where n is an odd integer.

For MQ,

$ \varphi(r)=\mathrm{e}^{-c^{2} r^{2}} $

(8)

For GS,

$ \varphi(r)=r^{n} $

(9)

2 RBFs for PDEs with Variable Coefficients

Consider a linear PDE with variable coefficient operators on a plane domain Ω⊂R².

$ L u(X)=f(X), X=(x, y) \in \varOmega $

(10)

$ B u(X)=g(X), X=(x, y) \in \partial \varOmega $

(11)

where

$ L=a \frac{\partial^{2}}{\partial x^{2}}+b \frac{\partial^{2}}{\partial y^{2}}+c \frac{\partial^{2}}{\partial x \partial y}+d \frac{\partial}{\partial x}+e \frac{\partial}{\partial x}+l $

(12)

is an operator with a(X), b(X), c(X), d(X), e(X), l(X) as known coefficients. f(X) and g(X) are given functions. B is a linear boundary differential operator.

For the boundary value problem (10)-(11), the basic theory of RBF is given by

$ \hat{u}\left(X_{i}\right)=\sum\limits_{j=1}^{n} \xi_{j} \varphi\left(r_{i j}\right) $

(13)

where φ(·) is a prescribed RBF.

Choose n distinct collocation points {(x_j, y_j)}_j=1ⁿ from Ω =(Ω∪∂Ω) with {(x_j, y_j)}_j=1^n_i in Ω and {(x_j, y_j)}ⁿ_j=ni+1 on ∂Ω. To show the numerical solution process, the differentiation of the GS φ(r)=e^-c2r2, MQ φ(r)=(r²+c²)^n/2 and CT φ(r)=rⁿ are shown in Table 1.

where

$ r_{j}=\left\|X-X_{j}\right\|=\sqrt{\left(x-x_{j}\right)^{2}+\left(y-y_{j}\right)^{2}} $

(14)

Then, substitute Eq.(13) into Eqs.(10)-(11). Eqs.(15) and (16) are obtained:

$ \sum\limits_{j=1}^{n} \xi_{j} L \varphi\left(r_{i j}\right)=f_{i} $

(15)

$ \sum\limits_{j=1}^{n} \xi_{j} B \varphi\left(r_{i j}\right)=g_{i} $

(16)

To be specific, substitute differentiation of the three RBFs into Eqs.(15) and (16), which leads to

$ \sum\limits_{j=1}^{n} \xi_{j} L \mathrm{e}^{-c^{2} r_{i j}^{2}}=f_{i} $

(17)

$ \sum\limits_{j=1}^{n} \xi_{j} B \mathrm{e}^{-c^{2} r_{i j}^{2}}=g_{i} $

(18)

with

$ \begin{aligned} L \mathrm{e}^{-c^{2} r_{i j}^{2}}=& \mathrm{e}^{-c^{2} r_{i j}^{2}}\left[a\left(X_{i}\right)\left(4 c^{2}\left(x_{i}-x_{j}\right)^{2}-2 c\right)+\right.\\ & b\left(X_{i}\right)\left(4 c^{2}\left(y_{i}-y_{j}\right)^{2}-2 c\right)+\cdots \\ & c\left(X_{i}\right) 4 c^{2}\left(x_{i}-x_{j}\right)\left(y_{i}-y_{j}\right)-\\ & 2 c d\left(X_{i}\right)\left(x_{i}-x_{j}\right)-\\ &\left.2 c e\left(X_{i}\right)\left(y_{i}-y_{j}\right)+l\left(X_{i}\right)\right] \end{aligned} $

(19)

for collocating points {(x_i, y_i)}_i=1ⁿ. Once {ξ_j}_j=1ⁿ are known through solving Eq.(20) and (21), the approximation of u can be obtained. In this work, the numerical algorithm was realized by using MATLAB without iteration.

3 Numerical Examples

Three benchmark numerical examples were considered in this section. Unless otherwise stated, the solution domain is distributed with evenly interpolation points. The shape parameter c was chosen first, after which the quasi-optimal c was chosen. Numerical results of parameter related MQ and GS were compared with the parameter-free CT for all examples. Besides, n=20 was chosen for MQ and n=11 for CT.

The root mean square error (RMSE) considered is

$ \operatorname{RMSE}=\sqrt{\frac{1}{T} \sum\limits_{j=1}^{T}\left|\frac{u\left(X_{j}\right)-\hat{u}\left(X_{j}\right)}{u\left(X_{j}\right)}\right|^{2}} $

where u(X_j) is the exact solution, û(X_j) is the numerical solution, and T denotes the total interested point number.

3.1 Examples

Case 1 The equation is given by

$ \nabla^{2} u+\lambda^{2} u=2 \sin (\mu x) \cos (\mu y)+4 \mu x \cos (\mu x) \cos (\mu y) $

(20)

in a square domain. This is the case that a=1, b=1, c=0, d=0, e=0 and l=λ² and in Eq.(10). $\lambda = \mu \sqrt 2 $, and μ=1 were chosen in this example. The exact solution is

$ g(x, y)=x^{2} \sin (\mu x) \cos (\mu y) $

(21)

To seek for the quasi-optimal shape parameter c of GS and MQ, a relatively large point number parameter n=20 was initially fixed. The corresponding interior points were generated by MATLAB codes "0.05:1/n: 0.95" with interior point number n_i=361 and boundary point on each side was n-1 with boundary point number n_t=4n-4=76. Fig. 1 shows the RMSE curves versus the shape parameter c with step interval 0.05. The corresponding condition numbers of the two RBFs are shown in Fig. 2.

Here, the quasi-optimal errors for the GS was obtained by the shape parameter c=0.96 with corresponding RMSE=1.36×10^-4 and the quasi-optimal errors for the MQ was achieved with the shape parameter c=0.71 with corresponding RMSE=5.53×10^-9. This phenomenon reveals that the MQ performed better than the GS for this case. At the same time, it should be noted that the condition numbers for the two RBFs were almost the same for shape parameter c>0.5. This results show that the MQ has better characters than the GS.

For comparison, convergence curves of the three RBFs with different collocation point number parameters n are shown in Fig. 3. The numerical results of MQ performed better than the GS and CT. But the convergence of CT was more stable than MQ and GS. This phenomenon may be partially attributed to the parameter-free character of the CT case.

The corresponding condition numbers of the three RBFs are shown in Fig. 4, which shows that the larger condition numbers in Fig. 4 corresponded to the more accurate numerical solutions in Fig. 3. Also, it was found that the larger condition numbers of the MQ and GS corresponded to the better numerical solutions, i.e., the smaller RMSEs. The smaller condition numbers of the CT corresponded to the less accurate numerical solutions, i.e., the bigger RMSEs. Since the results of condition numbers were similar, this issue was ignored in the following two cases.

Case 2 Consider a heat transfer problem with variable coefficients depicted by Eq.(10). The variable coefficients are

$ \left\{\begin{array}{l} a=(2 x+y+2)^{2} \\ b=(x+2 y+2)^{2} \\ c=2\left(x-y^{2}\right)^{2} \\ d=10 x+2 y+8 \\ e=2 x+10 y+8 \end{array}\right. $

(22)

The Neumann boundary condition was prescribed on y=0 and the rest Dirichlet boundary was chosen properly from the exact solution

$ u(x, y)=x^{2}+y^{2}-5 x y $

(23)

respectively. The corresponding source term is

$ f(x, y)=-18(x+y)+16 $

(24)

A relatively large point number parameter n=20 was initially fixed, i.e., interior points n_i=361 and boundary points n_t=76. The curves of RMSE versus the shape parameter c are shown in Fig. 5 with step interval 0.05.

Here, the quasi-optimal error RMSE=1.71×10^-3 for the GS was obtained by the shape parameter c=0.95 and the quasi-optimal RMSE=8.19×10^-6 for the MQ was achieved with the corresponding shape parameter c=7.15. These results reveal that the quasi-optimal error for the MQ is more accurate than the GS one for this case.

For comparison, convergence curves of the three RBFs with different collocation point number parameters n are shown in Fig. 6, from which it can be found that the MQ performed better than the other two RBFs. However, the convergence of CT was more stable than the MQ and GS. This phenomenon is similar with Case 1.

Case 3 In this case, the problem

$ \Delta u-\left(x^{2} y\right) u=f(x, y) $

(25)

$ g(x, y)=\sin \left(y^{2}+x\right)-\cos \left(y-x^{2}\right) $

(26)

was considered with the exact solution

$ u(x, y)=\sin \left(y^{2}+x\right)-\cos \left(y-x^{2}\right) $

(27)

This is the case that a=1, b=1, c=0, d=0, e=0 and l=-x²y in Eq.(10). The calculation domain Ω=Ω₁∪Ω₂ was formed by (Fig. 7)

$ \varOmega_{1}=[0, 2] \times[0, 2] $

(28)

$ \begin{aligned} \varOmega_{2}=&\{(x, y) \mid x=\cos \theta+2, y=\sin \theta+2, \\ &\left.-\frac{{\rm{ \mathit{ π} }}}{2} \leqslant \theta \leqslant \frac{{\rm{ \mathit{ π} }}}{2}\right\} \end{aligned} $

(29)

After fixing the 105 points distributed evenly on the boundary domain and 574 interior points spread evenly in the domain, the numerical results are presented with respect to different shape parameters in Fig. 8 for GS and MQ. Here, it is found that RMSE=0.0558 for GS with the quasi-optimal shape parameter c=1.58 and RMSE=0.2357 for the MQ with the corresponding quasi-optimal shape parameter c=0.64, respectively. Both GS and MQ performed not as well as the previous two cases. This may contribute to the irregular computational domain or governing equation. Besides, RMSE=0.0259 was computed for the CT case. Since the variation of collocation point number parameter is difficult to choose, the convergence curves of the three RBFs with different collocation point number parameter are not shown.

3.2 Discussions

It can be seen that the three RBFs can be directly applied to solve PDEs with variable coefficients. It is easy to implement and stable with respect to regular or irregular geometry. Regularization techniques are referred to for further investigations^[19-20]. Although this paper considers only three cases, the method can be similarly extended to the other cases with minor revision.

4 Conclusions

The main purpose of this work is to apply the three commonly-used RBFs, i.e., GS, MQ, and CT, for solving PDEs with variable coefficients. Numerical results were provided to show the good performance of the three radial basis functions as numerical tools for a wide range of problems. For some problems the CT one performed not well, but the convergence curves of the errors performed more stable than the other two RBFs. For irregular cases, the three RBFs performed almost the same.

However, a more theoretical and systematic study on the 3D and time-dependent problems is needed to further investigate all RBFs.