Fuzhang Wang, Congcong Li and Kehong Zheng

(1. School of Mathematics and Statistics, Xuzhou University of Technology, Xuzhou 221018, Jiangsu, China; 2. School of Mathematical Sciences, Huaibei Normal University, Huaibei 235000, Anhui, China; 3. Department of Mathematics, University of Macau, Macau 999078, China; 4. College of Water Conservancy and Ecological Engineering, Nanchang Institute of Technology, Nanchang 330099, China)

Abstract: The radial basis functions (RBFs) play an important role in the numerical simulation processes of partial differential equations. Since the radial basis functions are meshless algorithms, its approximation is easy to implement and mathematically simple. In this paper, the commonly-used multiquadric RBF, conical RBF, and Gaussian RBF were applied to solve boundary value problems which are governed by partial differential equations with variable coefficients. Numerical results were provided to show the good performance of the three RBFs as numerical tools for a wide range of problems. It is shown that the conical RBF numerical results were more stable than the other two radial basis functions. From the comparison of three commonly-used RBFs, one may obtain the best numerical solutions for boundary value problems.

Keywords: radial basis functions; partial differential equations; variable coefficient

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 functionΦ:Rt→Rin caseΦ(X)=φ(‖X‖) withφ: [0,∞)→RandX∈Rt, andtis the space dimensionality.

Interpolation with RBFs takes the following form:

(1)

The aim is to find the unknown coefficientsαifrom the system

s(Xi)=f(Xi)

(2)

Qα=f

(3)

Qij=φ‖Xi-Xj‖

(4)

with

f=[f(X1),...,f(XN)]T

(5)

α=[α1,α2,...,αN]T

(6)

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

φ(r)=(r2+c2)n/2

(7)

wherenis an odd integer.

For MQ,

φ(r)=e-c2r2

(8)

For GS,

φ(r)=rn

(9)

2 RBFs for PDEs with Variable Coefficients

Consider a linear PDE with variable coefficient operators on a plane domainΩ?R2.

Lu(X)=f(X),X=(x,y)∈Ω

(10)

Bu(X)=g(X),X=(x,y)∈?Ω

(11)

where

(12)

is an operator witha(X),b(X),c(X),d(X),e(X),l(X) as known coefficients.f(X) andg(X) are given functions.Bis a linear boundary differential operator.

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

(13)

whereφ(·) is a prescribed RBF.

Table 1 Differentiation of RBFs

where

(14)

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

(15)

(16)

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

(17)

(18)

with

b(Xi)(4c2(yi-yj)2-2c)+…

c(Xi)4c2(xi-xj)(yi-yj)-

2cd(Xi)(xi-xj)-

2ce(Xi)(yi-yj)+l(Xi)]

(19)

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 parametercwas chosen first, after which the quasi-optimalcwas 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 andn=11 for CT.

The root mean square error (RMSE) considered is

3.1 Examples

Case1The equation is given by

?2u+λ2u=2sin(μx)cos(μy)+4μxcos(μx)cos(μy)

(20)

g(x,y)=x2sin(μx)cos(μy)

(21)

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

Fig.1 Influence of the shape parameter c on RMSE for Case 1

Fig. 2 Condition numbers for Case 1

Here, the quasi-optimal errors for the GS was obtained by the shape parameterc=0.96 with corresponding RMSE=1.36×10-4and the quasi-optimal errors for the MQ was achieved with the shape parameterc=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 parameterc>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 parametersnare 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.

Fig.3 Convergence curves of the three RBFs with different point number parameter n for Case 1

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.

Fig.4 Condition numbers of the three RBFs with different point number parameter n for Case 1

Case2Consider a heat transfer problem with variable coefficients depicted by Eq.(10). The variable coefficients are

(22)

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

u(x,y)=x2+y2-5xy

(23)

respectively. The corresponding source term is

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

(24)

A relatively large point number parametern=20 was initially fixed, i.e., interior pointsni=361 and boundary pointsnt=76.The curves of RMSE versus the shape parametercare shown in Fig.5 with step interval 0.05.

Fig.5 RMSE for Case 2

Here, the quasi-optimal error RMSE=1.71×10-3for the GS was obtained by the shape parameterc=0.95 and the quasi-optimal RMSE=8.19×10-6for the MQ was achieved with the corresponding shape parameterc=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 parametersnare 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.

Case3In this case, the problem

Δu-(x2y)u=f(x,y)

(25)

g(x,y)=sin(y2+x)-cos(y-x2)

(26)

was considered with the exact solution

u(x,y)=sin(y2+x)-cos(y-x2)

(27)

This is the case thata=1,b=1,c=0,d=0,e=0 andl=-x2yin Eq.(10). The calculation domainΩ=Ω1∪Ω2was formed by (Fig.7)

Ω1=[0,2]×[0,2]

(28)

Ω2={(x,y)|x=cosθ+2,y=sinθ+2,

(29)

Fig.6 Convergence curves of the three RBFs with different point number parameter n for Case 2

Fig.7 The irregular domain for Case 3

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 parameterc=1.58 and RMSE=0.2357 for the MQ with the corresponding quasi-optimal shape parameterc=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.

Fig.8 RMSE for Case 3

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.