/ / Article

Research Article | Open Access

Volume 2012 |Article ID 286391 | https://doi.org/10.1155/2012/286391

Mohammad Mehdi Mazarei, Azim Aminataei, "Numerical Solution of Poisson's Equation Using a Combination of Logarithmic and Multiquadric Radial Basis Function Networks", Journal of Applied Mathematics, vol. 2012, Article ID 286391, 13 pages, 2012. https://doi.org/10.1155/2012/286391

# Numerical Solution of Poisson's Equation Using a Combination of Logarithmic and Multiquadric Radial Basis Function Networks

Accepted18 Sep 2011
Published04 Dec 2011

#### Abstract

This paper presents numerical solution of elliptic partial differential equations (Poisson's equation) using a combination of logarithmic and multiquadric radial basis function networks. This method uses a special combination between logarithmic and multiquadric radial basis functions with a parameter . Further, the condition number which arises in the process is discussed, and a comparison is made between them with our earlier studies and previously known ones. It is shown that the system is stable.

#### 1. Introduction

The organization of the present paper is as follows. In Section 2, we give the theories of the new method. In Section 3, we provide some numerical experiments on the two-dimensional Poisson’s equation with the Dirichlet, the Neumann, and curved boundary conditions and we propose to examine the stability of the method and its behavior towards input variations. Finally, in Section 4, some conclusions are presented.

#### 2. The New Method of the Present Study

The form of a Poisson’s equation is as follows: in which is a known function. Also, Poisson’s equation can be in the following two Dirichlet and Neumann boundary conditions: where and are boundaries of the domain. Also, the vector is the outer unit normal to the boundaries and , are two known functions of . In MQ approximation scheme, we approximate the unknown function by an expression. In this study, we consider two-dimensional Poisson’s equation.

In this method, we use the following expression: The derivatives in the two-dimensional Cartesian coordinates are presented in the following:

In expression (2.4), the set of weights is to be found. In the present study, the closed form of approximation function (2.4) is first obtained from a set of training points and the derivative functions are then calculated directly by differentiation of such closed activity. The nonzero parameter protects of having zero values inside the logarithmic function. Also, we decrease the parameter to improve approximate solutions. In fact, this parameter controls the accuracy and the stability of the system. In each experiments, we evaluate the condition number of the system (condition number , that is the coefficient matrix of the system) and try to inspect the affect of the parameter on the stability of our system.

#### 3. Illustrative Numerical Experiments

In this section, we present three experiments, wherein their numerical solutions illustrate some advantages of the new method with high accuracy and show that, in this new way, the system is not ill conditioned.

##### 3.1. Stability of the Solution

A method is said to be stable when the obtained solution undergoes small variations as there are slight variations in inputs and parameters and when probable perturbations in parameters that are effective in equations and conditions prevailing them do not introduce, in comparison to the physical reality of the problem, any perturbations in what is returned. We propose here to compare the new method with other numerical methods (i.e., DRBFN and IRBFN methods) by offering experiments and examining the stability of the new method (Tables 1 and 3).

 Exact solution Approximate solution of the new method Error of the new method 0.3333 0.2 0.4070935392947847 0.4070935392947850 0.3333 0.4 0.4972251717238353 0.4972251717238351 0.3333 0.6 0.6073121961701566 0.6073121961701570 0.3333 0.8 0.7417727914665396 0.7417727914665393 0.6667 0.2 0.8143092188653853 0.8143092188653852 0.6667 0.4 0.9945995259174347 0.9945995259174348 0.6667 0.6 1.214806604220352 1.214806604220355 0.6667 0.8 1.483768137025928 1.483768137025920 1.0000 0.2 1.221402758160170 1.221402758160172 1.0000 0.4 1.491824697641270 1.491824697641277 1.0000 0.6 1.822118800390509 1.822118800390500 1.0000 0.8 2.225540928492468 2.225540928492466 1.3333 0.2 1.628496297454955 1.628496297454971 1.3333 0.4 1.989049869365105 1.989049869365101 1.3333 0.6 2.429430996560666 2.429430996560662 1.3333 0.8 2.967313719959008 2.967313719959003 1.6666 0.2 2.035589836749739 2.035589836749740 1.6666 0.4 2.486275041088941 2.486275041088940 1.6666 0.6 3.036743192730822 3.036743192730820 1.6666 0.8 3.709086511425547 3.709086511425548

Experiment 1. Consider the following two-dimensional Poisson’s equation: with the following Dirichlet boundary conditions on and : The exact solution is: .
We denote the root-mean-square error by the RMSE from the following relation: where is exact solution and is approximate solution at points . We have considered those 20 points that we had used for IRBFN method on the polar coordinates . As we have shown in earlier work , the parameter influences on accuracy, partially. We have shown that increasing the values of this parameter causes instability, and usually can not affect on accuracy adequately (see Table 4). Although, in that method, we could improve our results by focusing on other parameters such as substituting the scattered point places, in this new way, we can improve our solutions by changing the values of parameter without having instability. In contrast of MQ-RBFs, in this special combination, as parameter decreases, the accuracy increases. Also, since, for values greater than , the condition number is not near to unity and we will not have a stable system sufficiently, so we have considered at least . For instance, for this value of , we have , and, when we continue to decrease this parameter over and over, we get a better solution. For instance, in the best position while in IRBFN method on the polar coordinates  in the best position, we had . Also, in this new way, when reaches to about and smaller values, our accuracy and condition number are almost fixed. As we are decreasing parameter , the accuracy is going to be better (see Table 2). Further, when we decrease values quite enough, the accuracy and condition number almost do not change (see Table 2). Note that, for smaller values of , condition number is very near to unity (equals unity in double precision).

 Condition number RMSE 1.616697379470667 1.005685744274181 1.000056810457828 1.000000568099880 1.000000005680999 1.000000000056810 1.000000000000585 1.000000000000003 1.000000000000000 1.000000000000000 1.000000000000000
 Exact solution Approximate solution of IRBFN method on the polar coordinate Error of IRBFN method on the polar coordinate 0.3333 0.2 0.4070935392947847 0.407093539278427 0.3333 0.4 0.4972251717238353 0.497225171776793 0.3333 0.6 0.6073121961701566 0.607919812163175 0.3333 0.8 0.7417727914665396 0.741772791449410 0.6667 0.2 0.8143092188653853 0.814309218844426 0.6667 0.4 0.9945995259174347 0.994599525931543 0.6667 0.6 1.214806604220352 1.214806604225190 0.6667 0.8 1.483768137025928 1.483768136992937 1.0000 0.2 1.221402758160170 1.221402758117891 1.0000 0.4 1.491824697641270 1.491824697612953 1.0000 0.6 1.822118800390509 1.822118800390206 1.0000 0.8 2.225540928492468 2.225540928445109 1.3333 0.2 1.628496297454955 1.628496297401382 1.3333 0.4 1.989049869365105 1.989049869316768 1.3333 0.6 2.429430996560666 2.429430996510109 1.3333 0.8 2.967313719959008 2.967313719902712 1.6666 0.2 2.035589836749739 2.035711976964034 1.6666 0.4 2.486275041088941 2.486424223498127 1.6666 0.6 3.036743192730822 3.036925404550813 1.6666 0.8 3.709086511425547 3.709309065459067
 Condition number RMSE 0.005 1.471481785169896 0.01 2.037248475218950 0.5 4.678768168755817 1.0 7.977675155476113 1.5 10.67180878932467 2.0 16.72051422331341 2.5 17.54142375428882 3.0 22.68488788104916 3.5 94.83497808866327 4.0 687.3099630448350 4.5 2251.857925768735

Experiment 2. Consider the following two-dimensional Poisson’s equation: with the following Neumann and Dirichlet boundary conditions on and : The exact solution is , where and are, respectively, 2 and 3. This experiment was solved using MQ approximation scheme by Kansa . The author used a total of 30 points, including 12 scattered data points in the interior and 18 along the boundary. The reported results showed that the norm of error is . Later, Mai-Duy and Tran-Cong  used IRBFN method and got a greater accuracy. They reported the norm of error is for this experiment. In this study, we have used those same points (see Figure 1) and have achieved better accuracy in comparison with those two previous works. By using present approach, the norm of error that we have gotten in the best position is (). The results are shown in Tables 5 and 6. There are the same properties and results about this new way that we have explained in Experiment 1 (more accuracy and stability).

 Exact solution Approximate solution of the new method 0.0 0.0 1.000000000000000 1.000000001520331 0.0 .25 2.117000016612675 2.116999962177082 0.0 .5 4.481689070338065 4.481689131011527 0.0 .75 9.487735836358526 9.487735908103340 0.0 1 20.08553692318767 20.08553702346013 1 0.0 7.389056098930650 7.389056124466515 1 .25 15.64263188418817 15.64263190711622 1 .5 33.11545195869231 33.11545243577679 1 .75 70.10541234668786 70.10541228133020 1 1 148.4131591025766 148.4131603446419 .2 0.0 1.491824697641270 1.491824709011273 .2 1 29.96410004739701 29.96410038594428 .4 0.0 2.225540928492468 2.225540978673106 .4 1 44.70118449330082 44.70118489448246 .6 0.0 3.320116922736547 3.320116658933115 .6 1 66.68633104092514 66.68633190025831 .8 0.0 4.953032424395115 4.953032677031149 .8 1 99.48431564193381 99.48431532815737 .05 .05 1.284025416687741 1.284025760755220 .13 .26 2.829217014351560 2.829217082663191 .46 .16 4.055199966844675 4.055199961753327 .31 .42 6.553504862191149 6.553504821832245 .07 .58 6.553504862191149 6.553504889746486 .12 .73 11.35888208000146 11.35888287993308 .42 .91 35.51659315162847 35.51659237035928 .51 .57 15.33288701990720 15.33288705221003 .68 .82 45.60420832084874 45.60420881755932 .84 .37 16.28101980178843 16.28101996810631 .97 .68 53.51703422749116 53.51703410219755 .17 .93 22.87397954244081 22.87397949274820
 Condition number RMSE 15.34291952699858 4.566804572945806 2.134108059285889 1.082012712119965 1.010904486607965 1.003900496513122 1.001720933454105 1.001551378516199 1.001534906804984 1.001537639564036 1.002332833767664

Experiment 3. Consider the following two-dimensional Poisson’s equation in the elliptical region: The great diameter of the ellipse is , and small diameter is . The boundary condition is on all of boundary points. The equation of ellipse is Since the above ellipse is a symmetrical region, then we have solved this equation on the first quarter. The Dirichlet and the Neumann boundary conditions are The analytical solution is .
The results have been computed for and . We have used 28 points that 17 of them are boundary points and 11 are interior points (see Figure 2) which were selected at random. In this new way, we have achieved a better accuracy in comparison with IRBFN method in the polar coordinates  (see Tables 7, 8, and 9). When we decrease parameter , RMSE decreases too, and so we have better accurate solution. Also, the condition number closes to unity more and more. For (), it almost equals to unity (see Table 8).
We have shown in , in using of MQ-RBFs on the polar coordinate, when we have used 28 data points and we have been increasing the width parameter , the accuracy of our solution has been increasing a little, though our system has been going to nonstability (see Table 9). In contrast, in this new way, when we are decreasing the parameter , the accuracy of our solution is increasing too. Also, the condition number of our system is decreasing and closes to unity (see Table 8).
Here, we would like to emphasize that this experiment had been also solved by  and , and the norm of error is in the DRBFN method , wherein the norm of errors in the DRBFN and the IRBFN methods are and , respectively.

 Exact solution Approximate solution of the new method Error of the new method 0.0 0.5 38.87195121951220 38.87195121952383 0.0 1.5 37.65243902439024 37.65243902440016 0.0 3.5 31.55487804878049 31.55487804879181 0.0 5.5 20.57926829268293 20.57926829269648 0.0 7.5 4.72560975609756 4.72560975611862 2.0 0.0 37.46341463414634 37.46341463417008 4.0 0.0 32.78048780487805 32.78048780491504 6.0 0.0 24.97560975609756 24.97560975615097 8.0 0.0 14.04878048780488 14.04878048786474 10.0 0.0 0.00000000000000 0.00000000005427 2.0 1.6 35.90243902439024 35.90243902440453 4.0 1.6 31.21951219512195 31.21951219515048 6.0 1.6 23.41463414634146 23.41463414638886 8.0 1.6 12.48780487804878 12.48780487810293 2.0 4.0 27.70731707317073 27.70731707318550 4.0 4.0 23.02439024390244 23.02439024392700 6.0 4.0 15.21951219512195 15.21951219516124 8.0 4.0 4.292682926829268 4.29268292687660 2.0 5.6 18.34146341463414 18.34146341465220 4.0 5.6 13.65853658536585 13.65853658539350 6.0 5.6 5.85365853658536 5.85365853662464 9.798 1.6 −0.00031375609756 −0.00031375604674 8.660 4.0 0.00171707317073 0.00171707321643 7.141 5.6 0.00238790243902 0.00238790247911 2.0 7.838 0.00350975609756 0.00350975612416 4.0 7.332 0.00108292682927 0.00108292685914 6.0 6.4 0.00000000000000 0.00000000003483 8.0 4.8 0.00000000000000 0.00000000004390
 Condition number RMSE 2.817474760824544 1.590396300485768 1.253627364959679 1.062100283211657 1.050234416869035 1.030320686741992 1.015443720207797 1.005553497698158 1.003855850797432 1.002467354513745 1.000616708637105
 r Condition number RMSE 0.001 1.433292009119422 0.01 1.833381468640938 0.5 1.641721037366621 1.0 2.368533981239234 1.5 6.742734977317889 2.0 7.268116615890860 2.5 19.12284221789496 3.0 56.72352660443333 3.5 83.76935856474893 4.0 101.1762326758897 4.5 164.9216307945755

#### 4. Conclusion

In the present paper, we have introduced a new way for numerical solution of Poisson’s partial differential equation by a special combination between logarithmic and MQ-RBFs. We have showed that by this new method it does not need to control the parameter (the width parameter) all times for preventing inaccuracy of solutions or increasing the value of condition number and having an ill-conditioned system. In this new way that is enough to consider the value of the parameter smaller than . In the aforesaid experiments, the accuracy is better than those before results obtained by [1, 3, 5, 16, 17] and the condition number of the systems is equal to unity. So we have some complete stable systems and more accurate solutions.

It should be noted that the computations associated with the experiments discussed above were performed by using Maple 13 on a PC, CPU 2.4 GHz.

#### Acknowledgments

This research paper has been financially supported by the office of vice chancellor for research of Islamic Azad University, Bushehr Branch, for the first author. The authors are very grateful to both reviewers for carefully reading the paper and for their comments and suggestions which have improved the paper. Also, authors acknowledge the editor professor Roberto Natalini, for managing the review process for this paper.

1. E. J. Kansa, “Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics. II. Solutions to parabolic, hyperbolic and elliptic partial differential equations,” Computers & Mathematics with Applications, vol. 19, no. 8-9, pp. 147–161, 1990.
2. G. J. Moridis and E. J. Kansa, “The Laplace transform multiquadrics method: a highly accurate scheme for the numerical solution of linear partial differential equations,” Journal of Applied Science and Computations, vol. 1, no. 2, pp. 375–407, 1994. View at: Google Scholar
3. M. Sharan, E. J. Kansa, and S. Gupta, “Application of the multiquadric method for numerical solution of elliptic partial differential equations,” Applied Mathematics and Computation, vol. 10, pp. 175–302, 1997. View at: Google Scholar | Zentralblatt MATH
4. E. J. Kansa and Y. C. Hon, “Circumventing the ill-conditioning problem with multiquadric radial basis functions: applications to elliptic partial differential equations,” Computers & Mathematics with Applications, vol. 39, no. 7-8, pp. 123–137, 2000.
5. N. Mai-Duy and T. Tran-Cong, “Numerical solution of differential equations using multiquadric radial basis function networks,” Neural Networks, vol. 14, no. 2, pp. 185–199, 2001. View at: Publisher Site | Google Scholar
6. A. I. Fedoseyev, M. J. Friedman, and E. J. Kansa, “Improved multiquadric method for elliptic partial differential equations via PDE collocation on the boundary,” Computers & Mathematics with Applications, vol. 43, no. 3–5, p. 439, 2002.
7. E. A. Galperin and E. J. Kansa, “Application of global optimization and radial basis functions to numerical solutions of weakly singular Volterra integral equations,” Computers & Mathematics with Applications, vol. 43, no. 3–5, pp. 491–499, 2002.
8. N. Mai-Duy and T. Tran-Cong, “Approximation of function and its derivatives using radial basis function networks,” Applied Mathematical Modelling, vol. 27, no. 3, pp. 197–220, 2003.
9. L. Mai-Cao, “Solving time-dependent PDEs with a meshless IRBFN-based method,” in Proceedings of the International Workshop on Meshfree Methods, 2003. View at: Google Scholar
10. M. D. Buhmann, Radial Basis Functions: Theory and Implementations, vol. 12, Cambridge University Press, Cambridge, UK, 2003. View at: Publisher Site
11. L. Ling and E. J. Kansa, “Preconditioning for radial basis functions with domain decomposition methods,” Mathematical and Computer Modelling, vol. 40, no. 13, pp. 1413–1427, 2004.
12. A. Aminataei and M. M. Mazarei, “Numerical solution of elliptic partial differential equations using direct and indirect radial basis function networks,” European Journal of Scientific Research, vol. 2, no. 2, pp. 2–11, 2005. View at: Google Scholar
13. A. Aminataei and M. Sharan, “Using multiquadric method in the numerical solution of ODEs with a singularity point and PDEs in one and two-dimensions,” European Journal of Scientific Research, vol. 10, no. 2, pp. 19–45, 2005. View at: Google Scholar
14. D. Brown, L. Ling, E. J. Kansa, and J. Levesley, “On approximate cardinal preconditioning methods for solving PDEs with radial basis functions,” Engineering Analysis with Boundary Elements, vol. 29, pp. 343–353, 2005.
15. J. A. Munoz-Gomez, P. Gonzalez-Casanova, and G. Rodriguez-Gomez, “Domain decomposition by radial basis functions for time-dependent partial differential equations, advances in computer science and technology,” in Proceedings of the IASTED International Conference, pp. 105–109, 2006. View at: Google Scholar
16. M. M. Mazarei and A. Aminataei, “Numerical solution of elliptic PDEs using radial basis function networks and comparison between RBFN and Adomian method,” Far East Journal of Applied Mathematics, vol. 32, no. 1, pp. 113–126, 2008. View at: Google Scholar | Zentralblatt MATH
17. A. Aminataei and M. M. Mazarei, “Numerical solution of Poisson's equation using radial basis function networks on the polar coordinate,” Computers & Mathematics with Applications, vol. 56, no. 11, pp. 2887–2895, 2008.
18. S. K. Vanani and A. Aminataei, “Multiquadric approximation scheme on the numerical solution of delay differential systems of neutral type,” Mathematical and Computer Modelling, vol. 49, no. 1-2, pp. 234–241, 2009.
19. S. Karimi Vanani and A. Aminataei, “Numerical solution of differential algebraic equations using a multiquadric approximation scheme,” Mathematical and Computer Modelling, vol. 53, no. 5-6, pp. 659–666, 2011.
20. M. A. Jafari and A. Aminataei, “Application of RBFs collocation method for solving integral equations,” Journal of Interdiscplinary Mathematics, vol. 14, no. 1, pp. 57–66, 2011. View at: Google Scholar