## Journal of Applied Mathematics

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

#### 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 $r$. 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.

#### Article information

Source
J. Appl. Math., Volume 2012 (2012), Article ID 286391, 13 pages.

Dates
First available in Project Euclid: 15 March 2012

https://projecteuclid.org/euclid.jam/1331817621

Digital Object Identifier
doi:10.1155/2012/286391

Mathematical Reviews number (MathSciNet)
MR2854978

Zentralblatt MATH identifier
1235.65139

#### Citation

Mazarei, Mohammad Mehdi; Aminataei, Azim. Numerical Solution of Poisson's Equation Using a Combination of Logarithmic and Multiquadric Radial Basis Function Networks. J. Appl. Math. 2012 (2012), Article ID 286391, 13 pages. doi:10.1155/2012/286391. https://projecteuclid.org/euclid.jam/1331817621

#### References

• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• L. Mai-Cao, “Solving time-dependent PDEs with a meshless IRBFN-based method,” in Proceedings of the International Workshop on Meshfree Methods, 2003.
• M. D. Buhmann, Radial Basis Functions: Theory and Implementations, vol. 12, Cambridge University Press, Cambridge, UK, 2003.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.
• 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.