##
**Numerical solution of Poisson’s equation using a combination of logarithmic and multiquadric radial basis function networks.**
*(English)*
Zbl 1235.65139

Summary: 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.

### MSC:

65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

PDFBibTeX
XMLCite

\textit{M. M. Mazarei} and \textit{A. Aminataei}, J. Appl. Math. 2012, Article ID 286391, 13 p. (2012; Zbl 1235.65139)

Full Text:
DOI

### References:

[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. · Zbl 0850.76048 · doi:10.1016/0898-1221(90)90271-K |

[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. |

[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. · Zbl 0883.65083 · doi:10.1016/S0096-3003(96)00109-9 |

[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. · Zbl 0955.65086 · doi:10.1016/S0898-1221(00)00071-7 |

[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. · Zbl 1047.76101 · doi:10.1016/S0893-6080(00)00095-2 |

[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. · Zbl 0999.65137 · doi:10.1016/S0898-1221(01)00297-8 |

[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. · Zbl 0999.65150 · doi:10.1016/S0898-1221(01)00300-5 |

[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. · Zbl 1024.65012 · doi:10.1016/S0307-904X(02)00101-4 |

[9] | L. Mai-Cao, “Solving time-dependent PDEs with a meshless IRBFN-based method,” in Proceedings of the International Workshop on Meshfree Methods, 2003. |

[10] | M. D. Buhmann, Radial Basis Functions: Theory and Implementations, vol. 12, Cambridge University Press, Cambridge, UK, 2003. · Zbl 1038.41001 · doi:10.1017/CBO9780511543241 |

[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. · Zbl 1077.41008 · doi:10.1016/j.mcm.2005.01.002 |

[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. |

[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. |

[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. · Zbl 1182.65174 · doi:10.1016/j.enganabound.2004.05.006 |

[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. |

[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. · Zbl 1153.65371 |

[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. · Zbl 1165.65401 · doi:10.1016/j.camwa.2008.07.026 |

[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. · Zbl 1165.65366 · doi:10.1016/j.mcm.2008.03.016 |

[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. · Zbl 1217.65161 · doi:10.1016/j.mcm.2010.10.002 |

[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. · Zbl 1269.45001 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.