Numerical solution of Poisson’s equation using radial basis function networks on the polar coordinate.(English)Zbl 1165.65401

Summary: We introduce a variant of direct and indirect radial basis function networks (DRBFNs and IRBFNs) for the numerical solution of Poisson’s equation. We use transformation from Cartesian coordinates to polar ones and use DRBFN and IRBFN methods on the basis of a multiquadric approximation scheme. We have experienced that the result shows better accuracy than previously known ones. Also, our new way of solution does not influence the condition number.

MSC:

 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
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 [1] Mai-Duy, N.; Tran-Cong, T., Numerical solution of differential equations using multiquadric radial basis function networks, Neural networks, 14, 185-199, (2001) [2] A.E. Tarwater, A parameter study of Hardy’s multiquadric method for scattered interpolation, Technical Report UCRL-563670, Lawrence Livermore National Laboratory, 1985 [3] Hardy, R.L., Theory and applications of the multiquadric-biharmonic method, Computers and mathematics with applications, 19, 163-208, (1990) · Zbl 0692.65003 [4] Franke, R., Scattered data interpolation: test of some methods, Mathematics of compute., 38, 181-200, (1982) · Zbl 0476.65005 [5] Carlson, R.E.; Foley, T.A., The parameter $$r^2$$ in multiquadric interpolation, Computers and mathematics with applications, 21, 29-42, (1991) · Zbl 0725.65009 [6] Kansa, E.J., Multiquadrics — A scattered data approximation scheme with applications to computational fluid dynamics-I surface approximations and partial derivative estimates, Computers and mathematics with applications, 19, 8-9, 127-145, (1990) · Zbl 0692.76003 [7] Moridis, G.J.; Kansa, E.J., The Laplace transform multiquadrics method: A highly accurate scheme for the numerical solution of linear partial differential equations, Journal of applied science and computation, 1, 2, 375-407, (1994) [8] Zerroukat, M.; Power, H.; Chen, C.S., A numerical method for heat transfer problems using collocation and radial basis functions, International journal for numerical methods in engineering, 42, 1263-1278, (1998) · Zbl 0907.65095 [9] Sharan, M.; Kansa, E.J.; Gupta, S., Application of the multiquadric method for numerical solution of elliptic partial differential equation, Applied mathematics and computation, 10, 175-302, (1997) · Zbl 0883.65083
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