×

Numerical solution of differential algebraic equations using a multiquadric approximation scheme. (English) Zbl 1217.65161

Summary: The objective of this paper is to solve differential algebraic equations using a multiquadric approximation scheme. Therefore, we present the notation and basic definitions of the Hessenberg forms of the differential algebraic equations. In addition, we present the properties of the proposed multiquadric approximation scheme and its advantages, which include using data points in arbitrary locations with arbitrary ordering. Moreover, error estimation and the run time of the method are also considered. Finally some experiments were performed to illustrate the high accuracy and efficiency of the proposed method, even when the data points are scattered and have a closed metric.

MSC:

65L80 Numerical methods for differential-algebraic equations

Software:

COLNEW; RODAS
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brenan, K. E.; Campbell, S. L.; Petzold, L. R., Numerical Solution of Initial Value Problems in Differential Algebraic Equations (1989), Elsevier: Elsevier New York · Zbl 0699.65057
[2] Hairer, E.; Norsett, S. P.; Wanner, G., Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems (1992), Springer: Springer New York
[3] Petzold, L. R., Numerical solution of differential-algebraic equations, Adv. Numer. Anal., 4 (1995) · Zbl 0843.65049
[4] Gear, C. W.; Petzold, L. R., ODE systems for the solution of differential-algebraic systems, SIAM J. Numer. Anal., 21, 716-728 (1984) · Zbl 0557.65053
[5] Ascher, U. M.; Petzold, L. R., Computer Methods for Ordinary Differential Equations and Differential Algebraic Equations (1998), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia · Zbl 0908.65055
[6] Celik, E., On the numerical solution of differential algebraic equation with index-2, Appl. Math. Comput., 156, 541-550 (2004) · Zbl 1061.65074
[7] Ascher, U. M.; Spiter, R. J., Collocation software for boundary-value differential algebraic equations, SIAM J. Sci. Comput., 15, 938-952 (1994) · Zbl 0804.65080
[8] Ascher, U. M.; Petzold, L. R., Projected implicit Runge-Kutta methods for differential algebraic equations, SIAM J. Numer. Anal., 28, 1097-1120 (1991) · Zbl 0732.65067
[9] Hardy, R. L., Multiquadric equations of topography and other irregular surfaces, J. Geophys. Res., 76, 1905 (1971)
[10] Hardy, R. L., Theory and applications of the multiquadric bi-harmonic method: 20 years of discovery, Comput. Math. Appl., 19, 8-9, 163 (1990) · Zbl 0692.65003
[11] Franke, R., Scattered data interpolation: tests of some methods, Math. Comp., 38, 181 (1971) · Zbl 0476.65005
[12] Ambrosetti, A.; Prodi, G., A Primer of Nonlinear Analysis (1993), Cambridge University Press · Zbl 0781.47046
[13] Zerroukut, M.; Power, H.; Chen, C. S., A numerical method for heat transfer problems using collocation and radial basis functions, Internat. J. Numer. Methods Engrg., 42, 1263 (1998) · Zbl 0907.65095
[14] Mai-Duy, N.; Tran-Cong, T., Numerical solution of differential equations using multiquadric radial basis function networks, Neural Netw., 14, 185 (2001)
[15] Aminataei, A.; Mazarei, M. M., Numerical solution of elliptic partial differential equations using direct and indirect radial basis function networks, Eur. J. Sci. Res., 2, 2, 5 (2005)
[16] Aminataei, A.; Mazarei, M. M., Numerical solution of Poisson’s equation using radial basis function networks on the polar coordinate, Comput. Math. Appl., 11, 56, 2887-2895 (2008) · Zbl 1165.65401
[17] Micchelli, C. A., Interpolation of scattered data: distance matrices and conditionally positive definite functions, Constr. Approx., 2, 11 (1986) · Zbl 0625.41005
[18] Madych, W. R.; Nelson, S. A., Multivariable interpolation and conditionally positive definite functions II, Math. Comp., 54, 211 (1990) · Zbl 0859.41004
[19] Kansa, E. J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics—I. Surface approximations and partial derivative estimates, Comput. Math. Appl., 19, 8-9, 127 (1990) · Zbl 0692.76003
[20] Kansa, E. J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics—II. Solutions to hyperbolic, parabolic and elliptic partial differential equations, Comput. Math. Appl., 19, 8-9, 147 (1990) · Zbl 0850.76048
[21] Aminataei, A.; Sharan, M., Using multiquadric method in the numerical solution of ODEs with a singularity point and PDEs in one and two dimensions, Eur. J. Sci. Res., 10, 2, 19 (2005)
[22] Madych, W. R., Miscellaneous error bounds for multiquadric and related interpolants, Comput. Math. Appl., 24, 12, 121 (1992) · Zbl 0766.41003
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.