Error Estimation and Assessment of an Approximation in a Wavelet Collocation Method

DOI: 10.4236/ajcm.2013.32019   PDF   HTML     4,612 Downloads   7,133 Views   Citations


This article describes how to assess an approximation in a wavelet collocation method which minimizes the sum of squares of residuals. In a research project several different types of differential equations were approximated with this method. A lot of parameters must be adjusted in the discussed method here. For example one parameter is the number of collocation points. In this article we show how we can detect whether this parameter is too small and how we can assess the error sum of squares of an approximation. In an example we see a correlation between the error sum of squares and a criterion to assess the approximation.

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M. Schuchmann and M. Rasguljajew, "Error Estimation and Assessment of an Approximation in a Wavelet Collocation Method," American Journal of Computational Mathematics, Vol. 3 No. 2, 2013, pp. 114-120. doi: 10.4236/ajcm.2013.32019.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] O. V. Vasilyev and C. Bowman, “Second-Generation Wavelet Collocation Method for the Solution of Partial Differential Equations,” Journal of Computational Physics, Vol. 165, No. 2, 2000, pp. 660-693. doi:10.1006/jcph.2000.6638
[2] S. Bertoluzza, “Adaptive Wavelet Collocation Method for the Solution of Burgers Equation,” Transport Theory and Statistical Physics, Vol. 25, No. 3-5, 2006, pp. 339-352. doi:10.1080/00411459608220705
[3] T. S. Carlson, J. Dockery and J. Lund, “A Sinc-Collocation Method for Initial Value Problems,” Mathematics of Computation, Vol. 66, No. 217, 1997, pp. 215-235. doi:10.1090/S0025-5718-97-00789-8
[4] K. Abdella, “Numerical Solution of Two-Point Boundary Value Problems Using Sinc Interpolation,” Proceedings of the American Conference on Applied Mathematics, Applied Mathematics in Electrical and Computer Engineering, 2012, pp. 157-162.
[5] A. Nurmuhammada, M. Muhammada, M. Moria and M. Sugiharab, “Double Exponential Transformation in the Sinc-Collocation Method for a Boundary Value Problem with Fourth-Order Ordinary Differential Equation,” Journal of Computational and Applied Mathematics, Vol. 162, No. 2, 2005, pp. 32-50. doi:10.1016/
[6] C. Blatter, “Wavelets—Eine Einführung,” 2nd Edition, Vieweg, Wiesbaden, 2003.
[7] G. Strang, “Wavelets and Dilation Equations: A Brief Introduction,” SIAM Review, Vol. 31, No. 4, 1989, pp. 614-627. doi:10.1137/1031128
[8] Z. Shi, D. J. Kouri, G. W. Wie and D. K. Hoffman, “Generalized Symmetric Interpolating Wavelets,” Computer Physics Communications, Vol. 119, No. 2-3, 1999, pp. 194-218. doi:10.1016/S0010-4655(99)00185-X
[9] D. L. Donoho, “Interpolating Wavelet Transforms,” Technical Report 408, Department of Statistics, Stanford University, Stanford, 1992.
[10] E. Hairer and G. Wanner, “Solving Ordinary Differential Equations I: Nonstiff Problems,” 2nd Edition, Springer, Berlin, 1993.

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