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Modification of Multiple Knot B-Spline Wavelet for Solving (Partially) Dirichlet Boundary Value Problem

Published online by Cambridge University Press:  03 June 2015

Fatemeh Pourakbari  and
Ali Tavakoli
Fatemeh Pourakbari*
Affiliation:
Department of Mathematics, Vali-e-Asr University of Rafsanjan, Iran
Ali Tavakoli*
Affiliation:
Department of Mathematics, Vali-e-Asr University of Rafsanjan, Iran
*
URL: http://tavakoli@vru.ac.ir, Email: f.pourakbari@stu.vru.ac.ir
Corresponding author. Email: tavakoli@mail.vru.ac.ir
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Abstract

A construction of multiple knot B-spline wavelets has been given in [C. K. Chui and E. Quak, Wavelet on a bounded interval, In: D. Braess and L. L. Schumaker, editors. Numerical methods of approximation theory. Basel: Birkhauser Verlag; (1992), pp. 57-76]. In this work, we first modify these wavelets to solve the elliptic (partially) Dirichlet boundary value problems by Galerkin and Petrov Galerkin methods. We generalize this construction to two dimensional case by Tensor product space. In addition, the solution of the system discretized by Galerkin method with modified multiple knot B-spline wavelets is discussed. We also consider a nonlinear partial differential equation for unsteady flows in an open channel called Saint-Venant. Since the solving of this problem by some methods such as finite difference and finite element produce unsuitable approximations specially in the ends of channel, it is solved by multiple knot B-spline wavelet method that yields a very well approximation. Finally, some numerical examples are given to support our theoretical results.

Keywords

65T6035L6035L04Galerkin methodsemi-orthogonalB-spline waveletmulti-resolution analysistensor producthyperbolic partial differential equationSaint-Venant equations
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 4 , Issue 6 , December 2012 , pp. 799 - 820
Copyright
Copyright © Global-Science Press 2012

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References

[1]Aghazadeh, N. and Mesgarani, H., Solving non-linear fredholm integro-differential equation, World Appl. Sci. J., 7 (2009), pp. 50–56.Google Scholar
[2]Akan, A. O., Open Channel Hydraulics, Elsevier, Oxford UK, 2006.Google Scholar
[3]Amein, M. and Chu, H. L., Implicit numerical modeling of unsteady flows, ASCE J. Hydr. Eng., 101 (1974), pp. 717–731.Google Scholar
[4]Audusse, E. and Bristeau, M. O., Finite-volume solvers for a multilayer Saint-Venant system, Int. J. Appl. Math. Comput. Sci., 17(3) (2007), pp. 311–320.Google Scholar
[5]Bittner, K., Biorthogonal spline wavelets on the interval, in Wavelets and splines: Athens2005, Mod. Methods Math., pp. 93–104, Nashboro press, Brentwood, TN, 2006.Google Scholar
[6]Chaudhry, M. H., Open-Channel Flow, 2nd ed., Springer, 2008.Google Scholar
[7]Chen, X., Xiang, J., Li, B. and He, Z., A study of multiscale wavelet-based elements for adaptive finite element analysis, Adv. Eng. Soft., 41 (2010), pp. 196–205.CrossRefGoogle Scholar
[8]Chow, V. T., Open Channel Hydraulics, McGraw-Hill, New York, 1959.Google Scholar
[9]Chui, C. K. and Quak, E., Wavelet on a bounded interval, in: Braess, D. and Schumaker, L. L., editors, Numerical Methods Of Approximation Theory, Basel: Birkhauser Verlag; (1992), pp. 57–76.Google Scholar
[10]Chui, C. K. and Wang, J. Z., A cardinal spline approach to wavelets, Proc. Amer. Math. Soc., 113 (1991), pp. 785–793.Google Scholar
[11]Ciarlet, P. G., The Finite Element Methods for Elliptic Problems, North Holland, Amesterdam, (1978).Google Scholar
[12]Daubechies, I., Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math., 41 (1988), pp. 909–996.CrossRefGoogle Scholar
[13]Hicks, F. E., Finite Element Modeling of Open Channel Flow, University of Alberta, Ph.D thesis, 1990.Google Scholar
[14]Granatowicz, J. and Szymkiewicz, R., Comparison of efficency of the solution of the Saint-Venant equations by finite element method and finite difference method, Arch. Hydr., 3-4 (1989), pp. 199–210.Google Scholar
[15]Szymkiewicz, R., Finite element method for the solution of the Saint-Venant equation in the open channel network, J. Hydr., 122 (1991), pp. 275–287.Google Scholar
[16]Tavakoli, A. and Jafari, S., New preconditioners for elliptic boundary value problems in multi-resolution space, Sci. Bull., Series A. Appl. Math. Phys., to appear.Google Scholar
[17]Tavakoli, A. and Zarmehi, F., Adaptive finite element methods for solving Saint-Venant equations, Scientia Iranica, Transactions B: Mech. Eng., 18 (2011), pp. 1321–1326.Google Scholar
[18]Urban, K., Wavelet Methods for Elliptic Partial Differential Equation, Golub, G. H., Stuart, A. M. and Suli, E., editors, University of Ulm, 2009.Google Scholar
[19]Zarmehi, F., Tavakoli, A. and Rahimpour, M., On numerical stabilization in the solution of Saint-Venant equations using finite element method Comput. Math. Appl., 62 (2011), pp. 1957–1968.Google Scholar