Modification of Multiple Knot $B$-Spline Wavelet for Solving (Partially) Dirichlet Boundary Value Problem
Year: 2012
Author: Fatemeh Pourakbari, Ali Tavakoli
Advances in Applied Mathematics and Mechanics, Vol. 4 (2012), Iss. 6 : pp. 799–820
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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.12-12S10
Advances in Applied Mathematics and Mechanics, Vol. 4 (2012), Iss. 6 : pp. 799–820
Published online: 2012-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 22
Keywords: Galerkin method semi-orthogonal B-spline wavelet multi-resolution analysis tensor product hyperbolic partial differential equation Saint-Venant equations.
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