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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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