Volume 4, Issue 6
Modification of Multiple Knot $B$-Spline Wavelet for Solving (Partially) Dirichlet Boundary Value Problem

Adv. Appl. Math. Mech., 4 (2012), pp. 799-820.

Published online: 2012-12

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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

Galerkin method, semi-orthogonal, B-spline wavelet, multi-resolution analysis, tensor product, hyperbolic partial differential equation, Saint-Venant equations.

65T60, 35L60, 35L04

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@Article{AAMM-4-799, author = {Pourakbari , Fatemeh and Tavakoli , Ali}, title = {Modification of Multiple Knot $B$-Spline Wavelet for Solving (Partially) Dirichlet Boundary Value Problem}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2012}, volume = {4}, number = {6}, pages = {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.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.12-12S10}, url = {http://global-sci.org/intro/article_detail/aamm/150.html} }
TY - JOUR T1 - Modification of Multiple Knot $B$-Spline Wavelet for Solving (Partially) Dirichlet Boundary Value Problem AU - Pourakbari , Fatemeh AU - Tavakoli , Ali JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 799 EP - 820 PY - 2012 DA - 2012/12 SN - 4 DO - http://doi.org/10.4208/aamm.12-12S10 UR - https://global-sci.org/intro/article_detail/aamm/150.html KW - Galerkin method, semi-orthogonal, B-spline wavelet, multi-resolution analysis, tensor product, hyperbolic partial differential equation, Saint-Venant equations. AB -

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.

Fatemeh Pourakbari & Ali Tavakoli. (1970). Modification of Multiple Knot $B$-Spline Wavelet for Solving (Partially) Dirichlet Boundary Value Problem. Advances in Applied Mathematics and Mechanics. 4 (6). 799-820. doi:10.4208/aamm.12-12S10
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