Haar Wavelet Solution of Poisson’s Equation and Their Block Structures

I. K. Youssef; M. H. El Dewaik

doi:doi:10.11648/j.ajmcm.20170203.13

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Haar Wavelet Solution of Poisson’s Equation and Their Block Structures

I. K. Youssef, M. H. El Dewaik

Published in American Journal of Mathematical and Computer Modelling (Volume 2, Issue 3)

Received: 1 March 2017 Accepted: 13 March 2017 Published: 29 March 2017

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Abstract

The structure of the algebraic system which results from the use of Haar wavelet when solving Poisson’s equation is studied. Haar wavelet technique is used to solve Poisson’s equation on a unit square domain. The form of collocation points are used at the mid points of the subintervals i.e at the odd multiple of the sub interval length labeling. It is proved that the coefficient matrix has symmetric block structure. Comparison with the tridagonal block structure obtained by the finite difference with the natural ordering is introduced. The numerical results have illustrated the superiority of the use of Haar wavelet technique. The matrices obtained can be used for any equations containing the Laplace operator.

Published in	American Journal of Mathematical and Computer Modelling (Volume 2, Issue 3)
DOI	10.11648/j.ajmcm.20170203.13
Page(s)	88-94
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Poisson’s Equation, Finite Difference, Wavelet, Haar Wavelet

References

[1]	Sumana R. Shesha, Tejuswini M. and Achala L. Nargund, Haar wavelet Methods for the solution of elliptic partial differential equations, International Journal of Mathematics and Computer Research, Vol. 4, No. 6, pp 1481-1492, 2016.
[2]	Gilbert Strang, Wavele transfors versus fourier transforms, appeared in bulletin of the American mathematical society Volume 28, Number 2, April 1993, Pages 288-305.
[3]	Gilbert Strang, Wavelets and Dilation Equations: A Brief Introduction SIAM Review, Vol. 31, No. 4. (Dec., 1989), pp. 614-627.
[4]	Memory Effects in Diffusion Like Equation Via Haar WaveletsI. K. YoussefA. R. A. AliPure and Applied Mathematics Journal Volume 5, Issue 4, August 2016, Pages: 130-140
[5]	Memory Effects Due to Fractional Time Derivative and Integral Space in Diffusion Like Equation Via Haar Wavelets I. K. YoussefA. R. A. Ali Applied and Computational Mathematics Volume 5, Issue 4, August 2016, Pages: 177-185.
[6]	Ulo Lepik, Helle Hein, Haar wavelets with applications, Springer international Publishing Switzerland, 2014.
[7]	C. F. Chen, C. H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proc.-Control Theory Appl. Vol. 144, No. 1, 1997.
[8]	Zhi Shi, Li-Youan Deng, Qing-Jiang Chen, Numerical solution of differential equation by using Haar wavelet, Proceedings of the 2007 Int. Conf. on Wavelet Analysis and Pattern Recognition, Beijing, 2007.
[9]	G. D. Smith, Numerical Solution of partial differential equations, Clarendon Press, Oxford, 1986.
[10]	David M. Young, Iterative solution of large linear system, Academic Press, 1971.
[11]	Abdollah Borhanifar and Sohrab Valizadeh, A fractional finite difference method for solving the fractional Poisson equation based on the shifted Grunwald estimate, Walailak J Sci & Tech, Vol. 10 No. 5, 2013.

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APA Style

I. K. Youssef, M. H. El Dewaik. (2017). Haar Wavelet Solution of Poisson’s Equation and Their Block Structures. American Journal of Mathematical and Computer Modelling, 2(3), 88-94. https://doi.org/10.11648/j.ajmcm.20170203.13

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ACS Style

I. K. Youssef; M. H. El Dewaik. Haar Wavelet Solution of Poisson’s Equation and Their Block Structures. Am. J. Math. Comput. Model. 2017, 2(3), 88-94. doi: 10.11648/j.ajmcm.20170203.13

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AMA Style

I. K. Youssef, M. H. El Dewaik. Haar Wavelet Solution of Poisson’s Equation and Their Block Structures. Am J Math Comput Model. 2017;2(3):88-94. doi: 10.11648/j.ajmcm.20170203.13

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@article{10.11648/j.ajmcm.20170203.13,
  author = {I. K. Youssef and M. H. El Dewaik},
  title = {Haar Wavelet Solution of Poisson’s Equation and Their Block Structures},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {2},
  number = {3},
  pages = {88-94},
  doi = {10.11648/j.ajmcm.20170203.13},
  url = {https://doi.org/10.11648/j.ajmcm.20170203.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20170203.13},
  abstract = {The structure of the algebraic system which results from the use of Haar wavelet when solving Poisson’s equation is studied. Haar wavelet technique is used to solve Poisson’s equation on a unit square domain. The form of collocation points are used at the mid points of the subintervals i.e at the odd multiple of the sub interval length labeling. It is proved that the coefficient matrix has symmetric block structure. Comparison with the tridagonal block structure obtained by the finite difference with the natural ordering is introduced. The numerical results have illustrated the superiority of the use of Haar wavelet technique. The matrices obtained can be used for any equations containing the Laplace operator.},
 year = {2017}
}

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TY  - JOUR
T1  - Haar Wavelet Solution of Poisson’s Equation and Their Block Structures
AU  - I. K. Youssef
AU  - M. H. El Dewaik
Y1  - 2017/03/29
PY  - 2017
N1  - https://doi.org/10.11648/j.ajmcm.20170203.13
DO  - 10.11648/j.ajmcm.20170203.13
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
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EP  - 94
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20170203.13
AB  - The structure of the algebraic system which results from the use of Haar wavelet when solving Poisson’s equation is studied. Haar wavelet technique is used to solve Poisson’s equation on a unit square domain. The form of collocation points are used at the mid points of the subintervals i.e at the odd multiple of the sub interval length labeling. It is proved that the coefficient matrix has symmetric block structure. Comparison with the tridagonal block structure obtained by the finite difference with the natural ordering is introduced. The numerical results have illustrated the superiority of the use of Haar wavelet technique. The matrices obtained can be used for any equations containing the Laplace operator.
VL  - 2
IS  - 3
ER  -

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Author Information

I. K. Youssef

Department of Mathematics, Ain Shams University, Cairo, Egypt
M. H. El Dewaik

Department of Basic Science, The British University, Cairo, Egypt

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APA Style

I. K. Youssef, M. H. El Dewaik. (2017). Haar Wavelet Solution of Poisson’s Equation and Their Block Structures. American Journal of Mathematical and Computer Modelling, 2(3), 88-94. https://doi.org/10.11648/j.ajmcm.20170203.13

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ACS Style

I. K. Youssef; M. H. El Dewaik. Haar Wavelet Solution of Poisson’s Equation and Their Block Structures. Am. J. Math. Comput. Model. 2017, 2(3), 88-94. doi: 10.11648/j.ajmcm.20170203.13

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AMA Style

I. K. Youssef, M. H. El Dewaik. Haar Wavelet Solution of Poisson’s Equation and Their Block Structures. Am J Math Comput Model. 2017;2(3):88-94. doi: 10.11648/j.ajmcm.20170203.13

Copy | Download

@article{10.11648/j.ajmcm.20170203.13,
  author = {I. K. Youssef and M. H. El Dewaik},
  title = {Haar Wavelet Solution of Poisson’s Equation and Their Block Structures},
  journal = {American Journal of Mathematical and Computer Modelling},
  volume = {2},
  number = {3},
  pages = {88-94},
  doi = {10.11648/j.ajmcm.20170203.13},
  url = {https://doi.org/10.11648/j.ajmcm.20170203.13},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajmcm.20170203.13},
  abstract = {The structure of the algebraic system which results from the use of Haar wavelet when solving Poisson’s equation is studied. Haar wavelet technique is used to solve Poisson’s equation on a unit square domain. The form of collocation points are used at the mid points of the subintervals i.e at the odd multiple of the sub interval length labeling. It is proved that the coefficient matrix has symmetric block structure. Comparison with the tridagonal block structure obtained by the finite difference with the natural ordering is introduced. The numerical results have illustrated the superiority of the use of Haar wavelet technique. The matrices obtained can be used for any equations containing the Laplace operator.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - Haar Wavelet Solution of Poisson’s Equation and Their Block Structures
AU  - I. K. Youssef
AU  - M. H. El Dewaik
Y1  - 2017/03/29
PY  - 2017
N1  - https://doi.org/10.11648/j.ajmcm.20170203.13
DO  - 10.11648/j.ajmcm.20170203.13
T2  - American Journal of Mathematical and Computer Modelling
JF  - American Journal of Mathematical and Computer Modelling
JO  - American Journal of Mathematical and Computer Modelling
SP  - 88
EP  - 94
PB  - Science Publishing Group
SN  - 2578-8280
UR  - https://doi.org/10.11648/j.ajmcm.20170203.13
AB  - The structure of the algebraic system which results from the use of Haar wavelet when solving Poisson’s equation is studied. Haar wavelet technique is used to solve Poisson’s equation on a unit square domain. The form of collocation points are used at the mid points of the subintervals i.e at the odd multiple of the sub interval length labeling. It is proved that the coefficient matrix has symmetric block structure. Comparison with the tridagonal block structure obtained by the finite difference with the natural ordering is introduced. The numerical results have illustrated the superiority of the use of Haar wavelet technique. The matrices obtained can be used for any equations containing the Laplace operator.
VL  - 2
IS  - 3
ER  -

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