Precondition for Discretized Fractional Boundary Value Problem

I. K. Youssef; A. M. Shukur

doi:doi:10.11648/j.pamj.20140301.11

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Precondition for Discretized Fractional Boundary Value Problem

I. K. Youssef, A. M. Shukur

Published in Pure and Applied Mathematics Journal (Volume 3, Issue 1)

Received: 23 January 2014 Published: 20 February 2014

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Abstract

A precondition that uses the special structure of the algebraic system arising from the discretization of a fractional partial differential equation on the red black ordering grid is introduced. Comparison with the numerical solution of the classical Poisson’s equationis considered. A graphical representation for the precondition is illustrated. The performance of our treatment is calculated for different values of the fractional order. The results of the implementation of the SOR and the KSOR with the help of MATLAB are given in compact tables. The value of the relaxation parameter is chosen on the bases of the graphical representation of the behavior of the spectral radius of the iteration matrices. Also, the natural ordering grid is considered.

Published in	Pure and Applied Mathematics Journal (Volume 3, Issue 1)
DOI	10.11648/j.pamj.20140301.11
Page(s)	1-6
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), 2014. Published by Science Publishing Group

Keywords

Fractional Boundary Value Problem, SOR, KSOR and Precondition

References

[1]	D. M. Young, Iterative solution of large linear systems, Academic Press, New York (1970).
[2]	D. M. Young, Iterative methods for solving Partial difference equation of Elliptic type, Ph. D. thesis, Department of mathematics, Harvard university Cambridge, mass, may 1, 1950.
[3]	I. K,Youssef, on the successive overrelaxation method, Journal of math. and statistics 8 (2): 172-180,2012.
[4]	V. D.Beibalaev ,Ruslan P. Meilanov, the Dirihlet problem for the fractional Poisson’s equation with caputoDerivatives, afinite difference approximation and a numerical solution. Thermal scince:year, val,16, 2012.
[5]	K. B. Oldham and J. Spanier,the Fractional Calculus, Academic Press, New York, NY, USA, 1974.
[6]	Lijuan Su and Pei Cheng, A Weighted Average Finite Difference Method for the Fractional Convection-Diffusion Equation, Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2013.
[7]	N. G. Abrashina-Zhadaeva and I. A. Timoshchenko, Finite-Diﬀerence Schemes for a Diﬀusion Equation with Fractional Derivatives in a Multidimensional Domain, ISSN 0012-2661, Vol. 49, Differential Equations, 2013.
[8]	YousefSaad, Iterative Methods for Sparse Linear Systems, by the Society for Industrial and Applied Mathematics, 2003.
[9]	D. J. Evans, M. M. Martins, M. E. Trigo, On preconditioned MSOR Iterations, Intern, J. Computer Math, Vol. 59, pp. 251-257.
[10]	Michele Benzi, Preconditioning Techniques for Large Linear Systems, Journal of Computational Physics 182, 418–477 (2002).
[11]	O. Settle, C. C. Douglas, I. Kim, D. Sheen, On the derivation of highest-order compact finitedifference schemes for the one- and two-dimensional poisson equation with dirichlet boundary conditions, SIAM Journal on Numerical Analysis, Vol. 51, No. 4 , (2013).

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I. K. Youssef, A. M. Shukur. (2014). Precondition for Discretized Fractional Boundary Value Problem. Pure and Applied Mathematics Journal, 3(1), 1-6. https://doi.org/10.11648/j.pamj.20140301.11

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ACS Style

I. K. Youssef; A. M. Shukur. Precondition for Discretized Fractional Boundary Value Problem. Pure Appl. Math. J. 2014, 3(1), 1-6. doi: 10.11648/j.pamj.20140301.11

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AMA Style

I. K. Youssef, A. M. Shukur. Precondition for Discretized Fractional Boundary Value Problem. Pure Appl Math J. 2014;3(1):1-6. doi: 10.11648/j.pamj.20140301.11

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@article{10.11648/j.pamj.20140301.11,
  author = {I. K. Youssef and A. M. Shukur},
  title = {Precondition for Discretized Fractional Boundary Value Problem},
  journal = {Pure and Applied Mathematics Journal},
  volume = {3},
  number = {1},
  pages = {1-6},
  doi = {10.11648/j.pamj.20140301.11},
  url = {https://doi.org/10.11648/j.pamj.20140301.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20140301.11},
  abstract = {A precondition that uses the special structure of the algebraic system arising from the discretization of a fractional partial differential equation on the red black ordering grid is introduced. Comparison with the numerical solution of the classical Poisson’s equationis considered. A graphical representation for the precondition is illustrated. The performance of our treatment is calculated for different values of the fractional order. The results of the implementation of the SOR and the KSOR with the help of MATLAB are given in compact tables. The value of the relaxation parameter is chosen on the bases of the graphical representation of the behavior of the spectral radius of the iteration matrices. Also, the natural ordering grid is considered.},
 year = {2014}
}

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TY  - JOUR
T1  - Precondition for Discretized Fractional Boundary Value Problem
AU  - I. K. Youssef
AU  - A. M. Shukur
Y1  - 2014/02/20
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.20140301.11
DO  - 10.11648/j.pamj.20140301.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 1
EP  - 6
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20140301.11
AB  - A precondition that uses the special structure of the algebraic system arising from the discretization of a fractional partial differential equation on the red black ordering grid is introduced. Comparison with the numerical solution of the classical Poisson’s equationis considered. A graphical representation for the precondition is illustrated. The performance of our treatment is calculated for different values of the fractional order. The results of the implementation of the SOR and the KSOR with the help of MATLAB are given in compact tables. The value of the relaxation parameter is chosen on the bases of the graphical representation of the behavior of the spectral radius of the iteration matrices. Also, the natural ordering grid is considered.
VL  - 3
IS  - 1
ER  -

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Author Information

I. K. Youssef

Department of Mathematics, Ain Shams University, Cairo, Egypt
A. M. Shukur

Department of Applied Mathematics, University of Technology, Baghdad, Iraq

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APA Style

I. K. Youssef, A. M. Shukur. (2014). Precondition for Discretized Fractional Boundary Value Problem. Pure and Applied Mathematics Journal, 3(1), 1-6. https://doi.org/10.11648/j.pamj.20140301.11

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ACS Style

I. K. Youssef; A. M. Shukur. Precondition for Discretized Fractional Boundary Value Problem. Pure Appl. Math. J. 2014, 3(1), 1-6. doi: 10.11648/j.pamj.20140301.11

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AMA Style

I. K. Youssef, A. M. Shukur. Precondition for Discretized Fractional Boundary Value Problem. Pure Appl Math J. 2014;3(1):1-6. doi: 10.11648/j.pamj.20140301.11

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@article{10.11648/j.pamj.20140301.11,
  author = {I. K. Youssef and A. M. Shukur},
  title = {Precondition for Discretized Fractional Boundary Value Problem},
  journal = {Pure and Applied Mathematics Journal},
  volume = {3},
  number = {1},
  pages = {1-6},
  doi = {10.11648/j.pamj.20140301.11},
  url = {https://doi.org/10.11648/j.pamj.20140301.11},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20140301.11},
  abstract = {A precondition that uses the special structure of the algebraic system arising from the discretization of a fractional partial differential equation on the red black ordering grid is introduced. Comparison with the numerical solution of the classical Poisson’s equationis considered. A graphical representation for the precondition is illustrated. The performance of our treatment is calculated for different values of the fractional order. The results of the implementation of the SOR and the KSOR with the help of MATLAB are given in compact tables. The value of the relaxation parameter is chosen on the bases of the graphical representation of the behavior of the spectral radius of the iteration matrices. Also, the natural ordering grid is considered.},
 year = {2014}
}

Copy | Download

TY  - JOUR
T1  - Precondition for Discretized Fractional Boundary Value Problem
AU  - I. K. Youssef
AU  - A. M. Shukur
Y1  - 2014/02/20
PY  - 2014
N1  - https://doi.org/10.11648/j.pamj.20140301.11
DO  - 10.11648/j.pamj.20140301.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 1
EP  - 6
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20140301.11
AB  - A precondition that uses the special structure of the algebraic system arising from the discretization of a fractional partial differential equation on the red black ordering grid is introduced. Comparison with the numerical solution of the classical Poisson’s equationis considered. A graphical representation for the precondition is illustrated. The performance of our treatment is calculated for different values of the fractional order. The results of the implementation of the SOR and the KSOR with the help of MATLAB are given in compact tables. The value of the relaxation parameter is chosen on the bases of the graphical representation of the behavior of the spectral radius of the iteration matrices. Also, the natural ordering grid is considered.
VL  - 3
IS  - 1
ER  -

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