Memory Effects in Diffusion Like Equation Via Haar Wavelets

I. K. Youssef; A. R. A. Ali

doi:doi:10.11648/j.pamj.20160504.17

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Memory Effects in Diffusion Like Equation Via Haar Wavelets

I. K. Youssef, A. R. A. Ali

Published in Pure and Applied Mathematics Journal (Volume 5, Issue 4)

Received: 17 July 2016 Accepted: 26 July 2016 Published: 10 August 2016

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Abstract

The memory and hereditary effects of fractional derivatives as well as integral terms are considered in a diffusion like problem. The Haar wavelet operational matrix technique is employed to solve fractional order diffusion equation with time dependent integral term and time dependent boundary condition. The fractional derivative is described in the Caputo sense. The effect of using inverse fractional operator which combines the memory behaviors of the fractional derivatives to all other terms in the equation is disscused. Different Haar bases functions are used (8, 16, 32, 64) and comparison of the wavelet operational matrix is considered. Error analysis is considered. A general numerical example with four subproblems is considered, graphical representation of the different solutions as well as their errors are given.

Published in	Pure and Applied Mathematics Journal (Volume 5, Issue 4)
DOI	10.11648/j.pamj.20160504.17
Page(s)	130-140
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), 2016. Published by Science Publishing Group

Keywords

Haar Wavelet, Operational Matrix, Fractional Derivative, Fractional Order Diffusion Equation

References

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[2]	I. K. Youssef, A. M. Shukur, Modified variation iteration method for fraction space-time partial differential heat and wave equations, International Journal 2 (2) (2013) 1000–1013.
[3]	Ü. Lepik, Solving fractional integral equations by the haar wavelet method, Applied Mathematics and Computation 214 (2) (2009) 468–478.
[4]	I. Podlubny, Fractional Differential Equations, Camb. Academic Press, San Diego, CA, 1999.
[5]	K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, New York: Wiley-Interscience Publ., 1993.
[6]	I. K. Youssef, R. A. Ibrahim, Boundary Value Problems, Fredholm Integral equations, SOR and KSOR Methods, Life Science Journal 10 (2) (2013) 304-312.
[7]	I. K. Youssef, A. M. Shukur, Precondition for discretized fractional boundary value problem, Pure and Applied Mathematics Journal 3 (1) (2014) 1-6.
[8]	I. K. Youssef, A. M. Shukur, The line method combined with spectral chebyshev for space-time fractional diffusion equation, Applied and Computational Mathematics 3 (6) (2014) 330- 336.
[9]	G. D. Smith, Numerical Solution of Partial Differential Equations Finite Difference Methods, Oxford University Press, 1978.
[10]	C. K. Chui, An introduction to wavelets, Vol. 1, Academic press, 2014.
[11]	Ü. Lepik, Buckling of elastic beams by the haar wavelet method, Estonian Journal of Engineering 17 (3) (2011) 271-284.
[12]	C. F. Chen, C. H. Hsiao, Haar wavelet method for solving lumped and distributed-parameter systems, IEE Proc., Control Theory Appl. 144 (1) (1997) 87–94. doi: 10.1049/ip-cta:19970702.
[13]	C. F. Chen, C. H. Hsiao, Wavelet approach to optimising dynamic systems, IEE Proc., Control Theory Appl. 146 (2) (1999) 213–219. doi: 10.1049/ip-cta:19990516.
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[15]	Ü. Lepik, Numerical solution of differential equations using haar wavelets, Mathematics and computers in simulation 68 (2) (2005) 127-143.
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[17]	C. H. Hsiao, Haar wavelet direct method for solving variational problems, Mathematics and Computers in Simulation 64 (5) (2004) 569-585.
[18]	C. H. Hsiao, W. J. Wang, Haar wavelet approach to nonlinear stiff systems, Mathematics and computers in simulation 57 (6) (2001) 347-353

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APA Style

I. K. Youssef, A. R. A. Ali. (2016). Memory Effects in Diffusion Like Equation Via Haar Wavelets. Pure and Applied Mathematics Journal, 5(4), 130-140. https://doi.org/10.11648/j.pamj.20160504.17

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ACS Style

I. K. Youssef; A. R. A. Ali. Memory Effects in Diffusion Like Equation Via Haar Wavelets. Pure Appl. Math. J. 2016, 5(4), 130-140. doi: 10.11648/j.pamj.20160504.17

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AMA Style

I. K. Youssef, A. R. A. Ali. Memory Effects in Diffusion Like Equation Via Haar Wavelets. Pure Appl Math J. 2016;5(4):130-140. doi: 10.11648/j.pamj.20160504.17

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@article{10.11648/j.pamj.20160504.17,
  author = {I. K. Youssef and A. R. A. Ali},
  title = {Memory Effects in Diffusion Like Equation Via Haar Wavelets},
  journal = {Pure and Applied Mathematics Journal},
  volume = {5},
  number = {4},
  pages = {130-140},
  doi = {10.11648/j.pamj.20160504.17},
  url = {https://doi.org/10.11648/j.pamj.20160504.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160504.17},
  abstract = {The memory and hereditary effects of fractional derivatives as well as integral terms are considered in a diffusion like problem. The Haar wavelet operational matrix technique is employed to solve fractional order diffusion equation with time dependent integral term and time dependent boundary condition. The fractional derivative is described in the Caputo sense. The effect of using inverse fractional operator which combines the memory behaviors of the fractional derivatives to all other terms in the equation is disscused. Different Haar bases functions are used (8, 16, 32, 64) and comparison of the wavelet operational matrix is considered. Error analysis is considered. A general numerical example with four subproblems is considered, graphical representation of the different solutions as well as their errors are given.},
 year = {2016}
}

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TY  - JOUR
T1  - Memory Effects in Diffusion Like Equation Via Haar Wavelets
AU  - I. K. Youssef
AU  - A. R. A. Ali
Y1  - 2016/08/10
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160504.17
DO  - 10.11648/j.pamj.20160504.17
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 130
EP  - 140
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20160504.17
AB  - The memory and hereditary effects of fractional derivatives as well as integral terms are considered in a diffusion like problem. The Haar wavelet operational matrix technique is employed to solve fractional order diffusion equation with time dependent integral term and time dependent boundary condition. The fractional derivative is described in the Caputo sense. The effect of using inverse fractional operator which combines the memory behaviors of the fractional derivatives to all other terms in the equation is disscused. Different Haar bases functions are used (8, 16, 32, 64) and comparison of the wavelet operational matrix is considered. Error analysis is considered. A general numerical example with four subproblems is considered, graphical representation of the different solutions as well as their errors are given.
VL  - 5
IS  - 4
ER  -

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Author Information

I. K. Youssef

Department of Mathematics, Ain Shams University, Cairo, Egypt
A. R. A. Ali

Department of Mathematics, Baghdad University, Baghdad, Iraq

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APA Style

I. K. Youssef, A. R. A. Ali. (2016). Memory Effects in Diffusion Like Equation Via Haar Wavelets. Pure and Applied Mathematics Journal, 5(4), 130-140. https://doi.org/10.11648/j.pamj.20160504.17

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ACS Style

I. K. Youssef; A. R. A. Ali. Memory Effects in Diffusion Like Equation Via Haar Wavelets. Pure Appl. Math. J. 2016, 5(4), 130-140. doi: 10.11648/j.pamj.20160504.17

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AMA Style

I. K. Youssef, A. R. A. Ali. Memory Effects in Diffusion Like Equation Via Haar Wavelets. Pure Appl Math J. 2016;5(4):130-140. doi: 10.11648/j.pamj.20160504.17

Copy | Download

@article{10.11648/j.pamj.20160504.17,
  author = {I. K. Youssef and A. R. A. Ali},
  title = {Memory Effects in Diffusion Like Equation Via Haar Wavelets},
  journal = {Pure and Applied Mathematics Journal},
  volume = {5},
  number = {4},
  pages = {130-140},
  doi = {10.11648/j.pamj.20160504.17},
  url = {https://doi.org/10.11648/j.pamj.20160504.17},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20160504.17},
  abstract = {The memory and hereditary effects of fractional derivatives as well as integral terms are considered in a diffusion like problem. The Haar wavelet operational matrix technique is employed to solve fractional order diffusion equation with time dependent integral term and time dependent boundary condition. The fractional derivative is described in the Caputo sense. The effect of using inverse fractional operator which combines the memory behaviors of the fractional derivatives to all other terms in the equation is disscused. Different Haar bases functions are used (8, 16, 32, 64) and comparison of the wavelet operational matrix is considered. Error analysis is considered. A general numerical example with four subproblems is considered, graphical representation of the different solutions as well as their errors are given.},
 year = {2016}
}

Copy | Download

TY  - JOUR
T1  - Memory Effects in Diffusion Like Equation Via Haar Wavelets
AU  - I. K. Youssef
AU  - A. R. A. Ali
Y1  - 2016/08/10
PY  - 2016
N1  - https://doi.org/10.11648/j.pamj.20160504.17
DO  - 10.11648/j.pamj.20160504.17
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 130
EP  - 140
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20160504.17
AB  - The memory and hereditary effects of fractional derivatives as well as integral terms are considered in a diffusion like problem. The Haar wavelet operational matrix technique is employed to solve fractional order diffusion equation with time dependent integral term and time dependent boundary condition. The fractional derivative is described in the Caputo sense. The effect of using inverse fractional operator which combines the memory behaviors of the fractional derivatives to all other terms in the equation is disscused. Different Haar bases functions are used (8, 16, 32, 64) and comparison of the wavelet operational matrix is considered. Error analysis is considered. A general numerical example with four subproblems is considered, graphical representation of the different solutions as well as their errors are given.
VL  - 5
IS  - 4
ER  -

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