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Memory Effects Due to Fractional Time Derivative and Integral Space in Diffusion Like Equation Via Haar Wavelets

Received: 6 August 2016     Accepted: 15 August 2016     Published: 2 September 2016
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Abstract

Memory and hereditary effects due to fractional time derivative are combined with the global behaviours due to space integral term. Haar wavelet operational matrix is adjusted to solve diffusion like equations with time fractional derivative, space derivatives and integral terms. The fractional derivative is understood in the Caputo sense. The memory behaviours is included in all the points of the domain due to the existence of space integral term and the inverse fractional operator treatment and this is ilustrated in error graphs introduced. A general example with four subproblems ranging from the simple classical heat equation to the fractional time diffusion equation with global integral term is proposed and the calculated results are displayed graphically.

Published in Applied and Computational Mathematics (Volume 5, Issue 4)
DOI 10.11648/j.acm.20160504.12
Page(s) 177-185
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, Diffusion Like Equation

References
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[2] K. Rektorys, The Method of Discretization in Time and Partial Differential Equations, Mathematics and its Applications, Springer Netherlands, 1982.
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[4] 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.
[5] 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.
[6] I. K. Youssef, A. R. A. Ali, Memory Effects in Diffusion Like Equation Via Haar Wavelets, Pure and Applied Mathematics Journal, 5 (4), (2016) 130-140.
[7] I. Podlubny, Fractional Differential Equations, Camb. Academic Press, San Diego, CA, 1999.
[8] K. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, New York: Wiley-Interscience Publ., 1993.
[9] Ü. Lepik, Solving fractional integral equations by the haar wavelet method, Applied Mathematics and Computation 214 (2) (2009) 468–478.
[10] I. K. Youssef, A. M. Shukur, Precondition for discretized fractional boundary value problem, Pure and Applied Mathematics Journal 3 (1) (2014) 1-6.
[11] G. D. Smith, Numerical Solution of Partial Differential Equations Finite Difference Methods, Oxford University Press, 1978.
[12] C. K. Chui, An introduction to wavelets, Vol. 1, Academic press, 2014.
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[14] 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.
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  • APA Style

    I. K. Youssef, A. R. A. Ali. (2016). Memory Effects Due to Fractional Time Derivative and Integral Space in Diffusion Like Equation Via Haar Wavelets. Applied and Computational Mathematics, 5(4), 177-185. https://doi.org/10.11648/j.acm.20160504.12

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    ACS Style

    I. K. Youssef; A. R. A. Ali. Memory Effects Due to Fractional Time Derivative and Integral Space in Diffusion Like Equation Via Haar Wavelets. Appl. Comput. Math. 2016, 5(4), 177-185. doi: 10.11648/j.acm.20160504.12

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    AMA Style

    I. K. Youssef, A. R. A. Ali. Memory Effects Due to Fractional Time Derivative and Integral Space in Diffusion Like Equation Via Haar Wavelets. Appl Comput Math. 2016;5(4):177-185. doi: 10.11648/j.acm.20160504.12

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  • @article{10.11648/j.acm.20160504.12,
      author = {I. K. Youssef and A. R. A. Ali},
      title = {Memory Effects Due to Fractional Time Derivative and Integral Space in Diffusion Like Equation Via Haar Wavelets},
      journal = {Applied and Computational Mathematics},
      volume = {5},
      number = {4},
      pages = {177-185},
      doi = {10.11648/j.acm.20160504.12},
      url = {https://doi.org/10.11648/j.acm.20160504.12},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20160504.12},
      abstract = {Memory and hereditary effects due to fractional time derivative are combined with the global behaviours due to space integral term. Haar wavelet operational matrix is adjusted to solve diffusion like equations with time fractional derivative, space derivatives and integral terms. The fractional derivative is understood in the Caputo sense. The memory behaviours is included in all the points of the domain due to the existence of space integral term and the inverse fractional operator treatment and this is ilustrated in error graphs introduced. A general example with four subproblems ranging from the simple classical heat equation to the fractional time diffusion equation with global integral term is proposed and the calculated results are displayed graphically.},
     year = {2016}
    }
    

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    AB  - Memory and hereditary effects due to fractional time derivative are combined with the global behaviours due to space integral term. Haar wavelet operational matrix is adjusted to solve diffusion like equations with time fractional derivative, space derivatives and integral terms. The fractional derivative is understood in the Caputo sense. The memory behaviours is included in all the points of the domain due to the existence of space integral term and the inverse fractional operator treatment and this is ilustrated in error graphs introduced. A general example with four subproblems ranging from the simple classical heat equation to the fractional time diffusion equation with global integral term is proposed and the calculated results are displayed graphically.
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Author Information
  • Department of Mathematics, Ain Shams University, Cairo, Egypt

  • Department of Mathematics, Baghdad University, Baghdad, Iraq

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