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Positivity Preserving Nonstandard Finite Difference Schemes Applied to Cancer Growth Model

Received: 17 October 2016     Accepted: 4 January 2017     Published: 31 January 2017
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Abstract

When one solves differential equations, modeling biological or physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. In this work, we introduce explicit finite difference schemes based on the nonstandard discretization method to approximate solution of the cross-diffusion system from bioscience. The proposed schemes improve the accuracy and guarantee the positivity requirement, as is demanded for the solution of such system. We apply new methods for numerical integration of the cancer growth model for illustrating the performance of them.

Published in Journal of Cancer Treatment and Research (Volume 4, Issue 4)
DOI 10.11648/j.jctr.20160404.11
Page(s) 27-33
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

Partial Differential Equations, Cross-Diffusion Equations, Positivity, Nonstandard Finite Difference, Cancer Growth Model

References
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[10] Dimitrov D. T., H. V. Kojouharov, Stability-Preserving finite-difference methods for general multi-dimensional autonomous dynamical system, Int. J. Numer. Anal. Model. 4(2) (2007) 282–292.
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Cite This Article
  • APA Style

    M. Mehdizadeh Khalsaraei, Sh. Heydari, L. Davari Algoo. (2017). Positivity Preserving Nonstandard Finite Difference Schemes Applied to Cancer Growth Model. Journal of Cancer Treatment and Research, 4(4), 27-33. https://doi.org/10.11648/j.jctr.20160404.11

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    ACS Style

    M. Mehdizadeh Khalsaraei; Sh. Heydari; L. Davari Algoo. Positivity Preserving Nonstandard Finite Difference Schemes Applied to Cancer Growth Model. J. Cancer Treat. Res. 2017, 4(4), 27-33. doi: 10.11648/j.jctr.20160404.11

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    AMA Style

    M. Mehdizadeh Khalsaraei, Sh. Heydari, L. Davari Algoo. Positivity Preserving Nonstandard Finite Difference Schemes Applied to Cancer Growth Model. J Cancer Treat Res. 2017;4(4):27-33. doi: 10.11648/j.jctr.20160404.11

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  • @article{10.11648/j.jctr.20160404.11,
      author = {M. Mehdizadeh Khalsaraei and Sh. Heydari and L. Davari Algoo},
      title = {Positivity Preserving Nonstandard Finite Difference Schemes Applied to Cancer Growth Model},
      journal = {Journal of Cancer Treatment and Research},
      volume = {4},
      number = {4},
      pages = {27-33},
      doi = {10.11648/j.jctr.20160404.11},
      url = {https://doi.org/10.11648/j.jctr.20160404.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.jctr.20160404.11},
      abstract = {When one solves differential equations, modeling biological or physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. In this work, we introduce explicit finite difference schemes based on the nonstandard discretization method to approximate solution of the cross-diffusion system from bioscience. The proposed schemes improve the accuracy and guarantee the positivity requirement, as is demanded for the solution of such system. We apply new methods for numerical integration of the cancer growth model for illustrating the performance of them.},
     year = {2017}
    }
    

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    T1  - Positivity Preserving Nonstandard Finite Difference Schemes Applied to Cancer Growth Model
    AU  - M. Mehdizadeh Khalsaraei
    AU  - Sh. Heydari
    AU  - L. Davari Algoo
    Y1  - 2017/01/31
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    N1  - https://doi.org/10.11648/j.jctr.20160404.11
    DO  - 10.11648/j.jctr.20160404.11
    T2  - Journal of Cancer Treatment and Research
    JF  - Journal of Cancer Treatment and Research
    JO  - Journal of Cancer Treatment and Research
    SP  - 27
    EP  - 33
    PB  - Science Publishing Group
    SN  - 2376-7790
    UR  - https://doi.org/10.11648/j.jctr.20160404.11
    AB  - When one solves differential equations, modeling biological or physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. In this work, we introduce explicit finite difference schemes based on the nonstandard discretization method to approximate solution of the cross-diffusion system from bioscience. The proposed schemes improve the accuracy and guarantee the positivity requirement, as is demanded for the solution of such system. We apply new methods for numerical integration of the cancer growth model for illustrating the performance of them.
    VL  - 4
    IS  - 4
    ER  - 

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Author Information
  • Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran

  • Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran

  • Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran

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