Positivity Preserving Nonstandard Finite Difference Schemes Applied to Cancer Growth Model

M. Mehdizadeh Khalsaraei; Sh. Heydari; L. Davari Algoo

doi:doi:10.11648/j.jctr.20160404.11

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Positivity Preserving Nonstandard Finite Difference Schemes Applied to Cancer Growth Model

M. Mehdizadeh Khalsaraei, Sh. Heydari, L. Davari Algoo

Published in Journal of Cancer Treatment and Research (Volume 4, Issue 4)

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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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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TY  - JOUR
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
PY  - 2017
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

M. Mehdizadeh Khalsaraei

Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran
Sh. Heydari

Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran
L. Davari Algoo

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

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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

Copy | Download

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
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
PY  - 2017
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  -

Copy | Download