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Fractional-calculus diffusion equation
BACKGROUND: Sequel to the work on the quantization of nonconservative systems using fractional calculus and quantization of a system with Brownian motion, which aims to consider the dissipation effects in quantum-mechanical description of microscale systems. RESULTS: The canonical quantization of a...
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Formato: | Texto |
Lenguaje: | English |
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BioMed Central
2010
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Acceso en línea: | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2890680/ https://www.ncbi.nlm.nih.gov/pubmed/20492677 http://dx.doi.org/10.1186/1753-4631-4-3 |
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author | Ajlouni, Abdul-Wali MS Al-Rabai'ah, Hussam A |
author_facet | Ajlouni, Abdul-Wali MS Al-Rabai'ah, Hussam A |
author_sort | Ajlouni, Abdul-Wali MS |
collection | PubMed |
description | BACKGROUND: Sequel to the work on the quantization of nonconservative systems using fractional calculus and quantization of a system with Brownian motion, which aims to consider the dissipation effects in quantum-mechanical description of microscale systems. RESULTS: The canonical quantization of a system represented classically by one-dimensional Fick's law, and the diffusion equation is carried out according to the Dirac method. A suitable Lagrangian, and Hamiltonian, describing the diffusive system, are constructed and the Hamiltonian is transformed to Schrodinger's equation which is solved. An application regarding implementation of the developed mathematical method to the analysis of diffusion, osmosis, which is a biological application of the diffusion process, is carried out. Schrödinger's equation is solved. CONCLUSIONS: The plot of the probability function represents clearly the dissipative and drift forces and hence the osmosis, which agrees totally with the macro-scale view, or the classical-version osmosis. |
format | Text |
id | pubmed-2890680 |
institution | National Center for Biotechnology Information |
language | English |
publishDate | 2010 |
publisher | BioMed Central |
record_format | MEDLINE/PubMed |
spelling | pubmed-28906802010-06-24 Fractional-calculus diffusion equation Ajlouni, Abdul-Wali MS Al-Rabai'ah, Hussam A Nonlinear Biomed Phys Research BACKGROUND: Sequel to the work on the quantization of nonconservative systems using fractional calculus and quantization of a system with Brownian motion, which aims to consider the dissipation effects in quantum-mechanical description of microscale systems. RESULTS: The canonical quantization of a system represented classically by one-dimensional Fick's law, and the diffusion equation is carried out according to the Dirac method. A suitable Lagrangian, and Hamiltonian, describing the diffusive system, are constructed and the Hamiltonian is transformed to Schrodinger's equation which is solved. An application regarding implementation of the developed mathematical method to the analysis of diffusion, osmosis, which is a biological application of the diffusion process, is carried out. Schrödinger's equation is solved. CONCLUSIONS: The plot of the probability function represents clearly the dissipative and drift forces and hence the osmosis, which agrees totally with the macro-scale view, or the classical-version osmosis. BioMed Central 2010-05-21 /pmc/articles/PMC2890680/ /pubmed/20492677 http://dx.doi.org/10.1186/1753-4631-4-3 Text en Copyright ©2010 Ajlouni and Al-Rabai'ah; licensee BioMed Central Ltd. http://creativecommons.org/licenses/by/2.0 This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. |
spellingShingle | Research Ajlouni, Abdul-Wali MS Al-Rabai'ah, Hussam A Fractional-calculus diffusion equation |
title | Fractional-calculus diffusion equation |
title_full | Fractional-calculus diffusion equation |
title_fullStr | Fractional-calculus diffusion equation |
title_full_unstemmed | Fractional-calculus diffusion equation |
title_short | Fractional-calculus diffusion equation |
title_sort | fractional-calculus diffusion equation |
topic | Research |
url | https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2890680/ https://www.ncbi.nlm.nih.gov/pubmed/20492677 http://dx.doi.org/10.1186/1753-4631-4-3 |
work_keys_str_mv | AT ajlouniabdulwalims fractionalcalculusdiffusionequation AT alrabaiahhussama fractionalcalculusdiffusionequation |