Research
Fractional-calculus diffusion equation
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* Corresponding author: Abdul-Wali MS Ajlouni awajlouni@hotmail.com
1 Applied Physics Department, Tafila Technical University, P.O. Box: 179 66110 Tafila- Jordan
2 Mathematics Department, Tafila Technical University, P.O. Box: 179 66110 Tafila- Jordan
Nonlinear Biomedical Physics 2010, 4:3 doi:10.1186/1753-4631-4-3
Published: 21 May 2010Abstract
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.