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  4. Generalized Equations for Ion Transport in Aqueous Solutions Through Membrane Structures: І. Electromagnetic and Subdiffusion Processes

Generalized Equations for Ion Transport in Aqueous Solutions Through Membrane Structures: І. Electromagnetic and Subdiffusion Processes

MMC.
2026;
Volume 13, Number 1
: pp. 181–197
https://doi.org/10.23939/mmc2026.01.181
Received: July 07, 2025
Revised: February 20, 2026
Accepted: March 03, 2026

Markovych B. M., Viznovych O. V., Tokarchuk M. V.  Generalized Equations for Ion Transport in Aqueous Solutions Through Membrane Structures: І. Electromagnetic and Subdiffusion Processes.  Mathematical Modeling and Computing. Vol. 13, No. 1, pp. 181–197 (2026)

Authors:
  1. B. M. Markovych
  2. O. V. Viznovych
  3. M. V. Tokarchuk
1
Lviv Polytechnic National University
2
Lviv Polytechnic National University
3
Lviv Polytechnic National University; Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine

A system of generalized transport equations for ions and water molecules in the system "initial aqueous solution of electrolyte – membrane – filtrate" is obtained, which describes diffusion electromagnetic processes through Maxwell's equations.  The method of non-equilibrium statistical operator Zubarev is used.  Generalized transport equations for ions of the $\bar{a}$ variety, which must be retained by the membrane for the purpose of selectivity, are identified.  Using the method of fractional calculus and modeling the corresponding memory function, a generalized subdiffusion equation of the Cattaneo type is obtained for the non-equilibrium average value of the number density of ions of the $\bar{a}$ variety in the solution phase.  This equation takes into account the interdiffusion processes between ions of the type $\bar{a}$ (which should not pass through the membrane) and ions $b$ and water molecules (which should pass through the membrane into the filtrate) flows, which are considered "fast" during the characteristic relaxation time $\tau_{\bar{a}}$ of subdiffusion processes.

diffusion processes of ions and water molecules
non-equilibrium statistical operator
Maxwell's equations
aqueous electrolyte solutions
membrane
fractional calculus
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