A modified fixed point method for biochemical transport
Abstract
This work is devoted to a modified fixed point method applied to the bio-chemical transport equation. To have a good accuracy for the solution we treat, we apply an implicit scheme to this equation and use a modified fixed point technique to linearize the problem of transport equation with a generalized nonlinear reaction and diffusion equation. Next, we apply this methods in particular to the the dynamical system of a bio-chemical process. Eventually, we accelerate these algorithms by the optimized domain decomposition methods.
Several test-cases of analytical problems illustrate this approach and show the efficiency of the proposed
new method.
Downloads
References
M. R. Amattouch, H. Belhadj, Combined Optimized Domain Decomposition Method and a Modified Fixed Point Method for Non Linear Diffusion Equation, Applied Mathematics and Information Sciences, 11, No. 1, 201-207 (2017). https://doi.org/10.18576/amis/110125
M. R. Amattouch, N. Nagid, H. Belhadj, Optimized Domain Decomposition Method for Non Linear Reaction Advection Diffusion Equation, European Scientific Journal , Vol 12, No 26 (2016). https://doi.org/10.19044/esj.2016.v12n27p63
M. R. Amattouch, N. Nagid, H. Belhadj, a new splitting method for the Navier Stokes equation , Journal of space exploration, Vol 2, 24 august 2017.
M. R. Amattouch, N. Nagid, H. Belhadj, A modified fixed point method for The Perona Malik equation, Journal of Mathematics and System Science 7, 175-185, september 2017 https://doi.org/10.17265/2159-5291/2017.07.001
Fisher R. A, The wave of advance of adventage genes.,Ann. Eugenics, Vol. 7, pp. 353-369, 1937 https://doi.org/10.1111/j.1469-1809.1937.tb02153.x
Murray J. D, Mathematical Biology., Berlin: Springer. 1993. https://doi.org/10.1007/978-3-662-08542-4
T. Hillen , K. J. Painter , user's guide to PDE models for chemotaxis,Journal of Math. Biol.(2009) 58,183-217 https://doi.org/10.1007/s00285-008-0201-3
A. M. Turing, he chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London , B 737, 1953, pp.37-72. https://doi.org/10.1098/rstb.1952.0012
R. FitzHugh, pulses and physiological states in theoretical models of nerve membrane, Biophys. J. 1 (1961) 445-466. https://doi.org/10.1016/S0006-3495(61)86902-6
Chang, Raymond. Physical Chemistry for the Biosciences. Sansalito, CA: University Science, 2005. Page 363-371.
A. V. Hill, The possible effects of the aggregation of the molecules of homoglobin on its dissociation curves, J. Physiol., 40,iv-vii, 1910
Goldbeter A, Koshland D E, An Amplified Sensitivity Arising from Covalent Modification in Biological-Systems., Proc Na tl Acad Sci USA 78: 6840-6844, 1981 https://doi.org/10.1073/pnas.78.11.6840
Tyson J J, Modeling the cell division cycle: cdc2 and cyclin interactions,Proc. Natl. Acad. Sci. U.S.A. 1991;88:7328-7332. https://doi.org/10.1073/pnas.88.16.7328
Y Fang, G T Yeh, W D Burgos, A general paradigm to model reaction?based biogeochemical processes in batch systemsWater Resources Research, 2003 https://doi.org/10.1029/2002WR001694
Benzekry, S., Modeling and mathematical analysis of anti-cancer therapies for metastatic cancers,PhD thesis University of Aix-Marseille (2011).
J. Nagumo, S. Arimoto, S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE 50 (1962) 2061-2070. https://doi.org/10.1109/JRPROC.1962.288235
Copyright (c) 2022 Boletim da Sociedade Paranaense de Matemática
This work is licensed under a Creative Commons Attribution 4.0 International License.
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
