October  2012, 17(7): 2299-2311. doi: 10.3934/dcdsb.2012.17.2299

A Neumann Boundary Value Problem in Two-Ion Electro-Diffusion with Unequal Valencies

1. 

Departamento de Matemática, Universidad de Buenos Aires and CONICET, Ciudad Universitaria, Pabellón I, (1428) Buenos Aires

2. 

Department of Applied Mathematics, The Hong Kong Polytechnic University, Hunghom, Kowloon, Hong Kong

3. 

Australian Research Council Centre & Excellence for Mathematics & Statistics of Complex Systems, School of Mathematics and Statistics, The University of New South Wales, Sydney, Australia

Received  December 2011 Revised  April 2012 Published  July 2012

In prior work, a series of two-point boundary value problems have been investigated for a steady state two-ion electro-diffusion model system in which the sum of the valencies $\nu_+$ and $\nu_-$ is zero. In that case, reduction is obtained to the canonical Painlevé II equation for the scaled electric field. Here, a physically important Neumann boundary value problem in the generic case when $\nu_+ + \nu_-\neq 0$ is investigated. The problem is novel in that the model equation for the electric field involves yet to be determined boundary values of the solution. A reduction of the Neumann boundary value problem in terms of elliptic functions is obtained for privileged valency ratios. A topological index argument is used to establish the existence of a solution in the general case, under the assumption $\nu_+ + \nu_- \leq 0$.
Citation: Pablo Amster, Man Kam Kwong, Colin Rogers. A Neumann Boundary Value Problem in Two-Ion Electro-Diffusion with Unequal Valencies. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2299-2311. doi: 10.3934/dcdsb.2012.17.2299
References:
[1]

W. Nernst, Zur Kinetik der in Lösung befindlichen Körper: I Theorie der Diffusion, Z. Phys. Chem., 2 (1882), 613-637.

[2]

M. Planck, Über die Erregung von Elektricität und Wärme in Electrolyten, Ann. Phys. Chem., 39 (1890), 161-186.

[3]

K. S. Cole, "Membranes, Ions and Impulses," University of California Press, Berkeley, 1968.

[4]

T. L. Schwarz, "Biophysics and Physiology of Excitable Membranes" (ed. W. J. Adelman, Jr.), Van Rostrand, New York, 1971.

[5]

J. O'M Bokris and A. K. N. Reddy, "Modern Electrochemistry," Plenum, New York, 1971.

[6]

H. R. Leuchtag, A family of differential equations arising from multi-ion electrodiffusion, J. Mathematical Phys., 22 (1981), 1317-1320.

[7]

R. Conte, C. Rogers and W. K. Schief, Painlevé structure of a multi-ion electrodiffusion system, J. Phys. A, 40 (2007), F1031-F1040. doi: 10.1088/1751-8113/40/48/F01.

[8]

H. B. Thompson, Existence for two-point boundary value problems in two-ion electrodiffusion, J. Math. Anal. Appl, 184 (1994), 82-94. doi: 10.1006/jmaa.1994.1185.

[9]

B. M. Grafov and A. A. Chernenko, Theory of the passage of a constant current through a solution of a binary electrolyte, Dokl. Akad. Nauk. SSR, 146 (1962), 135-138.

[10]

L. Bass, Electrical structures of interfaces in steady electrolysis, Trans. Faraday Soc., 60 (1964), 1655-1663. doi: 10.1039/tf9646001656.

[11]

N. A. Kudryashov, The second Painlevé equation as a model for the electric field in a semiconductor, Phys. Lett. A, 233 (1997), 397-400. doi: 10.1016/S0375-9601(97)00545-8.

[12]

C. Rogers, A. Bassom and W. K. Schief, On a Painlevé II model in steady electrolysis: Application of a Bäcklund transformation, J. Math. Anal. Appl., 240 (1999), 367-381. doi: 10.1006/jmaa.1999.6589.

[13]

L. Bass, J. Nimmo, C. Rogers and W. K. Schief, Enhanced structures of interfaces: A Painlevé II model, Proc. Roy. Soc. London Ser. A Math. Phys. Eng. Sci., 466 (2010), 2117-2136. doi: 10.1098/rspa.2009.0620.

[14]

L. Bass, Irreversible interactions between metals and electrolytes, Proc. Roy. Soc. London A, 277 (1964), 125-136. doi: 10.1098/rspa.1964.0009.

[15]

P. Amster, M. K. Kwong and C. Rogers, On a Neumann boundary value problem for Painlevé II in two-ion electro-diffusion, Nonlinear Analysis, Theory, Methods and Applications, in press.

[16]

C. De Coster and P. Habets, Upper and lower solutions in the theory of ODE boundary value problems: Classical and recent results, in "Nonlinear Analysis and Boundary Value Problems for ODEs," CISM Courses and Lectures, 371, Springer, Vienna, (1996), 1-78.

show all references

References:
[1]

W. Nernst, Zur Kinetik der in Lösung befindlichen Körper: I Theorie der Diffusion, Z. Phys. Chem., 2 (1882), 613-637.

[2]

M. Planck, Über die Erregung von Elektricität und Wärme in Electrolyten, Ann. Phys. Chem., 39 (1890), 161-186.

[3]

K. S. Cole, "Membranes, Ions and Impulses," University of California Press, Berkeley, 1968.

[4]

T. L. Schwarz, "Biophysics and Physiology of Excitable Membranes" (ed. W. J. Adelman, Jr.), Van Rostrand, New York, 1971.

[5]

J. O'M Bokris and A. K. N. Reddy, "Modern Electrochemistry," Plenum, New York, 1971.

[6]

H. R. Leuchtag, A family of differential equations arising from multi-ion electrodiffusion, J. Mathematical Phys., 22 (1981), 1317-1320.

[7]

R. Conte, C. Rogers and W. K. Schief, Painlevé structure of a multi-ion electrodiffusion system, J. Phys. A, 40 (2007), F1031-F1040. doi: 10.1088/1751-8113/40/48/F01.

[8]

H. B. Thompson, Existence for two-point boundary value problems in two-ion electrodiffusion, J. Math. Anal. Appl, 184 (1994), 82-94. doi: 10.1006/jmaa.1994.1185.

[9]

B. M. Grafov and A. A. Chernenko, Theory of the passage of a constant current through a solution of a binary electrolyte, Dokl. Akad. Nauk. SSR, 146 (1962), 135-138.

[10]

L. Bass, Electrical structures of interfaces in steady electrolysis, Trans. Faraday Soc., 60 (1964), 1655-1663. doi: 10.1039/tf9646001656.

[11]

N. A. Kudryashov, The second Painlevé equation as a model for the electric field in a semiconductor, Phys. Lett. A, 233 (1997), 397-400. doi: 10.1016/S0375-9601(97)00545-8.

[12]

C. Rogers, A. Bassom and W. K. Schief, On a Painlevé II model in steady electrolysis: Application of a Bäcklund transformation, J. Math. Anal. Appl., 240 (1999), 367-381. doi: 10.1006/jmaa.1999.6589.

[13]

L. Bass, J. Nimmo, C. Rogers and W. K. Schief, Enhanced structures of interfaces: A Painlevé II model, Proc. Roy. Soc. London Ser. A Math. Phys. Eng. Sci., 466 (2010), 2117-2136. doi: 10.1098/rspa.2009.0620.

[14]

L. Bass, Irreversible interactions between metals and electrolytes, Proc. Roy. Soc. London A, 277 (1964), 125-136. doi: 10.1098/rspa.1964.0009.

[15]

P. Amster, M. K. Kwong and C. Rogers, On a Neumann boundary value problem for Painlevé II in two-ion electro-diffusion, Nonlinear Analysis, Theory, Methods and Applications, in press.

[16]

C. De Coster and P. Habets, Upper and lower solutions in the theory of ODE boundary value problems: Classical and recent results, in "Nonlinear Analysis and Boundary Value Problems for ODEs," CISM Courses and Lectures, 371, Springer, Vienna, (1996), 1-78.

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