Series solution of Painlevé II in electrodiffusion: conjectured convergence
Abstract
A perturbation series solution is constructed with the use of Airy functions, for a nonlinear two-point boundary-value problem arising in an established model of steady electrodiffusion in one dimension, with two ionic species carrying equal and opposite charges. The solution includes a formal determination of the associated electric field, which is known to satisfy a form of the Painlevé II differential equation. Comparisons with the numerical solution of the boundary-value problem show excellent agreement following termination of the series after a sufficient number of terms, for a much wider range of values of the parameters in the model than suggested by previously presented analysis, or admitted by previously presented approximation schemes. These surprising results suggest that for a wide variety of cases, a convergent series expansion for the Painlevé transcendent describing the electric field has been obtained. A suitable weighting of error measures for the approximations to the field and its first derivative provides a monotonically decreasing overall measure of the error in a subset of these cases. It is conjectured that the series does converge for this subset.
- Publication:
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Journal of Physics A Mathematical General
- Pub Date:
- January 2018
- DOI:
- 10.1088/1751-8121/aa9bb0
- arXiv:
- arXiv:1708.03091
- Bibcode:
- 2018JPhA...51c5202B
- Keywords:
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- Mathematics - Classical Analysis and ODEs;
- Mathematical Physics;
- 34A05;
- 34B15;
- 34B60;
- 82C70
- E-Print:
- 30 pages, 9 figures. Typos corrected, figures modified, extra references added