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Journal of Engineering Mathematics
Erschienen in: Journal of Engineering Mathematics 1/2019

17.04.2019

Exact, approximate and asymptotic solutions of the Klein–Gordon integral equation

Erschienen in: Journal of Engineering Mathematics | Ausgabe 1/2019

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Abstract

Interaction of the highly localized probe of scanning probe microscopy with solid surfaces with mobile electronic or ionic carriers leads to the redistribution of mobile carriers at the tip surface junction. For small probe biases, this problem is equivalent to the Debye screening, described by Klein–Gordon (K–G) integral equation. Here, an exact solution to the K–G equation is derived for the case of a circle in the form of a convergent series expansion of the solution, which is effective for relatively small values of the inverse Debye length, k. Also, a reasonably accurate solution is derived for large values of parameter k by using the method of collocation. A surprisingly simple asymptotic solution is derived for very large values of k, which is valid for the arbitrary right-hand side of the equation. The same methods can be used for the case of elliptic domain.
Vorheriger Artikel Reformulation of wavenumber integrals describing transient waves
Nächster Artikel Numerical modelling of steady detonations with the CREST reactive burn model

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Metadaten
Titel
Exact, approximate and asymptotic solutions of the Klein–Gordon integral equation
Publikationsdatum
17.04.2019
Erschienen in
Journal of Engineering Mathematics / Ausgabe 1/2019
Print ISSN: 0022-0833
Elektronische ISSN: 1573-2703
DOI
https://doi.org/10.1007/s10665-019-09996-4

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