On the solutions of linear ordinary differential equations and Bessel-type special functions on the Levi-Civita field

Because of the disconnectedness of a non-Archimedean ordered field in the topology induced by the order, it is possible to have non-constant functions with zero derivatives everywhere. In fact the solution space of the differential equation y ′ = 0 is infinite dimensional. In this paper, we give suf...

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Bibliographic Details
Published in:Journal of contemporary mathematical analysis 2015-03, Vol.50 (2), p.53-62
Main Authors: Mészáros, A. R., Shamseddine, K.
Format: Article
Language:English
Subjects:
Differential Equations
Mathematics
Mathematics and Statistics
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Summary:Because of the disconnectedness of a non-Archimedean ordered field in the topology induced by the order, it is possible to have non-constant functions with zero derivatives everywhere. In fact the solution space of the differential equation y ′ = 0 is infinite dimensional. In this paper, we give sufficient conditions for a function on an open subset of the Levi-Civita field to have zero derivative everywhere and we use the nonconstant zero-derivative functions to obtain non-analytic solutions of systems of linear ordinary differential equations with analytic coefficients. Then we use the results to introduce Bessel-type special functions on the Levi-Civita field and to study some of their properties.
ISSN:1068-3623
1934-9416
DOI:10.3103/S1068362315020016