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2020 | OriginalPaper | Chapter

3. Mathematical Model

Authors : Vladimir Danilov, Roman Gaydukov, Vadim Kretov

Published in: Mathematical Modeling of Emission in Small-Size Cathode

Publisher: Springer Singapore

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Abstract

This chapter is a “mathematical” one. Here we collect the mathematical background related to the mathematical model of phase transition based on the phase field system introduced by G. Caginalp. Sections 3.1 and 3.2 of the chapter contain some preliminaries and considerations about mathematical models from the physical viewpoint. In Sect. 3.3, we give the results of asymptotic analysis applied to the phase field system. In Sect. 3.4, we discuss a new definition of the generalized solution to the phase field system which is stable under passing to the limiting Stefan–Gibbs–Thomson problem. Finally, in Sect. 3.5, we discuss an approach which is a combination of mathematical (asymptotic) investigation and numerical analysis.

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previous chapter Physical Basis for Field Emission
next chapter Numerical Simulation and its Results
Footnotes
1
For the rigorous asymptotic analysis of the phase field system, see in the next sections.
 
2
For more details about the problems considered here and references to the literature, see [19].
 
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Metadata
Title
Mathematical Model
Authors
Vladimir Danilov
Roman Gaydukov
Vadim Kretov
Copyright Year
2020
Publisher
Springer Singapore
Book
Mathematical Modeling of Emission in Small-Size Cathode

Print ISBN: 978-981-15-0194-4

Electronic ISBN: 978-981-15-0195-1

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-981-15-0195-1_3

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