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Erschienen in:
Basic Principle of the Verification Kapitel lesen Erstes Kapitel lesen

2019 | OriginalPaper | Buchkapitel

9. Other Problem Types

verfasst von : Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe

Erschienen in: Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations

Verlag: Springer Singapore

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Abstract

In this chapter we will study some extensions of the types of problems considered so far. The first two sections are concerned with parameter-dependent problems in an abstract formulation, which is tailored for applications to elliptic boundary value problems. Section 9.1 provides a computer-assisted approach for proving existence of smooth branches of solutions to such problems. In Sect. 9.2 we consider situations where solution branches contain turning points or bifurcation points , requiring modifications of our Newton-type computer-assisted approach to be still applicable. With modifications similar to the turning-point case, we also treat non-self-adjoint eigenvalue problems in Sect. 9.3, with applications to the famous Orr–Sommerfeld equation. Finally, in Sect. 9.4 we are concerned with systems of second-order elliptic boundary value problems, where the linearized operator L lacks symmetry, whence a norm bound for L −1 cannot be computed via the spectrum of L or Φ −1L.
In this chapter we concentrate on the main ideas and partially will be a bit less extensive with technical details.

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Vorheriges Kapitel Fourth-Order Problems
Nächstes Kapitel Eigenvalue Bounds for Self-Adjoint Eigenvalue Problems
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Metadaten
Titel
Other Problem Types
verfasst von
Mitsuhiro T. Nakao
Michael Plum
Yoshitaka Watanabe
Copyright-Jahr
2019
Verlag
Springer Singapore
Buch
Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations

Print ISBN: 978-981-13-7668-9

Electronic ISBN: 978-981-13-7669-6

Copyright-Jahr: 2019

DOI
https://doi.org/10.1007/978-981-13-7669-6_9

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