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Other Problem Types

  • Mitsuhiro T. Nakao
  • Michael Plum
  • Yoshitaka Watanabe
Chapter
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 53)

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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Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Mitsuhiro T. Nakao
    • 1
  • Michael Plum
    • 2
  • Yoshitaka Watanabe
    • 3
  1. 1.Faculty of MathematicsKyushu UniversityFukuokaJapan
  2. 2.Faculty of MathematicsKarlsruhe Institute of TechnologyKarlsruheGermany
  3. 3.Research Institute for Information TechnologyKyushu UniversityFukuokaJapan

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