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Verified Computations for Solutions to Semilinear Parabolic Equations Using the Evolution Operator

  • Akitoshi TakayasuEmail author
  • Makoto Mizuguchi
  • Takayuki Kubo
  • Shin’ichi Oishi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9582)

Abstract

This article presents a theorem for guaranteeing existence of a solution for an initial-boundary value problem of semilinear parabolic equations. The sufficient condition of our main theorem is derived by a fixed-point formulation using the evolution operator. We note that the sufficient condition can be checked by verified numerical computations.

References

  1. 1.
    Tanabe, H.: On the equations of evolution in a Banach space. Osaka Math. J. 12(2), 363–376 (1960)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Sobolevskii, P.E.: On equations of parabolic type in Banach space with unbounded variable operator having a constant domain. Akad. Nauk Azerbaidzan. SSR Doki, 17:6 (1961). (in Russian)Google Scholar
  3. 3.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)CrossRefzbMATHGoogle Scholar
  4. 4.
    Fujita, H., Saito, N., Suzuki, T.: Operator Theory and Numerical Methods. North Holland, Amsterdam (2001)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Akitoshi Takayasu
    • 1
    Email author
  • Makoto Mizuguchi
    • 2
  • Takayuki Kubo
    • 3
  • Shin’ichi Oishi
    • 4
  1. 1.Research Institute for Science and EngineeringWaseda UniversityTokyoJapan
  2. 2.Graduate School of Fundamental Science and EngineeringWaseda UniversityTokyoJapan
  3. 3.Institute of MathematicsUniversity of TsukubaIbarakiJapan
  4. 4.Department of Applied MathematicsWaseda University and CREST, JSTTokyoJapan

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