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Springer Series in Computational Mathematics

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

Authors: Nakao, Mitsuhiro T., Plum, Michael, Watanabe, Yoshitaka

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  • Is the first published monograph on computer-assisted proofs
  • Presents pioneering work on the numerical verification method of solution for partial differential equations
  • Provides verification techniques for partial differential equations and applications for computer-assisted proofs
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eBook 93,08 €
price for Spain (gross)
  • ISBN 978-981-13-7669-6
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  • Included format: EPUB, PDF
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  • Immediate eBook download after purchase
Hardcover 114,39 €
price for Spain (gross)
  • ISBN 978-981-13-7668-9
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  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
About this book

In the last decades, various mathematical problems have been solved by computer-assisted proofs, among them the Kepler conjecture, the existence of chaos, the existence of the Lorenz attractor, the famous four-color problem, and more. In many cases, computer-assisted proofs have the remarkable advantage (compared with a “theoretical” proof) of additionally providing accurate quantitative information.

The authors have been working more than a quarter century to establish methods for the verified computation of solutions for partial differential equations, mainly for nonlinear elliptic problems of the form -∆u=f(x,u,∇u) with Dirichlet boundary conditions. Here, by “verified computation” is meant a computer-assisted numerical approach for proving the existence of a solution in a close and explicit neighborhood of an approximate solution. The quantitative information provided by these techniques is also significant from the viewpoint of a posteriori error estimates for approximate solutions of the concerned partial differential equations in a mathematically rigorous sense.

In this monograph, the authors give a detailed description of the verified computations and computer-assisted proofs for partial differential equations that they developed. In Part I, the methods mainly studied by the authors Nakao and Watanabe are presented. These methods are based on a finite dimensional projection and constructive a priori error estimates for finite element approximations of the Poisson equation. In Part II, the computer-assisted approaches via eigenvalue bounds developed by the author Plum are explained in detail. The main task of this method consists of establishing eigenvalue bounds for the linearization of the corresponding nonlinear problem at the computed approximate solution. Some brief remarks on other approaches are also given in Part III. Each method in Parts I and II is accompanied by appropriate numerical examples that confirm the actual usefulness of the authors’ methods. Also in some examples practical computer algorithms are supplied so that readers can easily implement the verification programs by themselves.

Table of contents (12 chapters)

Table of contents (12 chapters)

Buy this book

eBook 93,08 €
price for Spain (gross)
  • ISBN 978-981-13-7669-6
  • Digitally watermarked, DRM-free
  • Included format: EPUB, PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover 114,39 €
price for Spain (gross)
  • ISBN 978-981-13-7668-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
  • The final prices may differ from the prices shown due to specifics of VAT rules
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Bibliographic Information

Bibliographic Information
Book Title
Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations
Authors
Series Title
Springer Series in Computational Mathematics
Series Volume
53
Copyright
2019
Publisher
Springer Singapore
Copyright Holder
Springer Nature Singapore Pte Ltd.
eBook ISBN
978-981-13-7669-6
DOI
10.1007/978-981-13-7669-6
Hardcover ISBN
978-981-13-7668-9
Series ISSN
0179-3632
Edition Number
1
Number of Pages
XIII, 467
Number of Illustrations
222 b/w illustrations, 11 illustrations in colour
Topics