Overview
- 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
Part of the book series: Springer Series in Computational Mathematics (SSCM, volume 53)
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Table of contents (12 chapters)
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Front Matter
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Verification by Finite-Dimensional Projection
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Front Matter
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Computer-Assisted Proofs for Nonlinear Elliptic Boundary Value Problems via Eigenvalue Bounds
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Front Matter
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Related Work and Tools
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Front Matter
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Back Matter
Keywords
About this book
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 theauthors’ methods. Also in some examples practical computer algorithms are supplied so that readers can easily implement the verification programs by themselves.
Reviews
“Each chapter of the book is written in an excellent, easy-to-understand way, both in terms of items, proofs, and codes. The reader himself can easily check the simplicity and correctness of the codes.” (Rózsa Horváth-Bokor, zbMATH 1462.65004, 2021)
Authors and Affiliations
Bibliographic Information
Book Title: Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations
Authors: Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe
Series Title: Springer Series in Computational Mathematics
DOI: https://doi.org/10.1007/978-981-13-7669-6
Publisher: Springer Singapore
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Singapore Pte Ltd. 2019
Hardcover ISBN: 978-981-13-7668-9Published: 20 November 2019
Softcover ISBN: 978-981-13-7671-9Published: 20 November 2020
eBook ISBN: 978-981-13-7669-6Published: 11 November 2019
Series ISSN: 0179-3632
Series E-ISSN: 2198-3712
Edition Number: 1
Number of Pages: XIII, 467
Number of Illustrations: 222 b/w illustrations, 11 illustrations in colour
Topics: Numerical Analysis, Mathematical Applications in Computer Science, Partial Differential Equations