Advertisement

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

  • Mitsuhiro T. Nakao
  • Michael Plum
  • Yoshitaka Watanabe
Book

Part of the Springer Series in Computational Mathematics book series (SSCM, volume 53)

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Verification by Finite-Dimensional Projection

    1. Front Matter
      Pages 1-1
    2. Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe
      Pages 3-42
    3. Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe
      Pages 43-71
    4. Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe
      Pages 73-101
    5. Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe
      Pages 103-131
    6. Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe
      Pages 133-176
  3. Computer-Assisted Proofs for Nonlinear Elliptic Boundary Value Problems via Eigenvalue Bounds

    1. Front Matter
      Pages 177-177
    2. Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe
      Pages 179-213
    3. Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe
      Pages 215-250
    4. Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe
      Pages 251-269
    5. Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe
      Pages 271-347
    6. Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe
      Pages 349-411
  4. Related Work and Tools

    1. Front Matter
      Pages 413-413
    2. Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe
      Pages 415-421
    3. Mitsuhiro T. Nakao, Michael Plum, Yoshitaka Watanabe
      Pages 423-445
  5. Back Matter
    Pages 447-467

About this book

Introduction

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.

Keywords

Numerical verification methods for nonlinear problems Computer-assisted proofs in analysis Nonlinear partial differential equations Mathematically rigorous a posteriori error estimates Numerical enclosure of solutions for PDEs

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

Bibliographic information

  • DOI https://doi.org/10.1007/978-981-13-7669-6
  • Copyright Information Springer Nature Singapore Pte Ltd. 2019
  • Publisher Name Springer, Singapore
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-981-13-7668-9
  • Online ISBN 978-981-13-7669-6
  • Series Print ISSN 0179-3632
  • Series Online ISSN 2198-3712
  • Buy this book on publisher's site