Numerical Verification Methods and ComputerAssisted Proofs for Partial Differential Equations
Authors: Nakao, Mitsuhiro T., Plum, Michael, Watanabe, Yoshitaka
Free Preview Is the first published monograph on computerassisted 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 computerassisted proofs
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Recently, various mathematical problems have been solved by computerassisted proofs, among them the Kepler conjecture, the existence of chaos, the existence of the Lorenz attractor, the famous fourcolor problem, and more. In many cases, computerassisted proofs have the remarkable advantage (compared with a “theoretical” proof) of providing accurate quantitative information.The authors have been working more than a quarter century to establish the verified computations of solutions for partial differential equations, mainly to the nonlinear elliptic problems of the form ∆u=f(x,u,∇u) with Dirichlet boundary conditions. Here, by "verified computation" is meant a computerassisted numerical approach to proving the existence of a solution in a close and explicit neighborhood of an approximate solution. Therefore, the quantitative information by the technique shown here should also be significant from the viewpoint of the a posteriori error estimates for approximate solution of concerned partial differential equations with mathematically rigorous sense.In this monograph, the authors describe a survey on the verified computations or computerassisted proofs for partial differential equations that they developed. In Part I, the methods mainly studied by authors Nakao and Watanabe are presented. These methods are based on the finite dimensional projection and the constructive a priori error estimates for the finite element approximation of the Poisson equation. In Part II, the computerassisted approaches via eigenvalue bounds developed by the second author, Plum, are explained in detail. The main task of this method consists of eigenvalue bounds for the corresponding nonlinear problems of the linearized operators. Some brief remarks are also given on other approaches in Part III. Each method in Parts I and II is followed by appropriate numerical examples that confirm the actual usefulness of the authors’ methods. Also in some examples the practical computer algorithms are supplied so that readers can easily implement the verification program by themselves.
 Table of contents (12 chapters)


Basic Principle of the Verification
Pages 342

NewtonType Approaches in Finite Dimension
Pages 4371

InfiniteDimensional NewtonType Method
Pages 73101

Applications to the ComputerAssisted Proofs in Analysis
Pages 103131

Evolutional Equations
Pages 133176

Table of contents (12 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 Numerical Verification Methods and ComputerAssisted Proofs for Partial Differential Equations
 Authors

 Mitsuhiro T. Nakao
 Michael Plum
 Yoshitaka Watanabe
 Series Title
 Springer Series in Computational Mathematics
 Series Volume
 53
 Copyright
 2019
 Publisher
 Springer Singapore
 Copyright Holder
 Springer Nature Singapore Pte Ltd.
 eBook ISBN
 9789811376696
 DOI
 10.1007/9789811376696
 Hardcover ISBN
 9789811376689
 Series ISSN
 01793632
 Edition Number
 1
 Number of Pages
 XIII, 467
 Number of Illustrations
 42 b/w illustrations, 17 illustrations in colour
 Topics