Mitsuhiro T. Nakao; Michael Plum; Yoshitaka Watana Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations 9789811376689

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Автор: Mitsuhiro T. Nakao; Michael Plum; Yoshitaka Watana
Название: Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations
ISBN: 9789811376689
Издательство: Springer
Классификация:

Основы высшей математики и математический анализ

Дифференциальное исчисление и дифференциальные уравнения

Вычислительная математика

Математическое моделирование

Математическая теория вычислений

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ISBN-10: 9811376689
Обложка/Формат: Hardcover
Страницы: 467
Вес: 0.87 кг.
Дата издания: 2019
Серия: Springer Series in Computational Mathematics
Язык: English
Издание: 1st ed. 2019
Иллюстрации: 11 illustrations, color; 222 illustrations, black and white; xiii, 467 p. 233 illus., 11 illus. in color.
Размер: 243 x 164 x 28
Читательская аудитория: Professional & vocational
Основная тема: Mathematics
Ссылка на Издательство: Link
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Поставляется из: Германии
Описание: 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.
Дополнительное описание: 1. Basic principle of the verification.- 2. Newton-type approaches in finite dimension.- 3. Infinite dimensional Newton-type method.- 4. Applications to the computer-assisted proof in analysis.- 5. Evolutional equations.- 6. Eigenvalue enclosures for self

Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations, Mitsuhiro T. Nakao; Michael Plum; Yoshitaka Watana