Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations (Springer Series in Computational Mathematics)

Nakao, Mitsuhiro T.; Plum, Michael; Watanabe, Yoshitaka

ISBN 10: 9811376719 ISBN 13: 9789811376719

Editorial: Springer, 2020

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This book is the first publication in the world as a monograph on the concerned research fieldNakao and Plum are pioneers of the numerical verification method of solution for PDEs
This book offers the basic principle of verification techniques for PDEs as well as interested applications for computer assisted proofs of nonlinear problems

De la contraportada:

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 actualusefulness of the authors’ methods. Also in some examples practical computer algorithms are supplied so that readers can easily implement the verification programs by themselves.

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T�tulo: Numerical Verification Methods and ...
Editorial: Springer
A�o de publicaci�n: 2020
Encuadernaci�n: Encuadernaci�n de tapa blanda
Condici�n: New

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

Mitsuhiro T. Nakao|Michael Plum|Yoshitaka Watanabe

Publicado por Springer Nature Singapore, 2020

ISBN 10: 9811376719 ISBN 13: 9789811376719

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Condici�n: New. Dieser Artikel ist ein Print on Demand Artikel und wird nach Ihrer Bestellung fuer Sie gedruckt. Is the first published monograph on computer-assisted proofsPresents pioneering work on the numerical verification method of solution for partial differential equationsProvides verification techniques. N� de ref. del art�culo: 449938573

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

Mitsuhiro T. Nakao (u. a.)

Publicado por Springer Singapore, 2020

ISBN 10: 9811376719 ISBN 13: 9789811376719

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Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations (Springer Series in Computational Mathematics)

Nakao, Mitsuhiro T.; Plum, Michael; Watanabe, Yoshitaka

Publicado por Springer, 2020

ISBN 10: 9811376719 ISBN 13: 9789811376719

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

Mitsuhiro T. Nakao

Publicado por Springer Nature Singapore Nov 2020, 2020

ISBN 10: 9811376719 ISBN 13: 9789811376719

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Taschenbuch. Condici�n: Neu. This item is printed on demand - it takes 3-4 days longer - Neuware -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 theauthors' methods. Also in some examples practical computer algorithms are supplied so that readers can easily implement the verification programs by themselves. 484 pp. Englisch. N� de ref. del art�culo: 9789811376719

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

Mitsuhiro T. Nakao

Publicado por Springer Nature Singapore, Springer Nature Singapore Nov 2020, 2020

ISBN 10: 9811376719 ISBN 13: 9789811376719

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Librería: buchversandmimpf2000, Emtmannsberg, BAYE, Alemania

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Taschenbuch. Condici�n: Neu. Neuware -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 theauthors� methods. Also in some examples practical computer algorithms are supplied so that readers can easily implement the verification programs by themselves.Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg 484 pp. Englisch. N� de ref. del art�culo: 9789811376719

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

Michael Plum, Yoshitaka Watanabe, Mitsuhiro T. Nakao

Publicado por Springer Verlag, Singapore, SG, 2020

ISBN 10: 9811376719 ISBN 13: 9789811376719

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Paperback. Condici�n: New. 2019 ed. 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 theauthors' methods. Also in some examples practical computer algorithms are supplied so that readers can easily implement the verification programs by themselves. N� de ref. del art�culo: LU-9789811376719

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

Mitsuhiro T. Nakao

Publicado por Springer Nature Singapore, Springer Nature Singapore, 2020

ISBN 10: 9811376719 ISBN 13: 9789811376719

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Librería: AHA-BUCH GmbH, Einbeck, Alemania

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Taschenbuch. Condici�n: Neu. Druck auf Anfrage Neuware - Printed after ordering - 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 theauthors' methods. Also in some examples practical computer algorithms are supplied so that readers can easily implement the verification programs by themselves. N� de ref. del art�culo: 9789811376719

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