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Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations: 53 (Springer Series in Computational Mathematics, 53) - Hardcover
Synopsis
Recently, 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 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 computer-assisted 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 computer-assisted 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 computer-assisted 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.
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- PublisherSpringer
- Publication date2019
- ISBN 10 9811376689
- ISBN 13 9789811376689
- BindingHardcover
- LanguageEnglish
- Edition number1
- Number of pages480
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Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations (Springer Series in Computational Mathematics, 53)
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Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations
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Springer Nature Singapore Nov 2019, 2019
ISBN 10: 9811376689
ISBN 13: 9789811376689
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Buch. Condition: 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 the authors' methods. Also in some examples practical computer algorithms are supplied so that readers can easily implement the verification programs by themselves. 484 pp. Englisch. Seller Inventory # 9789811376689
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Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations
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Springer Nature Singapore, 2019
ISBN 10: 9811376689
ISBN 13: 9789811376689
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Condition: 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. Seller Inventory # 280959876
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Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations (Springer Series in Computational Mathematics, 53)
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Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations
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ISBN 13: 9789811376689
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Buch. Condition: 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. Seller Inventory # 9789811376689
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Numerical Verification Methods And Computer-Assisted Proofs For Partial Differential Equations
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Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations (Springer Series in Computational Mathematics, 53)
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Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations
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