Computer-assisted Proofs for a Nonlinear Laplace-Beltrami Equation on the Sphere PDF Download
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Author: Gabriel Duchesne
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
"In this study, we prove the existence and local uniqueness of radially symmetric solutions of nonlinear partial differential equations via a rigorous numerical method. We introduce the concepts of Banach spaces of geometrically decaying sequences and explore the domains of convergence of the Taylor and Chebyshev expansions. These notions constitute the basis for the Radii Polynomial Theorem on Banach spaces of infinite sequences, which is the main tool employed to obtain the proofs by using a combination of Taylor and Chebyshev coefficients of the solutions. We also introduce briefly the notions of interval analysis to justify the rigor of our computer-assisted results"--
Author: Gabriel Duchesne
Publisher:
ISBN:
Category :
Languages : en
Pages :
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Book Description
"In this study, we prove the existence and local uniqueness of radially symmetric solutions of nonlinear partial differential equations via a rigorous numerical method. We introduce the concepts of Banach spaces of geometrically decaying sequences and explore the domains of convergence of the Taylor and Chebyshev expansions. These notions constitute the basis for the Radii Polynomial Theorem on Banach spaces of infinite sequences, which is the main tool employed to obtain the proofs by using a combination of Taylor and Chebyshev coefficients of the solutions. We also introduce briefly the notions of interval analysis to justify the rigor of our computer-assisted results"--
Author: Mitsuhiro T. Nakao
Publisher: Springer Nature
ISBN: 9811376697
Category : Mathematics
Languages : en
Pages : 469
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Book Description
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.
Author: Oswald Fogelklou
Publisher:
ISBN: 9789150622690
Category :
Languages : en
Pages :
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Author: Walter A. Strauss
Publisher: John Wiley & Sons
ISBN: 0470054565
Category : Mathematics
Languages : en
Pages : 467
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Book Description
Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of Partial Differential Equations provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject, illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations. In this book mathematical jargon is minimized. Our focus is on the three most classical PDEs: the wave, heat and Laplace equations. Advanced concepts are introduced frequently but with the least possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics.
Author: American Mathematical Society
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 726
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Book Description
Author: Mark A. Pinsky
Publisher: American Mathematical Soc.
ISBN: 0821868896
Category : Mathematics
Languages : en
Pages : 545
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Book Description
Building on the basic techniques of separation of variables and Fourier series, the book presents the solution of boundary-value problems for basic partial differential equations: the heat equation, wave equation, and Laplace equation, considered in various standard coordinate systems--rectangular, cylindrical, and spherical. Each of the equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system, which makes the mathematics especially transparent. Bessel and Legendre functions are studied and used whenever appropriate throughout the text. The notions of steady-state solution of closely related stationary solutions are developed for the heat equation; applications to the study of heat flow in the earth are presented. The problem of the vibrating string is studied in detail both in the Fourier transform setting and from the viewpoint of the explicit representation (d'Alembert formula). Additional chapters include the numerical analysis of solutions and the method of Green's functions for solutions of partial differential equations. The exposition also includes asymptotic methods (Laplace transform and stationary phase). With more than 200 working examples and 700 exercises (more than 450 with answers), the book is suitable for an undergraduate course in partial differential equations.
Author: Willard Miller
Publisher: Cambridge University Press
ISBN: 9780521177399
Category : Mathematics
Languages : en
Pages : 0
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Book Description
Originally published in 1977, this volume is concerned with the relationship between symmetries of a linear second-order partial differential equation of mathematical physics, the coordinate systems in which the equation admits solutions via separation of variables, and the properties of the special functions that arise in this manner. Some group-theoretic twists in the ancient method of separation of variables that can be used to provide a foundation for much of special function theory are shown. In particular, it is shown explicitly that all special functions that arise via separation of variables in the equations of mathematical physics can be studied using group theory.
Author: E. C. Zachmanoglou
Publisher: Courier Corporation
ISBN: 048613217X
Category : Mathematics
Languages : en
Pages : 434
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Book Description
This text explores the essentials of partial differential equations as applied to engineering and the physical sciences. Discusses ordinary differential equations, integral curves and surfaces of vector fields, the Cauchy-Kovalevsky theory, more. Problems and answers.
Author: Haim Brezis
Publisher: Springer Science & Business Media
ISBN: 0387709142
Category : Mathematics
Languages : en
Pages : 600
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Book Description
This textbook is a completely revised, updated, and expanded English edition of the important Analyse fonctionnelle (1983). In addition, it contains a wealth of problems and exercises (with solutions) to guide the reader. Uniquely, this book presents in a coherent, concise and unified way the main results from functional analysis together with the main results from the theory of partial differential equations (PDEs). Although there are many books on functional analysis and many on PDEs, this is the first to cover both of these closely connected topics. Since the French book was first published, it has been translated into Spanish, Italian, Japanese, Korean, Romanian, Greek and Chinese. The English edition makes a welcome addition to this list.
Author:
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 1208
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