The Action Principle And Partial Differential Equations

Download or read book The Action Principle and Partial Differential Equations written by Demetrios Christodoulou and published by Princeton University Press. This book was released on 2000-01-17 with total page 332 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces new methods in the theory of partial differential equations derivable from a Lagrangian. These methods constitute, in part, an extension to partial differential equations of the methods of symplectic geometry and Hamilton-Jacobi theory for Lagrangian systems of ordinary differential equations. A distinguishing characteristic of this approach is that one considers, at once, entire families of solutions of the Euler-Lagrange equations, rather than restricting attention to single solutions at a time. The second part of the book develops a general theory of integral identities, the theory of "compatible currents," which extends the work of E. Noether. Finally, the third part introduces a new general definition of hyperbolicity, based on a quadratic form associated with the Lagrangian, which overcomes the obstacles arising from singularities of the characteristic variety that were encountered in previous approaches. On the basis of the new definition, the domain-of-dependence theorem and stability properties of solutions are derived. Applications to continuum mechanics are discussed throughout the book. The last chapter is devoted to the electrodynamics of nonlinear continuous media.

Download or read book The Action Principle and Partial Differential Equations. (AM-146), Volume 146 written by Demetrios Christodoulou and published by Princeton University Press. This book was released on 2016-03-02 with total page 328 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces new methods in the theory of partial differential equations derivable from a Lagrangian. These methods constitute, in part, an extension to partial differential equations of the methods of symplectic geometry and Hamilton-Jacobi theory for Lagrangian systems of ordinary differential equations. A distinguishing characteristic of this approach is that one considers, at once, entire families of solutions of the Euler-Lagrange equations, rather than restricting attention to single solutions at a time. The second part of the book develops a general theory of integral identities, the theory of "compatible currents," which extends the work of E. Noether. Finally, the third part introduces a new general definition of hyperbolicity, based on a quadratic form associated with the Lagrangian, which overcomes the obstacles arising from singularities of the characteristic variety that were encountered in previous approaches. On the basis of the new definition, the domain-of-dependence theorem and stability properties of solutions are derived. Applications to continuum mechanics are discussed throughout the book. The last chapter is devoted to the electrodynamics of nonlinear continuous media.

Download or read book Partial Differential Equations in Action written by Sandro Salsa and published by Springer. This book was released on 2015-04-24 with total page 714 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book is intended as an advanced undergraduate or first-year graduate course for students from various disciplines, including applied mathematics, physics and engineering. It has evolved from courses offered on partial differential equations (PDEs) over the last several years at the Politecnico di Milano. These courses had a twofold purpose: on the one hand, to teach students to appreciate the interplay between theory and modeling in problems arising in the applied sciences, and on the other to provide them with a solid theoretical background in numerical methods, such as finite elements. Accordingly, this textbook is divided into two parts. The first part, chapters 2 to 5, is more elementary in nature and focuses on developing and studying basic problems from the macro-areas of diffusion, propagation and transport, waves and vibrations. In turn the second part, chapters 6 to 11, concentrates on the development of Hilbert spaces methods for the variational formulation and the analysis of (mainly) linear boundary and initial-boundary value problems.

Download or read book Partial Differential Equations written by Walter A. Strauss and published by John Wiley & Sons. This book was released on 2007-12-21 with total page 467 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Download or read book Elliptic Partial Differential Equations written by Qing Han and published by American Mathematical Soc.. This book was released on 2011 with total page 161 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume is based on PDE courses given by the authors at the Courant Institute and at the University of Notre Dame, Indiana. Presented are basic methods for obtaining various a priori estimates for second-order equations of elliptic type with particular emphasis on maximal principles, Harnack inequalities, and their applications. The equations considered in the book are linear; however, the presented methods also apply to nonlinear problems.

Download or read book A Course on Partial Differential Equations written by Walter Craig and published by American Mathematical Soc.. This book was released on 2018-12-12 with total page 217 pages. Available in PDF, EPUB and Kindle. Book excerpt: Does entropy really increase no matter what we do? Can light pass through a Big Bang? What is certain about the Heisenberg uncertainty principle? Many laws of physics are formulated in terms of differential equations, and the questions above are about the nature of their solutions. This book puts together the three main aspects of the topic of partial differential equations, namely theory, phenomenology, and applications, from a contemporary point of view. In addition to the three principal examples of the wave equation, the heat equation, and Laplace's equation, the book has chapters on dispersion and the Schrödinger equation, nonlinear hyperbolic conservation laws, and shock waves. The book covers material for an introductory course that is aimed at beginning graduate or advanced undergraduate level students. Readers should be conversant with multivariate calculus and linear algebra. They are also expected to have taken an introductory level course in analysis. Each chapter includes a comprehensive set of exercises, and most chapters have additional projects, which are intended to give students opportunities for more in-depth and open-ended study of solutions of partial differential equations and their properties.

Download or read book Functional Analysis, Sobolev Spaces and Partial Differential Equations written by Haim Brezis and published by Springer Science & Business Media. This book was released on 2010-11-02 with total page 600 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.