Undergraduate Texts in Mathematics Peter J. Olver Introduction to Partial Differential Equations Undergraduate Texts in Mathematics Undergraduate Texts in Mathematics Series Editors: Sheldon Axler San Francisco State University, San Francisco, CA, USA Kenneth Ribet University of California, Berkeley, CA, USA Advisory Board: Colin Adams, Williams College, Williamstown, MA, USA Alejandro Adem, University of British Columbia, Vancouver, BC, Canada Ruth Charney, Brandeis University, Waltham, MA, USA Irene M. Gamba, The University of Texas at Austin, Austin, TX, USA Roger E. Howe, Yale University, New Haven, CT, USA David Jerison, Massachusetts Institute of Technology, Cambridge, MA, USA Jeffrey C.
Lagarias, University of Michigan, Ann Arbor, MI, USA Jill Pipher, Brown University, Providence, RI, USA Fadil Santosa, University of Minnesota, Minneapolis, MN, USA Amie Wilkinson, University of Chicago, Chicago, IL, USA Undergraduate Texts in Mathematics are generally aimed at third- and fourth-year undergraduate mathematics students at North American universities. These texts strive to provide students and teachers with new perspectives and novel approaches. The books include motivation that guides the reader to an appreciation of interrelations among different aspects of the subject. They feature examples that illustrate key concepts as well as exercises that strengthen understanding.
For further volumes: http://www.com/series/666 Peter J. Olver Introduction to Partial Differential Equations Peter J. Olver School of Mathematics University of Minnesota Minneapolis, MN USA ISSN 0172-6056 ISSN 2197-5604 (electronic) ISBN 978-3-319-02098-3 ISBN 978-3-319-02099-0 (eBook) DOI 10.1007/978-3-319-02099-0 Springer Cham Heidelberg New York Dordrecht London Library of Congress Control Number: 2013954394 Mathematics Subject Classification: 35-01, 42-01, 65-01 © Springer International Publishing Switzerland 2014 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.
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Printed on acid-free paper Springer is part of Springer Science+Business Media (www.com) To the memory of my father, Frank W. Olver (1924-2013) and mother, Grace E. Olver (née Smith, 1927-1980), whose love, patience, and guidance formed the heart of it all. Preface The momentous revolution in science precipitated by Isaac Newton’s calculus soon re- vealed the central role of partial differential equations throughout mathematics and its manifold applications.
Notable examples of fundamental physical phenomena modeled by partial differential equations, most of which are named after their discovers or early proponents, include quantum mechanics (Schrödinger, Dirac), relativity (Einstein), elec- tromagnetism (Maxwell), optics (eikonal, Maxwell–Bloch, nonlinear Schrödinger), fluid me- chanics (Euler, Navier–Stokes, Korteweg–de Vries, Kadomstev–Petviashvili), superconduc- tivity (Ginzburg–Landau), plasmas (Vlasov), magneto-hydrodynamics (Navier–Stokes + Maxwell), elasticity (Lamé, von Karman), thermodynamics (heat), chemical reactions (Kolmogorov–Petrovsky–Piskounov), finance (Black–Scholes), neuroscience (FitzHugh– Nagumo), and many, many more. The challenge is that, while their derivation as physi- cal models — classical, quantum, and relativistic — is, for the most part, well established, [57, 69], most of the resulting partial differential equations are notoriously difficult to solve, and only a small handful can be deemed to be completely understood. In many cases, the only means of calculating and understanding their solutions is through the design of so- phisticated numerical approximation schemes, an important and active subject in its own right. However, one cannot make serious progress on their numerical aspects without a deep understanding of the underlying analytical properties, and thus the analytical and numerical approaches to the subject are inextricably intertwined.
This textbook is designed for a one-year course covering the fundamentals of partial differential equations, geared towards advanced undergraduates and beginning graduate students in mathematics, science, and engineering. No previous experience with the subject is assumed, while the mathematical prerequisites for embarking on this course of study will be listed below. For many years, I have been teaching such a course to students from mathematics, physics, engineering, statistics, chemistry, and, more recently, biology, finance, economics, and elsewhere. Over time, I realized that there is a genuine need for a well-written, systematic, modern introduction to the basic theory, solution techniques, qualitative properties, and numerical approximation schemes for the principal varieties of partial differential equations that one encounters in both mathematics and applications.
It is my hope that this book will fill this need, and thus help to educate and inspire the next generation of students, researchers, and practitioners. While the classical topics of separation of variables, Fourier analysis, Green’s functions, and special functions continue to form the core of an introductory course, the inclusion of nonlinear equations, shock wave dynamics, dispersion, symmetry and similarity meth- ods, the Maximum Principle, Huygens’ Principle, quantum mechanics and the Schrödinger equation, and mathematical finance makes this book more in tune with recent developments and trends. Numerical approximation schemes should also play an essential role in an in- troductory course, and this text covers the two most basic approaches: finite differences and finite elements. vii viii Preface On the other hand, modeling and the derivation of equations from physical phenomena and principles, while not entirely absent, has been downplayed, not because it is unimpor- tant, but because time constraints limit what one can reasonably cover in an academic year’s course.
My own belief is that the primary purpose of a course in partial differential equations is to learn the principal solution techniques and to understand the underlying mathematical analysis. Thus, time devoted to modeling effectively lessens what can be ad- equately covered in the remainder of the course. For this reason, modeling is better left to a separate course that covers a wider range of mathematics, albeit at a more cursory level.) Nevertheless, this book continually makes contact with the physical applications that spawn the partial differential equations under consideration, and appeals to physical intuition and familiar phenomena to motivate, predict, and understand their mathematical properties, solutions, and applications. Nor do I attempt to cover stochastic differential equations — see [83] for this increasingly im- portant area — although I do work through one important by-product: the Black–Scholes equation, which underlies the modern financial industry.
I have tried throughout to bal- ance rigor and intuition, thus giving the instructor flexibility with their relative emphasis and time to devote to solution techniques versus theoretical developments. The course material has now been developed, tested, and revised over the past six years here at the University of Minnesota, and has also been used by several other universities in both the United States and abroad. It consists of twelve chapters along with two appendices that review basic complex numbers and some essential linear algebra. See below for further details on chapter contents and dependencies, and suggestions for possible semester and year-long courses that can be taught from the book.
Prerequisites The initial prerequisite is a reasonable level of mathematical sophistication, which includes the ability to assimilate abstract constructions and apply them in concrete situations. Some physical insight and familiarity with basic mechanics, continuum physics, elemen- tary thermodynamics, and, occasionally, quantum mechanics is also very helpful, but not essential. Since partial differential equations involve the partial derivatives of functions, the most fundamental prerequisite is calculus — both univariate and multivariate. Fluency in the basics of differentiation, integration, and vector analysis is absolutely essential.
Thus, the student should be at ease with limits, including one-sided limits, continuity, differentiation, integration, and the Fundamental Theorem. Key techniques include the chain rule, product rule, and quotient rule for differentiation, integration by parts, and change of variables in integrals. In addition, I assume some basic understanding of the convergence of sequences and series, including the standard tests — ratio, root, integral — along with Taylor’s theorem and elementary properties of power series. (On the other hand, Fourier series will be developed from scratch.) When dealing with several space dimensions, some familiarity with the key construc- tions and results from two- and three-dimensional vector calculus is helpful: rectangular (Cartesian), polar, cylindrical, and spherical coordinates; dot and cross products; partial derivatives; the multivariate chain rule; gradient, divergence, and curl; parametrized curves and surfaces; double and triple integrals; line and surface integrals, culminating in Green’s Theorem and the Divergence Theorem — as well as very basic point set topology: notions of Preface ix open, closed, bounded, and compact subsets of Euclidean space; the boundary of a domain and its normal direction; etc.
However, all the required concepts and results will be quickly reviewed in the text at the appropriate juncture: Section 6.3 covers the two-dimensional material, while Section 12.1 deals with the three-dimensional counterpart. Many solution techniques for partial differential equations, e., separation of variables and symmetry methods, rely on reducing them to one or more ordinary differential equa- tions. In order the make progress, the student should therefore already know how to find the general solution to first-order linear equations, both homogeneous and inhomogeneous, along with separable nonlinear first-order equations, linear constant-coefficient equations, particularly those of second order, and first-order linear systems with constant-coefficient matrices, in particular the role of eigenvalues and the construction of a basis of solutions. The student should also be familiar with initial value problems, including statements of the basic existence and uniqueness theorems, but not necessarily their proofs.
Basic ref- erences include [18, 20, 23], while more advanced topics can be found in [52, 54, 59]. On the other hand, while boundary value problems for ordinary differential equations play a central role in the analysis of partial differential equations, the book does not assume any prior experience, and will develop solution techniques from the beginning. Students should also be familiar with the basics of complex numbers, including real and imaginary parts; modulus and phase (or argument); and complex exponentials and Euler’s formula. These are reviewed in Appendix A.
In the numerical chapters, some familiarity with basic computer arithmetic, i., floating-point and round-off errors, is as- sumed. Also, on occasion, basic numerical root finding algorithms, e., Newton’s Method; numerical linear algebra, e., Gaussian Elimination and basic iterative methods; and nu- merical solution schemes for ordinary differential equations, e., Runge–Kutta Methods, are mentioned. Students who have forgotten the details can consult a basic numerical analysis textbook, e., [24, 60], or reference volume, e. Finally, knowledge of the basic results and conceptual framework provided by modern linear algebra will be essential throughout the text.
Students should already be on familiar terms with the fundamental concepts of vector space, both finite- and infinite-dimensional, linear independence, span, and basis, inner products, orthogonality, norms, and Cauchy– Schwarz and triangle inequalities, eigenvalues and eigenvectors, determinants, and linear systems. These are all covered in Appendix B; a more comprehensive and recommended reference is my previous textbook, [89], coauthored with my wife, Cheri Shakiban, which provides a firm grounding in the key ideas, results, and methods of modern applied linear algebra.