Introduction to Partial Differential Equations (Undergraduate Texts in Mathematics)

Description

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, engineering, and elsewhere. The exposition carefully balances solution techniques, mathematical rigor, and significant applications, all illustrated by numerous examples. Extensive exercise sets appear at the end of almost every subsection, and include straightforward computational problems to develop and reinforce new techniques and results, details on theoretical developments and proofs, challenging projects both computational and conceptual, and supplementary material that motivates the student to delve further into the subject. No previous experience with the subject of partial differential equations or Fourier theory is assumed, the main prerequisites being undergraduate calculus, both one- and multi-variable, ordinary differential equations, and basic linear algebra. While the classical topics of separation of variables, Fourier analysis, boundary value problems, Green's functions, and special functions continue to form the core of an introductory course, the inclusion of nonlinear equations, shock wave dynamics, symmetry and similarity, the Maximum Principle, financial models, dispersion and solutions, Huygens' Principle, quantum mechanical systems, and more make this text well attuned to recent developments and trends in this active field of contemporary research. Numerical approximation schemes are an important component of any introductory course, and the text covers the two most basic approaches: finite differences and finite elements. Preface Table of Contents Chapter 1 What Are Partial Differential Equations? Exercises Classical Solutions Initial Conditions and Boundary Conditions Exercises Linear and Nonlinear Equations Exercises Chapter 2 Linear and Nonlinear Waves 2.1 StationaryWaves Exercises 2.2 Transport and TravelingWaves Uniform Transport Transport with Decay Exercises Nonuniform Transport Exercises 2.3 Nonlinear Transport and Shocks Shock Dynamics More General Wave Speeds Exercises 2.4 TheWave Equation: d’Alembert’s Formula d’Alembert’s Solution External Forcing and Resonance Exercises Chapter 3 Fourier Series 3.1 Eigensolutions of Linear Evolution Equations The Heated Ring Exercises 3.2 Fourier Series Exercises Periodic Extensions Exercises Piecewise Continuous Functions Exercises The Convergence Theorem Exercises Even and Odd Functions Exercises Complex Fourier Series Exercises 3.3 Differentiation and Integration Integration of Fourier Series Differentiation of Fourier Series Exercises 3.4 Change of Scale Exercises 3.5 Convergence of Fourier Series Pointwise and Uniform Convergence Exercises Smoothness and Decay Exercises Hilbert Space Convergence in Norm Completeness Pointwise Convergence Exercises Chapter 4 Separation of Variables 4.1 The Diffusion and Heat Equations The Heat Equation Smoothing and Long–Time Behavior The Heated Ring Redux Inhomogeneous Boundary Conditions Robin Boundary Conditions The Root Cellar Problem Exercises 4.2 TheWave Equation Separation of Variables and Fourier Series Solutions Exercises The d’Alembert Formula for Bounded Intervals Exercises 4.3 The Planar Laplace and Poisson Equations Exercises Separation of Variables Exercises Polar Coordinates Averaging, the Maximum Principle, and Analyticity Exercises 4.4 Classification of Linear Partial Differential Equations Exercises Characteristics and the Cauchy Problem Exercises Chapter 5 Finite Differences 5.1 Finite Difference Approximations Exercises 5.2 Numerical Algorithms for the Heat Equation Stability Analysis Implicit and Crank–Nicolson Methods Exercises 5.3 Numerical Algorithms for First–Order Partial Differential Equations The CFL Condition Upwind and Lax–Wendroff Schemes Exercises 5.4 Numerical Algorithms for theWave Equation Exercises 5.5 Finite Difference Algorithms for the Laplace and Poisson Equations Solution Strategies Exercises Chapter 6 Generalized Functions and Green’s Functions 6.1 Generalized Functions The Delta Function Calculus of Generalized Functions Exercises The Fourier Series of the Delta Function Exercises 6.2 Green’s Functions for One–Dimensional Boundary Value Problems Exercises 6.3 Green’s Functions for the Planar Poisson Equation Calculus in the Plane The Two–Dimensional Delta Function The Green’s Function Exercises The Method of Images Exercises Chapter 7 Fourier Transforms 7.1 The Fourier Transform Exercises 7.2 Derivatives and Integrals Differentiation Integration Exercises 7.3 Green’s Functions and Convolution Solution of Boundary Value Problems Exercises Convolution Exercises 7.4 The Fourier Transform on Hilbert Space Quantum Mechanics and the Uncertainty Principle Exercises Chapter 8 Linear and Nonlinear Evolution Equations 8.1 The Fundamental Solution to the Heat Equation The Forced Heat Equation and Duhamel’s Principle The Black–Scholes Equation and Mathematical Finance Exercises 8.2 Symmetry and Similarity Similarity Solutions Exercises 8.3 The Maximum Principle Exercises 8.4 Nonlinear Diffusion Burgers’ Equation The Hopf–Cole Transformation Exercises 8.5 Dispersion and Solitons Linear Dispersion The Dispersion Relation Exercises The Korteweg–deVries Equation Exercises Chapter 9 A General Framework for Linear Partial Differential Equations 9.1 Adjoints Differential Operators Higher–Dimensional Operators Exercises The Fredholm Alternative Exercises 9.2 Self–Adjoint and Positive Definite Linear Functions Self–Adjointness Positive Definiteness Two–Dimensional Boundary Value Problems Exercises 9.3 Minimization Principles Sturm–Liouville Boundary Value Problems Exercises The Dirichlet Principle Exercises 9.4 Eigenvalues and Eigenfunctions Self–Adjoint Operators The Rayleigh Quotient Eigenfunction Series Green’s Functions and Completeness Exercises 9.5 A General Framework for Dynamics Evolution Equations Exercises Vibration Equations Forcing and Resonance Exercises The Schr¨odinger Equation Exercises Chapter 10 Finite Elements and Weak Solutions 10.1 Minimization and Finite Elements Exercises 10.2 Finite Elements for Ordinary Differential Equations Exercises 10.3 Finite Elements in Two Dimensions Triangulation Exercises The Finite Element Equations Exercises Assembling the Elements The Coefficient Vector and the Boundary Conditions Inhomogeneous Boundary Conditions Exercises 10.4 Weak Solutions Weak Formulations of Linear Systems Finite Elements Based on Weak Solutions Shock Waves as Weak Solutions Exercises Chapter 11 Dynamics of Planar Media 11.1 Diffusion in Planar Media Derivation of the Diffusion and Heat Equations Separation of Variables Qualitative Properties Inhomogeneous Boundary Conditions and Forcing The Maximum Principle Exercises 11.2 Explicit Solutions of the Heat Equation Heating of a Rectangle Exercises Heating of a Disk — Preliminaries 11.3 Series Solutions of Ordinary Differential Equations The Gamma Function Regular Points The Airy Equation Exercises Regular Singular Points Bessel’s Equation Exercises 11.4 The Heat Equation in a Disk, Continued Exercises 11.5 The Fundamental Solution to the Planar Heat Equation Exercises 11.6 The PlanarWave Equation Separation of Variables Vibration of a Rectangular Drum Vibration of a Circular Drum Exercises Scaling and Symmetry Exercises Chladni Figures and Nodal Curves Exercises Chapter 12 Partial Differential Equations in Space 12.1 The Three–Dimensional Laplace and Poisson Equations Self–Adjoint Formulation and Minimum Principle Exercises 12.2 Separation of Variables for the Laplace Equation Laplace’s Equation in a Ball The Legendre Equation and Ferrers Functions Spherical Harmonics Harmonic Polynomials Averaging, the Maximum Principle, and Analyticity Exercises 12.3 Green’s Functions for the Poisson Equation The Free–Space Green’s Function Bounded Domains and the Method of Images Exercises 12.4 The Heat Equation for Three–Dimensional Media Exercises Heating of a Ball Spherical Bessel Functions Exercises The Fundamental Solution of the Heat Equation Exercises 12.5 TheWave Equation for Three–Dimensional Media Vibration of Balls and Spheres Exercises 12.6 Spherical Waves and Huygens’ Principle Spherical Waves Kirchhoff’s Formula and Huygens’ Principle Exercises Descent to Two Dimensions Exercises 12.7 The Hydrogen Atom Bound States Atomic Eigenstates and Quantum Numbers Exercises Appendix A Complex Numbers Appendix B Linear Algebra B.1 Vector Spaces and Subspaces B.2 Bases and Dimension B.3 Inner Products and Norms B.4 Orthogonality B.5 Eigenvalues and Eigenvectors B.6 Linear Iteration B.7 Linear Functions and Systems References Symbol Index Author Index Subject Index Selected Solutions Manual for Instructors Chapter 1. What Are Partial Differential Equations? Chapter 2. Linear and Nonlinear Waves Chapter 3. Fourier Series Chapter 4. Separation of Variables Chapter 5. Finite Differences Chapter 6. Generalized Functions and Green’s Functions Chapter 7. Fourier Transforms Chapter 8. Linear and Nonlinear Evolution Equations Chapter 9. A General Framework for Linear Partial Differential Equations Chapter 10. Finite Elements and Weak Solutions Chapter 11. Dynamics of Planar Media Chapter 12. Partial Differential Equations in Space