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Introduction to Partial Differential Equations (Undergraduate Texts in Mathematics)

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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.

Author(s): Peter J. Olver
Edition: 1
Publisher: Springer
Year: 2013

Language: English
Commentary: Solutions Manual Included
Pages: 661

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