Partial Differential Equations: An Introduction, 2nd Edition
December 2007, ©2008
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.
Chapter 1: Where PDEs Come From
1.1 What is a Partial Differential Equation?
1.2 First-Order Linear Equations
1.3 Flows, Vibrations, and Diffusions
1.4 Initial and Boundary Conditions
1.5 Well-Posed Problems
1.6 Types of Second-Order Equations
Chapter 2: Waves and Diffusions
2.1 The Wave Equation
2.2 Causality and Energy
2.3 The Diffusion Equation
2.4 Diffusion on the Whole Line
2.5 Comparison of Waves and Diffusions
Chapter 3: Reflections and Sources
3.1 Diffusion on the Half-Line
3.2 Reflections of Waves
3.3 Diffusion with a Source
3.4 Waves with a Source
3.5 Diffusion Revisited
Chapter 4: Boundary Problems
4.1 Separation of Variables, The Dirichlet Condition
4.2 The Neumann Condition
4.3 The Robin Condition
Chapter 5: Fourier Series
5.1 The Coefficients
5.2 Even, Odd, Periodic, and Complex Functions
5.3 Orthogonality and the General Fourier Series
5.5 Completeness and the Gibbs Phenomenon
5.6 Inhomogeneous Boundary Conditions
Chapter 6: Harmonic Functions
6.1 Laplace’s Equation
6.2 Rectangles and Cubes
6.3 Poisson’s Formula
6.4 Circles, Wedges, and Annuli
Chapter 7: Green’s Identities and Green’s
7.1 Green’s First Identity
7.2 Green’s Second Identity
7.3 Green’s Functions
7.4 Half-Space and Sphere
Chapter 8: Computation of Solutions
8.1 Opportunities and Dangers
8.2 Approximations of Diffusions
8.3 Approximations of Waves
8.4 Approximations of Laplace’s Equation
8.5 Finite Element Method
Chapter 9: Waves in Space
9.1 Energy and Causality
9.2 The Wave Equation in Space-Time
9.3 Rays, Singularities, and Sources
9.4 The Diffusion and Schrodinger Equations
9.5 The Hydrogen Atom
Chapter 10: Boundaries in the Plane and in Space
10.1 Fourier’s Method, Revisited
10.2 Vibrations of a Drumhead
10.3 Solid Vibrations in a Ball
10.5 Bessel Functions
10.6 Legendre Functions
10.7 Angular Momentum in Quantum Mechanics
Chapter 11: General Eigenvalue Problems
11.1 The Eigenvalues Are Minima of the Potential Energy
11.2 Computation of Eigenvalues
11.4 Symmetric Differential Operators
11.5 Completeness and Separation of Variables
11.6 Asymptotics of the Eigenvalues
Chapter 12: Distributions and Transforms
12.2 Green’s Functions, Revisited
12.3 Fourier Transforms
12.4 Source Functions
12.5 Laplace Transform Techniques
Chapter 13: PDE Problems for Physics
13.2 Fluids and Acoustics
13.4 Continuous Spectrum
13.5 Equations of Elementary Particles
Chapter 14: Nonlinear PDEs
14.1 Shock Waves
14.3 Calculus of Variations
14.4 Bifurcation Theory
14.5 Water Waves
A.1 Continuous and Differentiable Functions
A.2 Infinite Sets of Functions
A.3 Differentiation and Integration
A.4 Differential Equations
A.5 The Gamma Function
Answers and Hints to Selected Exercises
- Eigenvalue problems (Chapters 10 and 11) are covered in appropriate depth.
- Treatment of distributions and Green’s functions eliminates student confusion by giving instructors the option of going directly from Green's functions in Chapter 7 to distributions and Fourier transforms in Chapter 12.
- Numerous exercises, varying in difficulty.
- Frequent mention of wave propagation, heat and diffusion, electrostatics, and quantum mechanics puts PDE into context, which is especially important for engineering and science majors.
- Rational organization of material: from science to mathematics, from one dimension to multidimensions, from full-line to half-line to finite interval, etc.
- Companion solutions manual allows students to see detailed worked out solutions.
- Introduction to nonlinear PDEs (Chapter 14) provides the student with a taste of the most important problems studied today by researchers in mathematics and science.
- Provides appropriate introduction to numerical analysis (Chapter 8).
- Organization gives instructors flexibility in chapter coverage; for example, one can go from Chapter 6 to Chapters 7 and 12 (Green’s functions and distributions), Chapters 13 and 14 (science and nonlinear PDEs), Chapter 8 (numerical), Chapter 9 (waves), or to Chapter 10 (disks, spheres).
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