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• COVER
• PREFACE
• PREFACE TO SECOND EDITION
• 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
• 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
• 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
• 4 BOUNDARY PROBLEMS
• 4.1 SEPARATION OF VARIABLES, THE DIRICHLET CONDITION
• 4.2 THE NEUMANN CONDITION
• 4.3 THE ROBIN CONDITION
• 5 FOURIER SERIES
• 5.1 THE COEFFICIENTS
• 5.2 EVEN, ODD, PERIODIC, AND COMPLEX FUNCTIONS
• 5.3 ORTHOGONALITY AND GENERAL FOURIER SERIES
• 5.4 COMPLETENESS
• 5.5 COMPLETENESS AND THE GIBBS PHENOMENON
• 5.6 INHOMOGENEOUS BOUNDARY CONDITIONS
• 6 HARMONIC FUNCTIONS
• 6.1 LAPLACE’S EQUATION
• 6.2 RECTANGLES AND CUBES
• 6.3 POISSON’S FORMULA
• 6.4 CIRCLES, WEDGES, AND ANNULI
• 7 GREEN’S IDENTITIES AND GREEN’S FUNCTIONS
• 7.1 GREEN’S FIRST IDENTITY
• 7.2 GREEN’S SECOND IDENTITY
• 7.3 GREEN’S FUNCTIONS
• 7.4 HALF-SPACE AND SPHERE
• 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
• 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 SCHRŐDINGER EQUATIONS
• 9.5 THE HYDROGEN ATOM
• 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
• 10.4 NODES
• 10.5 BESSEL FUNCTIONS
• 10.6 LEGENDRE FUNCTIONS
• 10.7 ANGULAR MOMENTUM IN QUANTUM MECHANICS
• 11 GENERAL EIGENVALUE PROBLEMS
• 11.1 THE EIGENVALUES ARE MINIMA OF THE POTENTIAL ENERGY
• 11.2 COMPUTATION OF EIGENVALUES
• 11.3 COMPLETENESS
• 11.4 SYMMETRIC DIFFERENTIAL OPERATORS
• 11.5 COMPLETENESS AND SEPARATION OF VARIABLES
• 11.6 ASYMPTOTICS OF THE EIGENVALUES
• 12 DISTRIBUTIONS AND TRANSFORMS
• 12.1 DISTRIBUTIONS
• 12.2 GREEN’S FUNCTIONS, REVISITED
• 12.3 FOURIER TRANSFORMS
• 12.4 SOURCE FUNCTIONS
• 12.5 LAPLACE TRANSFORM TECHNIQUES
• 13 PDE PROBLEMS FROM PHYSICS
• 13.1 ELECTROMAGNETISM
• 13.2 FLUIDS AND ACOUSTICS
• 13.3 SCATTERING
• 13.4 CONTINUOUS SPECTRUM
• 13.5 EQUATIONS OF ELEMENTARY PARTICLES
• 14 NONLINEAR PDES
• 14.1 SHOCK WAVES
• 14.2 SOLITONS
• 14.3 CALCULUS OF VARIATIONS
• 14.4 BIFURCATION THEORY
• 14.5 WATER WAVES
• APPENDIX
• A.1 CONTINUOUS AND DIFFERENTIABLE FUNCTIONS
• A.2 INFINITE SERIES OF FUNCTIONS
• A.3 DIFFERENTIATION AND INTEGRATION
• A.4 DIFFERENTIAL EQUATIONS
• A.5 THE GAMMA FUNCTION
• REFERENCES
• INDEX
• END USER LICENSE AGREEMENT