Linear partial differential equations and Fourier theory

Marcus Pivato9780521199704, 0521199700

Do you want a rigorous book that remembers where PDEs come from and what they look like? This highly visual introduction to linear PDEs and initial/boundary value problems connects the math to physical reality, all the time providing a rigorous mathematical foundation for all solution methods. Readers are gradually introduced to abstraction – the most powerful tool for solving problems – rather than simply drilled in the practice of imitating solutions to given examples. The book is therefore ideal for students in mathematics and physics who require a more theoretical treatment than given in most introductory texts. Also designed with lecturers in mind, the fully modular presentation is easily adapted to a course of one-hour lectures, and a suggested 12-week syllabus is included to aid planning. Downloadable files for the hundreds of figures, hundreds of challenging exercises, and practice problems that appear in the book are available online, as are solutions.

---

Table of contents :
I Motivating Examples & Major Applications……Page 13
Sets and Functions……Page 14
Derivatives —Notation……Page 18
Complex Numbers……Page 19
Vector Calculus……Page 21
Divergence……Page 22
Even and Odd Functions……Page 23
Coordinate Systems and Domains……Page 24
Polar Coordinates on R2……Page 25
Cylindrical Coordinates on R3……Page 26
Differentiation of Function Series……Page 28
Differentiation of Integrals……Page 29
…in many dimensions……Page 32
The Heat Equation……Page 33
…in one dimension……Page 34
…in many dimensions……Page 35
Laplace’s Equation……Page 37
The Poisson Equation……Page 40
Practice Problems……Page 43
Properties of Harmonic Functions……Page 44
(*) Reaction and Diffusion……Page 46
(*) Conformal Maps……Page 48
The Laplacian and Spherical Means……Page 53
…in one dimension: the string……Page 56
…in two dimensions: the drum……Page 60
…in higher dimensions:……Page 62
Practice Problems……Page 63
Basic Framework……Page 65
The Schrödinger Equation……Page 68
Miscellaneous Remarks……Page 70
Some solutions to the Schrödinger Equation……Page 72
Stationary Schrödinger ; Hamiltonian Eigenfunctions……Page 76
The Momentum Representation……Page 84
Practice Problems……Page 85
II General Theory……Page 87
Functions and Vectors……Page 88
…on finite dimensional vector spaces……Page 90
…on C……Page 91
Eigenvalues, Eigenvectors, and Eigenfunctions……Page 93
Homogeneous vs. Nonhomogeneous……Page 94
Practice Problems……Page 96
Evolution vs. Nonevolution Equations……Page 98
…in two dimensions, with constant coefficients……Page 99
…in general……Page 100
Practice Problems……Page 102
Boundary Value Problems……Page 103
Dirichlet boundary conditions……Page 105
Neumann Boundary Conditions……Page 107
Mixed (or Robin) Boundary Conditions……Page 112
Periodic Boundary Conditions……Page 114
Uniqueness of Solutions……Page 116
Practice Problems……Page 121
III Fourier Series on Bounded Domains……Page 123
Inner Products (Geometry)……Page 124
L2 space (finite domains)……Page 125
Orthogonality……Page 128
L2 convergence……Page 132
Pointwise Convergence……Page 135
Uniform Convergence……Page 137
Convergence of Function Series……Page 142
Self-Adjoint Operators and their Eigenfunctions (*)……Page 145
Appendix: Symmetric Elliptic Operators……Page 152
Practice Problems……Page 153
Sine Series on [ 0, ]……Page 157
Cosine Series on [ 0, ]……Page 161
Sine Series on [ 0,L ]……Page 164
Cosine Series on [ 0,L ]……Page 166
Computing Fourier (co)sine coefficients……Page 167
Polynomials……Page 168
Step Functions……Page 172
Piecewise Linear Functions……Page 175
Differentiating Fourier (co)sine Series……Page 178
Practice Problems……Page 179
Real Fourier Series on [ -, ]……Page 181
Polynomials……Page 182
Step Functions……Page 183
Piecewise Linear Functions……Page 185
Differentiating Real Fourier Series……Page 186
(*)Relation between (Co)sine series and Real series……Page 187
(*) Complex Fourier Series……Page 189
(*) Relation between Real and Complex Fourier Coefficients……Page 190
…in two dimensions……Page 192
…in many dimensions……Page 198
Practice Problems……Page 200
IV BVPs in Cartesian Coordinates……Page 202
The Heat Equation on a Line Segment……Page 203
The Wave Equation on a Line (The Vibrating String)……Page 207
The Poisson Problem on a Line Segment……Page 211
Practice Problems……Page 212
The (nonhomogeneous) Dirichlet problem on a Square……Page 215
Homogeneous Boundary Conditions……Page 221
Nonhomogeneous Boundary Conditions……Page 226
Homogeneous Boundary Conditions……Page 229
Nonhomogeneous Boundary Conditions……Page 232
The Wave Equation on a Square (The Square Drum)……Page 233
Practice Problems……Page 236
BVP’s on a Cube……Page 239
The Heat Equation on a Cube……Page 240
The (nonhomogeneous) Dirichlet problem on a Cube……Page 242
The Poisson Problem on a Cube……Page 244
V BVPs in other Coordinate Systems……Page 246
Introduction……Page 247
Polar Harmonic Functions……Page 248
Boundary Value Problems on a Disk……Page 251
Boundary Value Problems on a Codisk……Page 256
Boundary Value Problems on an Annulus……Page 259
Poisson’s Solution to the Dirichlet Problem on the Disk……Page 262
Bessel’s Equation; Eigenfunctions of in Polar Coordinates……Page 264
Boundary conditions; the roots of the Bessel function……Page 266
Initial conditions; Fourier-Bessel Expansions……Page 269
The Poisson Equation in Polar Coordinates……Page 270
The Heat Equation in Polar Coordinates……Page 272
The Wave Equation in Polar Coordinates……Page 274
The power series for a Bessel Function……Page 276
Properties of Bessel Functions……Page 280
Practice Problems……Page 285
VI Miscellaneous Solution Methods……Page 288
…in Cartesian coordinates on R2……Page 290
…in Cartesian coordinates on RD……Page 292
…in polar coordinates: Bessel’s Equation……Page 293
…in spherical coordinates: Legendre’s Equation……Page 295
Separated vs. Quasiseparated……Page 304
The Polynomial Formalism……Page 305
Boundedness……Page 307
Boundary Conditions……Page 308
Introduction……Page 310
…in one dimension……Page 313
…in many dimensions……Page 317
…in one dimension……Page 319
…in many dimensions……Page 326
Poisson’s Solution (Dirichlet Problem on the Half-plane)……Page 327
(*) Properties of Convolution……Page 331
Unbounded Domain……Page 333
Bounded Domain……Page 339
Poisson’s Solution (Dirichlet Problem on the Disk)……Page 342
Practice Problems……Page 344
VII Fourier Transforms on Unbounded Domains……Page 348
One-dimensional Fourier Transforms……Page 349
Properties of the (one-dimensional) Fourier Transform……Page 353
Two-dimensional Fourier Transforms……Page 359
Three-dimensional Fourier Transforms……Page 361
Fourier (co)sine Transforms on the Half-Line……Page 363
Practice Problems……Page 364
Fourier Transform Solution……Page 367
Fourier Transform Solution……Page 370
Poisson’s Spherical Mean Solution; Huygen’s Principle……Page 373
The Dirichlet Problem on a Half-Plane……Page 376
Impulse-Response solution……Page 377
PDEs on the Half-Line……Page 378
(*) The Big Idea……Page 379
Practice Problems……Page 380
Solutions……Page 383
Bibliography……Page 411
Notation……Page 413
Useful Formulae……Page 430