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.
Author(s): Marcus Pivato
Edition: 1
Year: 2010
Language: English
Pages: 630
Cover page......Page 1
Half title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 17
Conventions in the text......Page 18
Acknowledgements......Page 19
Physical motivation......Page 20
Flat dependency lattice......Page 21
Many ‘practice problems’ (with complete solutions and source code available online)......Page 22
Appropriate rigour......Page 23
Expositional clarity......Page 24
Clear and explicit statements of solution techniques......Page 25
Suggested 12-week syllabus......Page 27
Part I Motivating examples and major applications......Page 31
1A(ii) …in many dimensions......Page 35
1B(i) …in one dimension......Page 37
1B(ii) …in many dimensions......Page 38
1C The Laplace equation......Page 41
1D The Poisson equation......Page 44
1E Properties of harmonic functions......Page 48
1G Reaction and diffusion......Page 51
1I Practice problems......Page 53
2A The Laplacian and spherical means......Page 55
2B(i) …in one dimension: the string......Page 59
2B(ii) …in two dimensions: the drum......Page 63
2B(iii) …in higher dimensions......Page 65
2C The telegraph equation......Page 66
2D Practice problems......Page 67
3A Basic framework......Page 69
3B The Schrödinger equation......Page 73
3C Stationary Schrödinger equation......Page 77
3E Practice problems......Page 86
Part II General theory......Page 89
4A Functions and vectors......Page 91
4B(i) …on finite-dimensional vector spaces......Page 93
4B(ii) …on C(Infinity)......Page 95
4B(iv) Eigenvalues, eigenvectors, and eigenfunctions......Page 97
4C Homogeneous vs.nonhomogeneous......Page 98
4D Practice problems......Page 100
5A Evolution vs. nonevolution equations......Page 103
5B Initial value problems......Page 104
5C Boundary value problems......Page 105
5C(i) Dirichlet boundary conditions......Page 106
5C(ii) Neumann boundary conditions......Page 110
5C(iii) Mixed (or Robin) boundary conditions......Page 113
5C(iv) Periodic boundary conditions......Page 115
5D Uniqueness of solutions......Page 117
5D(i) Uniqueness for the Laplace and Poisson equations......Page 118
5D(ii) Uniqueness for the heat equation......Page 121
5D(iii) Uniqueness for the wave equation......Page 124
5E(i) …in two dimensions, with constant coefficients......Page 127
5E(ii) …in general......Page 129
5F Practice problems......Page 130
Part III Fourier series on bounded domains......Page 135
6A Inner products......Page 137
6B L2-space......Page 139
6C(i) Complex L2-space......Page 143
6C(ii) Riemann vs. Lebesgue integrals......Page 144
6D Orthogonality......Page 146
6E Convergence concepts......Page 151
6E(i) L2-convergence......Page 152
6E(ii) Pointwise convergence......Page 156
6E(iii) Uniform convergence......Page 159
6E(iv) Convergence of function series......Page 164
6F Orthogonal and orthonormal bases......Page 166
6G Further reading......Page 167
6H Practice problems......Page 168
7A(i) Sine series on [0, π]......Page 171
7A(ii) Cosine series on [0, π]......Page 175
7B(i) Sine series on [ 0,L ]......Page 178
7B(ii) Cosine series on [ 0,L ]......Page 179
7C(i) Integration by parts......Page 180
7C(ii) Polynomials......Page 181
7C(iii) Step functions......Page 185
7C(iv) Piecewise linear functions......Page 189
7C(v) Differentiating Fourier (co)sine series......Page 192
7D Practice problems......Page 193
8A Real Fourier series on [-π, π]......Page 195
8B(i) Polynomials......Page 197
8B(ii) Step functions......Page 198
8B(iii) Piecewise linear functions......Page 200
8C Relation between (co)sine series and real series......Page 202
8D Complex Fourier series......Page 205
9A …in two dimensions......Page 211
9B …in many dimensions......Page 217
9C Practice problems......Page 225
10A Bessel, Riemann, and Lebesgue......Page 227
10B Pointwise convergence......Page 229
10C Uniform convergence......Page 236
Remarks......Page 238
10D L2-convergence......Page 239
10D(i) Integrable functions and step functions in L2[-π, π]......Page 240
10D(ii) Convolutions and mollifiers......Page 245
10D(iii) Proofs of Theorems 8A.1(a) and 10D.1......Page 253
Remarks......Page 255
Part IV BVP solutions via eigenfunction expansions......Page 257
11A The heat equation on a line segment......Page 259
11B The wave equation on a line (vibrating string)......Page 263
11C The Poisson problem on a line segment......Page 268
11D Practice problems......Page 270
12 Boundary value problems on a square......Page 273
12A The Dirichlet problem on a square......Page 274
12B(i) Homogeneous boundary conditions......Page 281
12B(ii) Nonhomogeneous boundary conditions......Page 285
12C(i) Homogeneous boundary conditions......Page 289
12C(ii) Nonhomogeneous boundary conditions......Page 292
12D The wave equation on a square (square drum)......Page 293
12E Practice problems......Page 296
13 Boundary value problems on a cube......Page 299
13A The heat equation on a cube......Page 300
13B The Dirichlet problem on a cube......Page 303
13C The Poisson problem on a cube......Page 306
14A Introduction......Page 309
14B(i) Polar harmonic functions......Page 310
14B(ii) Boundary value problems on a disk......Page 314
Physical interpretations......Page 319
14B(iv) Boundary value problems on an annulus......Page 323
14B(v) Poisson's solution to the Dirichlet problem on the disk......Page 326
14C(i) Bessel's equation; eigenfunctions of in polar coordinates......Page 328
14C(ii) Boundary conditions; the roots of the Bessel function......Page 331
14C(iii) Initial conditions; Fourier--Bessel expansions......Page 334
14D The Poisson equation in polar coordinates......Page 336
14E The heat equation in polar coordinates......Page 338
14F The wave equation in polar coordinates......Page 339
14G The power series for a Bessel function......Page 343
Remarks......Page 346
14H Properties of Bessel functions......Page 347
14I Practice problems......Page 352
15A Solution to Poisson, heat, and wave equation BVPs......Page 355
15B General solution to Laplace equation BVPs......Page 361
15C Eigenbases on Cartesian products......Page 367
15D General method for solving I/BVPs......Page 373
15E(i) Self-adjoint operators......Page 376
15E(ii) Eigenfunctions and eigenbases......Page 381
15E(iii) Symmetric elliptic operators......Page 384
15F Further reading......Page 385
Part V Miscellaneous solution methods......Page 387
16A Separation of variables in Cartesian coordinates on R2......Page 389
16B Separation of variables in Cartesian coordinates on RD......Page 391
16C Separation of variables in polar coordinates: Bessel’s equation......Page 393
16D Separation of variables in spherical coordinates: Legendre’s equation......Page 395
16E Separated vs. quasiseparated......Page 405
16F The polynomial formalism......Page 406
16G(i) Boundedness......Page 408
16G(ii) Boundary conditions......Page 409
17A Introduction......Page 411
17B(i) …in one dimension......Page 415
17B(ii) …in many dimensions......Page 419
17C(i) …in one dimension......Page 421
17C(ii) …in many dimensions......Page 427
17D d’Alembert’s solution (one-dimensional wave equation)......Page 428
17D(i) Unbounded domain......Page 429
17D(ii) Bounded domain......Page 434
17E Poisson's solution (Dirichlet problem on half-plane)......Page 438
17F Poisson's solution (Dirichlet problem on the disk)......Page 441
17G Properties of convolution......Page 444
17H Practice problems......Page 446
18A Holomorphic functions......Page 451
Remark......Page 453
18B Conformal maps......Page 458
18C Contour integrals and Cauchy's theorem......Page 470
18D Analyticity of holomorphic maps......Page 485
18E Fourier series as Laurent series......Page 489
18F Abel means and Poisson kernels......Page 496
18G Poles and the residue theorem......Page 499
18H Improper integrals and Fourier transforms......Page 507
18I Homological extension of Cauchy's theorem......Page 516
Part VI Fourier transforms on unbounded domains......Page 519
19A One-dimensional Fourier transforms......Page 521
19B Properties of the (one-dimensional) Fourier transform......Page 527
19C Parseval and Plancherel......Page 536
19D Two-dimensional Fourier transforms......Page 538
19E Three-dimensional Fourier transforms......Page 541
19F Fourier (co)sine transforms on the half-line......Page 544
19G Momentum representation and Heisenberg uncertainty......Page 545
19H Laplace transforms......Page 550
19J Practice problems......Page 557
20A(i) Fourier transform solution......Page 561
20A(ii) The Gaussian convolution formula, revisited......Page 564
20B(i) Fourier transform solution......Page 565
20B(ii) Poisson's spherical mean solution; Huygens' principle......Page 567
20C The Dirichlet problem on a half-plane......Page 570
20C(i) Fourier solution......Page 571
20C(ii) Impulse-response solution......Page 572
20E General solution to PDEs using Fourier transforms......Page 573
20F Practice problems......Page 575
A(i) Sets......Page 577
A(ii) Functions......Page 578
Appendix B: Derivatives - notation......Page 581
Appendix C: Complex numbers......Page 583
D(ii) Polar coordinates on R2......Page 587
D(iii) Cylindrical coordinates on R3......Page 588
D(v) What is a `domain'?......Page 589
…in many dimensions......Page 591
. . . in one dimension......Page 592
. . . in many dimensions......Page 593
. . . in two dimensions......Page 594
. . . in many dimensions......Page 595
Appendix F: Differentiation of function series......Page 599
Remarks......Page 600
Appendix G: Differentiation of integrals......Page 601
H(i) Taylor polynomials in one dimension......Page 603
H(ii) Taylor series and analytic functions......Page 604
H(iii) Using the Taylor series to solve ordinary differential equations......Page 605
H(iv) Taylor polynomials in two dimensions......Page 608
H(v) Taylor polynomials in many dimensions......Page 609
References......Page 611
Subject index......Page 615
Sets and domains......Page 627
Spaces of functions......Page 628
Norms and inner products......Page 629
Special functions......Page 630
