Linear Partial Differential Equations and Fourier Theory

"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"--Provided by publisher. 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 Machine generated contents note: Preface; Notation; What's good about this book?; Suggested twelve-week syllabus; Part I. Motivating Examples and Major Applications: 1. Heat and diffusion; 2. Waves and signals; 3. Quantum mechanics; Part II. General Theory: 4. Linear partial differential equations; 5. Classification of PDEs and problem types; Part III. Fourier Series on Bounded Domains: 6. Some functional analysis; 7. Fourier sine series and cosine series; 8. Real Fourier series and complex Fourier series; 9. Mulitdimensional Fourier series; 10. Proofs of the Fourier convergence theorems; Part IV. BVP Solutions Via Eigenfunction Expansions: 11. Boundary value problems on a line segment; 12. Boundary value problems on a square; 13. Boundary value problems on a cube; 14. Boundary value problems in polar coordinates; 15. Eigenfunction methods on arbitrary domains; Part V. Miscellaneous Solution Methods: 16. Separation of variables; 17. Impulse-response methods; 18. Applications of complex analysis; Part VI. Fourier Transforms on Unbounded Domains: 19. Fourier transforms; 20. Fourier transform solutions to PDEs; Appendices; References; Index.

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.