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. 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 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.