Partial differential equations : an introduction

Our understanding of the fundamental processes of the natural world is based to a large extent on partial differential equations (PDEs). The second edition of Partial Differential Equations provides an introduction to the basic properties of PDEs and the ideas and techniques that have proven useful in analyzing them. It provides the student a broad perspective on the subject, illustrates the incredibly rich variety of phenomena encompassed by it, and imparts a working knowledge of the most important techniques of analysis of the solutions of the equations.In this book mathematical jargon is minimized. Our focus is on the three most classical PDEs, the wave, heat and Lapace equations. Advanced concepts are introduced frequently but with the least possible technicalities. The book is flexibly designed for juniors, seniors or beginning graduate students in science, engineering or mathematics. Cover......Page 1 Title......Page 2 Title page......Page 3 Date-line......Page 4 Preface......Page 5 CONTENTS......Page 7 Title......Page 11 1.1* What Is a Partial Differential Equation?......Page 13 1.2* First-Order Linear Equations......Page 18 1.3* Flows, Vibrations, and Diffusions......Page 22 1.4* Initial and Boundary Conditions......Page 31 1.5 Well-Posed Problems......Page 37 1.6 Types of Second-Order Equations......Page 39 2.1* The Wave Equation......Page 44 2.2* Causality and Energy......Page 49 2.3* The Diffusion Equation......Page 53 2.4* Diffusion on the Whole Line......Page 57 2.5* Comparison of Waves and Diffusions......Page 64 3.1 Diffusion on the Half-Line......Page 67 3.2 Reflections of Waves......Page 71 3.3 Diffusion with a Source......Page 77 3.4 Waves with a Source......Page 81 3.5 Diffusion Revisited......Page 90 4.1* Separation of Variables, the Dirichlet Condition......Page 94 4.2* The Neumann Condition......Page 99 4.3* The Robin Condition......Page 102 5.1* The Coefficients......Page 113 5.2* Even, Odd, Periodic, and Complex Functions......Page 121 5.3* Orthogonality and General Fourier Series......Page 126 5.4* Completeness......Page 132 5.5 Completeness and the Gibbs Phenomenon......Page 144 5.6 Inhomogeneous Boundary Conditions......Page 152 6.1 * Laplace's Equation......Page 158 6.2* Rectangles and Cubes......Page 167 6.3* Poisson's Formula......Page 171 6.4 Circles, Wedges, and Annuli......Page 176 7.1 Green's First Identity......Page 181 7.2 Green's Second Identity......Page 188 7.3 Green's Functions......Page 190 7.4 Half-Space and Sphere......Page 193 8.1 Opportunities and Dangers......Page 201 8.2 Approximations of Diffusions......Page 205 8.3 Approximations of Waves......Page 213 8.4 Approximations of Laplace's Equation......Page 220 8.5 Finite Element Method......Page 224 9.1 Energy and Causality......Page 228 9.2 The Wave Equation in Space-Time......Page 234 9.3 Rays, Singularities, and Sources......Page 241 9.4 The Diffusion and Schrodinger Equations......Page 247 9.5 The Hydrogen Atom......Page 253 10.1 Fourier's Method, Revisited......Page 257 10.2 Vibrations of a Drumhead......Page 263 10.3 Solid Vibrations in a Ball......Page 269 10.4 Nodes......Page 276 10.5 Bessel Functions......Page 280 10.6 Legendre Functions......Page 286 10.7 Angular Momentum in Quantum Mechanics......Page 291 11.1 The Eigenvalues Are Minima of the Potential Energy......Page 295 11.2 Computation of Eigenvalues......Page 300 11.3 Completeness......Page 305 11.4 Symmetric Differential Operators......Page 309 11.5 Completeness and Separation of Variables......Page 313 11.6 Asymptotics of the Eigenvalues......Page 316 12.1 Distributions......Page 326 12.2 Green's Functions, Revisited......Page 332 12.3 Fourier Transforms......Page 337 12.4 Source Functions......Page 342 12.5 Laplace Transform Techniques......Page 346 13.1 Electromagnetism......Page 351 13.2 Fluids and Acoustics......Page 354 13.3 Scattering......Page 358 13.4 Continuous Spectrum......Page 362 13.5 Equations of Elementary Particles......Page 365 14.1 ShockWaves......Page 371 14.2 Solitons......Page 379 14.3 Calculus of Variations......Page 386 14.4 Bifurcation Theory......Page 391 A.1 Continuous and Differentiate Functions......Page 396 A.2 Infinite Series of Functions......Page 400 A.3 Differentiation and Integration......Page 402 A.4 Differential Equations......Page 405 A.5 The Gamma Function......Page 407 References......Page 409 Answers and Hints to Selected Exercises......Page 412 Index......Page 427 Publisher description: Covers the fundamental properties of partial differential equations (PDEs) and proven techniques useful in analyzing them. Uses a broad approach to illustrate the rich diversity of phenomena such as vibrations of solids, fluid flow, molecular structure, photon and electron interactions, radiation of electromagnetic waves encompassed by this subject as well as the role PDEs play in modern mathematics, especially geometry and analysis Containing both realistic exercises and advanced topics, this undergraduate introduction to the field provides an analysis of the Robin boundary condition and the need for Fourier expansions. Schrodinger equations are also discussed, to illustrate the connection with chemistry and physics.