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0 (hodnocen0 x )

(1.3) Půjčeno:5x 

BK

Cham : Springer, c2014

xxv, 635 s. : il. ; 26 cm

ISBN 978-3-319-02098-3 (váz.)

Undergraduate texts in mathematics, ISSN 0172-6056

Opravený 2. dotisk 2016

Obsahuje bibliografii na s. 589-594 a rejstříky

000252284

Undergraduate Texts in Mathematics // Peter J. Olver // Introduction to Partial Differential Equations // This textbook is designed for a one year course covering the fundamentals of partial // differential equations, geared towards advanced undergraduates and beginning // graduate students in mathematics, science, engineering, and elsewhere. The exposition // carefully balances solution techniques, mathematical rigor, and significant applications, // all illustrated by numerous examples. Extensive exercise sets appear at the end of almost // every subsection, and include straightforward computational problems to develop and // reinforce new techniques and results, details on theoretical developments and proofs, // challenging projects both computational and conceptual, and supplementary material // that motivates the student to delve further into the subject. // No previous experience with the subject of partial differential equations or Fourier // theory is assumed, the main prerequisites being undergraduate calculus, both oneand multi-variable, ordinary differential equations, and basic linear algebra. While the // classical topics of separation of variables, Fourier analysis, boundary value problems, // Green’s functions, and special functions continue to form the core of an introductory // course, the inclusion of nonlinear equations, shock wave dynamics, symmetry and // similarity, the Maximum Principle, financial models, dispersion and solitons, Huygens’

Principle, quantum mechanical systems, and more make this text well attuned to recent // developments and trends in this active field of contemporary research. Numerical // approximation schemes are an important component of any introductory course, and // the text covers the two most basic approaches: finite differences and finite elements. // Peter J. Olver is professor of mathematics at the University of Minnesota. His wideranging research interests are centered on the development of symmetry-based // methods for differential equations and their manifold applications. He is the author // of over 130 papers published in major scientific research journals as well as 4 other // books, including the definitive Springer graduate text, Applications of Lie Groups to // Differential Equations, and another undergraduate text, Applied Linear Algebra. // Mathematics // ISBN 978-3-319-02098-3 // 9 783319020983 // â–ş springer.com // 9783319020983 // 185520107109924000200710832636 // Table of Contents // Preface...vii // Chapter 1. What Are Partial Differential Equations? ...1 // Classical Solutions...4 // Initial Conditions and Boundary Conditions ...6 // Linear and Nonlinear Equations...8 // Chapter 2. Linear and Nonlinear Waves ...15 // 2.1. Stationary Waves...16 // 2.2. Transport and Traveling Waves...19 // Uniform Transport...19 // Transport with Decay...22 // Nonuniform Transport...24 // 2.3. Nonlinear Transport and Shocks...31 // Shock Dynamics...37 // â–  More General Wave

Speeds ...46 // 2.4. The Wave Equation: d’Alembert’s Formula ...49 // d’Alembert’s Solution ...50 // External Forcing and Resonance ...56 // Chapter 3. Fourier Series ...63 // 3.1. Eigensolutions of Linear Evolution Equations...64 // The Heated Ring...69 // 3.2. Fourier Series...72 // •Periodic Extensions...77 // Piecewise Continuous Functions... 79 // The Convergence Theorem...82 // Even and Odd Functions ...85 // Complex Fourier Series ...88 // 3.3. Differentiation and Integration...92 // Integration of Fourier Series ...92 // Differentiation of Fourier Series...94 // 3.4. Change of Scale...95 // 3.5. Convergence of Fourier Series ...98 // Pointwise and Uniform Convergence ...99 // Smoothness and Decay ...104 // Hilbert Space...106 // Convergence in Norm...109 // Completeness...112 // the Pointwise Convergence...115 // Chapter 4. Separation of Variables...121 // 4.1. The Diffusion and Heat Equations...122 // The Heat Equation...124 // Smoothing and Long Time Behavior...126 // The Heated Ring Redux...130 // Inhomogeneous Boundary Conditions ...133 // Robin Boundary Conditions ...134 // The Root Cellar Problem...136 // 4.2. The Wave Equation ...140 // Separation of Variables and Fourier Series Solutions ...140 // The d’Alembert Formula for Bounded Intervals ...146 // 4.3. The Planar Laplace and Poisson Equations ...152 // Separation of Variables...155 // Polar Coordinates...160 // Averaging, the Maximum Principle, and Analyticity ...167

// 4.4. Classification of Linear Partial Differential Equations ...172 // Characteristics and the Cauchy Problem...175 // Chapter 5. Finite Differences ...181 // 5.1. Finite Difference Approximations ...182 // 5.2. Numerical Algorithms for the Heat Equation ...186 // Stability Analysis...188 // Implicit and Crank-Nicolson Methods...190 // 5.3. Numerical Algorithms for First Order Partial Differential Equations . 195 // The CFL Condition...196 // Upwind and Lax-Wendroff Schemes...198 // 5.4. Numerical Algorithms for the Wave Equation ...201 // 5.5. Finite Difference Algorithms for the Laplace and Poisson Equations . 207 // Solution Strategies...211 // Chapter 6. Generalized Functions and Green’s Functions ...215 // 6.1. Generalized Functions...216 // The Delta Function...217 // Calculus of Generalized Functions...221 // The Fourier Series of the Delta Function ...229 // 6.2. Green’s Functions for One-Dimensional Boundary Value Problems . . 234 // 6.3. Green’s Functions for the Planar Poisson Equation ...242 // Calculus in the Plane ...242 // The Two-Dimensional Delta Function ...246 // The Green’s Function...248 // The Method of Images...256 // Table of Contents xxiii // Chapter 7. Fourier Transforms...263 // 7.1. The Fourier Transform ...263 // Concise Table of Fourier Transforms...272 // 7.2. Derivatives and Integrals...275 // Differentiation...275 // Integration...276 // 7.3. Green’s Functions and Convolution ...278 // Solution of Boundary

Value Problems...278 // Convolution...281 // 7.4. The Fourier Transform on Hilbert Space ...284 // Quantum Mechanics and the Uncertainty Principle ...286 // Chapter 8. Linear and Nonlinear Evolution Equations ...291 // 8.1. The Fundamental Solution to the Heat Equation ...292 // The Forced Heat Equation and Duhamel’s Principle ...296 // The Black-Scholes Equation and Mathematical Finance ... 299 // 8.2. Symmetry and Similarity ...305 // Similarity Solutions...308 // 8.3. The Maximum Principle ...312 // 8.4. Nonlinear Diffusion...315 // Burgers’ Equation...315 // The Hopf-Cole Transformation...317 // 8.5. Dispersion and Solitons...323 // Linear Dispersion...324 // The Dispersion Relation...330 // The Korteweg-de Vries Equation...333 // Chapter 9. A General Framework for // Linear Partial Differential Equations . . . 339 // 9.1. Adjoints...340 // Differential Operators ...342 // Higher-Dimensional Operators...345 // The Fredholm Alternative...350 // 9.2. Self-Adjoint and Positive Definite Linear Functions ...353 // Self-Adjointness...354 // Positive Definiteness...355 // Two-Dimensional Boundary Value Problems ...359 // 9.3. Minimization Principles...362 // Sturm-Liouville Boundary Value Problems ...363 // The Dirichlet Principle ...368 // 9.4. Eigenvalues and Eigenfunctions ...371 // Self-Adjoint Operators...371 // The Rayleigh Quotient... 375 // Eigenfunction Series...378 // Green’s Functions and Completeness ...379 // 9.5. A General Framework for Dynamics

// Evolution Equations... // Vibration Equations... // Forcing and Resonance . . . // The Schrodinger Equation .. // Chapter 10. Finite Elements and Weak Solutions... // 10.1. Minimization and Finite Elements... // 10.2. Finite Elements for Ordinary Differential Equations . . .. // 10.3. Finite Elements in Two Dimensions... // Triangulation... // The Finite Element Equations... // Assembling the Elements ... // The Coefficient Vector and the Boundary Conditions . // Inhomogeneous Boundary Conditions... // 10.4. Weak Solutions... // Weak Formulations of Linear Systems... // Finite Elements Based on Weak Solutions... // Shock Waves as Weak Solutions... // Chapter 11. Dynamics of Planar Media... // 11.1. Diffusion in Planar Media... // Derivation of the Diffusion and Heat Equations... // Separation of Variables... // Qualitative Properties ...-... // Inhomogeneous Boundary Conditions and Forcing... // The Maximum Principle... // 11.2. Explicit Solutions of the Heat Equation... // Heating of a Rectangle ... // Heating of a Disk - Preliminaries ... // 11.3. Series Solutions of Ordinary Differential Equations... // The Gamma Function... // Regular Points... // The Airy Equation ... // Regular Singular Points... // Bessel’s Equation... // 11.4. The Heat Equation in a Disk, Continued... // 11.5. The Fundamental Solution to the Planar Heat Equation... // 11.6. The Planar Wave Equation ... // Separation of Variables... // Vibration of a Rectangular Drum... // Vibration of a Circular

Drum... // Scaling and Symmetry... // Chladni Figures and Nodal Curves ... // Table of Contents xxv // Chapter 12. Partial Differential Equations in Space ...503 // 12.1. The Three-Dimensional Laplace and Poisson Equations ...504 // Self-Adjoint Formulation and Minimum Principle ...505 // 12.2. Separation of Variables for the Laplace Equation ...506 // Laplace’s Equation in a Ball...507 // The Legendre Equation and Ferrers Functions ...510 // Spherical Harmonics...517 // Harmonic Polynomials...519 // Averaging, the Maximum Principle, and Analyticity ...521 // 12.3. Green’s Functions for the Poisson Equation ...527 // The Free-Space Green’s Function...528 // Bounded Domains and the Method of Images ...531 // 12.4. The Heat Equation for Three-Dimensional Media ...535 // Heating of a Ball...537 // Spherical Bessel Functions...538 // The Fundamental Solution to the Heat Equation in Space . . . 543 // 12.5. The Wave Equation for Three-Dimensional Media ...545 // Vibration of Balls and Spheres ...547 // 12.6. Spherical Waves and Huygens’ Principle ...551 // Spherical Waves...551 // Kirchhoff’s Formula and Huygens’ Principle ...558 // Descent to Two Dimensions ...561 // 12.7. The Hydrogen Atom ...564 // Bound States... 565 // Atomic Eigenstates and Quantum Numbers ...567 // Correction to: Introduction to Partial Differential Equations ...Cl // Appendix A. Complex Numbers // 571 // Appendix B. Linear Algebra ...575 // B.1. Vector Spaces and Subspaces...575

// B.2. Bases and Dimension...576 // .3. Inner Products and Norms...578 // .4. Orthogonality...581 // .5. Eigenvalues and Eigenvectors...582 // .6. Linear Iteration...583 // .7. Linear Functions and Systems...585 // References...589 // Symbol Index ...595 // Author Index ...603 // Subject Index...607