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Author O'Neil, Peter V
Title Beginning Partial Differential Equations
Imprint Somerset : John Wiley & Sons, Incorporated, 2014
©2014
book jacket
Edition 3rd ed
Descript 1 online resource (453 pages)
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
Series Pure and Applied Mathematics: a Wiley Series of Texts, Monographs and Tracts Ser
Pure and Applied Mathematics: a Wiley Series of Texts, Monographs and Tracts Ser
Note Intro -- Beginning Partial Differential Equations -- Copyright -- Contents -- Preface -- 1 First Ideas -- 1.1 Two Partial Differential Equations -- 1.1.1 The Heat, or Diffusion, Equati -- 1.1.2 The Wave Equation -- 1.2 Fourier Series -- 1.2.1 The Fourier Series of a Function -- 1.2.2 Fourier Sine and Cosine Series -- 1.3 Two Eigenvalue Problems -- 1.4 A Proof of the Fourier Convergence Theorem -- 1.4.1 The Role of Periodicity -- 1.4.2 Dirichlet's Formula -- 1.4.3 The Riemann-Lebesgue Lemma -- 1.4.4 Proof of the Convergence Theorem -- 2 Solutions of the Heat Equation -- 2.1 Solutions on an Interval [0, L] -- 2.1.1 Ends Kept at Temperature Zero -- 2.1.2 Insulated Ends -- 2.1.3 Ends at Different Temperatures -- 2.1.4 A Diffusion Equation with Additional Terms -- 2.1.5 One Radiating End -- 2.2 A Nonhomogeneous Problem -- 2.3 The Heat Equation in Two Space Variables -- 2.4 The Weak Maximum Principle -- 3 Solutions of the Wave Equation -- 3.1 Solutions on Bounded Intervals -- 3.1.1 Fixed Ends -- 3.1.2 Fixed Ends with a Forcing Term -- 3.1.3 Damped Wave Motion -- 3.2 The Cauchy Problem -- 3.2.1 d'Alembert's Solution -- 3.2.1.1 Forward and Backward Waves -- 3.2.2 The Cauchy Problem on a Half Line -- 3.2.3 Characteristic Triangles and Quadrilaterals -- 3.2.4 A Cauchy Problem with a Forcing Term -- 3.2.5 String with Moving Ends -- 3.3 The Wave Equation in Higher Dimensions -- 3.3.1 Vibrations in a Membrane with Fixed Frame -- 3.3.2 The Poisson Integral Solution -- 3.3.3 Hadamard's Method of Descent -- 4 Dirichlet and Neumann Problems -- 4.1 Laplace's Equation and Harmonic Functions -- 4.1.1 Laplace's Equation in Polar Coordinates -- 4.1.2 Laplace's Equation in Three Dimensions -- 4.2 The Dirichlet Problem for a Rectangle -- 4.3 The Dirichlet Problem for a Disk -- 4.3.1 Poisson's Integral Solution -- 4.4 Properties of Harmonic Functions
4.4.1 Topology of Rn -- 4.4.2 Representation Theorems -- 4.4.2.1 A Representation Theorem in R3 -- 4.4.2.2 A Representation Theorem in the Plane -- 4.4.3 The Mean Value Property and the Maximum Principle -- 4.5 The Neumann Problem -- 4.5.1 Existence and Uniqueness -- 4.5.2 Neumann Problem for a Rectangle -- 4.5.3 Neumann Problem for a Disk -- 4.6 Poisson's Equation -- 4. 7 Existence Theorem for a Dirichlet Problem -- 5 Fourier Integral Methods of Solution -- 5.1 The Fourier Integral of a Function -- 5.1.1 Fourier Cosine and Sine Integrals -- 5.2 The Heat Equation on the Real Line -- 5.2.1 A Reformulation of the Integral Solution -- 5.2.2 The Heat Equation on a Half Line -- 5.3 The Debate over the Age of the Earth -- 5.4 Burger's Equation -- 5.4.1 Traveling Wave Solutions of Burger's Equation -- 5.5 The Cauchy Problem for the Wave Equation -- 5.6 Laplace's Equation on Unbounded Domains -- 5.6.1 Dirichlet Problem for the Upper Half Plane -- 5.6.2 Dirichlet Problem for the Right Quarter Plane -- 5.6.3 A Neumann Problem for the Upper Half Plane -- 6 Solutions Using Eigenfunction Expansions -- 6.1 A Theory of Eigenfunction Expansions -- 6.1.1 A Closer Look at Expansion Coefficients -- 6.2 Bessel Functions -- 6.2.1 Variations on Bessel's Equation -- 6.2.2 Recurrence Relations -- 6.2.3 Zeros of Bessel Functions -- 6.2.4 Fourier-Bessel Expansions -- 6.3 Applications of Bessel Functions -- 6.3.1 Temperature Distribution in a Solid Cylinder -- 6.3.2 Vibrations of a Circular Drum -- 6.3.3 Oscillations of a Hanging Chain -- 6.3.4 Did Poe Get His Pendulum Right? -- 6.4 Legendre Polynomials and Applications -- 6.4.1 A Generating Function -- 6.4.2 A Recurrence Relation -- 6.4.3 Fourier-Legendre Expansions -- 6.4.4 Zeros of Legendre Polynomials -- 6.4.5 Steady-State Temperature in a Solid Sphere -- 6.4.6 Spherical Harmonics
7 Integral Transform Methods of Solution -- 7.1 The Fourier Transform -- 7.1.1 Convolution -- 7.1.2 Fourier Sine and Cosine Transforms -- 7.2 Heat and Wave Equations -- 7.2.1 The Heat Equation on the Real Line -- 7.2.2 Solution by Convolution -- 7.2.3 The Heat Equation on a Half Line -- 7.2.4 The Wave Equation by Fourier Transform -- 7.3 The Telegraph Equation -- 7.4 The Laplace Transform -- 7.4.1 Temperature Distribution in a Semi-Infinite Bar -- 7.4.2 A Diffusion Problem in a Semi-Infinite Medium -- 7.4.3 Vibrations in an Elastic Bar -- 8 First-Order Equations -- 8.1 Linear First-Order Equations -- 8.2 The Significance of Characteristics -- 8.3 The Quasi-Linear Equation -- 9 End Materials -- 9.1 Notation -- 9.2 Use of MAPLE -- 9.2.1 Numerical Computations and Graphing -- 9.2.2 Ordinary Differential Equations -- 9.2.3 Integral Transforms -- 9.2.4 Special Functions -- 9.3 Answers to Selected Problems -- Index
A broad introduction to PDEs with an emphasis on specialized topics and applications occurring in a variety of fields Featuring a thoroughly revised presentation of topics, Beginning Partial Differential Equations, Third Edition provides a challenging, yet accessible, combination of techniques, applications, and introductory theory on the subjectof partial differential equations. The new edition offers nonstandard coverageon material including Burger's equation, the telegraph equation, damped wavemotion, and the use of characteristics to solve nonhomogeneous problems. The Third Edition is organized around four themes: methods of solution for initial-boundary value problems; applications of partial differential equations; existence and properties of solutions; and the use of software to experiment with graphics and carry out computations. With a primary focus on wave and diffusion processes, Beginning Partial Differential Equations, Third Edition also includes: Proofs of theorems incorporated within the topical presentation, such as the existence of a solution for the Dirichlet problem The incorporation of Maple™ to perform computations and experiments Unusual applications, such as Poe's pendulum Advanced topical coverage of special functions, such as Bessel, Legendre polynomials, and spherical harmonics Fourier and Laplace transform techniques to solve important problems Beginning of Partial Differential Equations, Third Edition is an ideal textbook for upper-undergraduate and first-year graduate-level courses in analysis and applied mathematics, science, and engineering
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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2020. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries
Link Print version: O'Neil, Peter V. Beginning Partial Differential Equations Somerset : John Wiley & Sons, Incorporated,c2014 9781118629949
Subject Differential equations, Partial.;MATHEMATICS / Differential Equations.;Mathematics
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