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Beginning Partial Differential Equations, 3rd Edition

Book Description

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.

    Table of Contents

    1. Cover
    2. Half Title page
    3. Title page
    4. Copyright page
    5. Preface
    6. Chapter 1: First Ideas
      1. 1.1 Two Partial Differential Equations
      2. 1.2 Fourier Series
      3. 1.3 Two Eigenvalue Problems
      4. 1.4 A Proof of the Convergence Theorem
    7. Chapter 2: Solutions of the Heat Equation
      1. 2.1 Solutions on an Interval [0, L]
      2. 2.2 A Nonhomogeneous Problem
      3. 2.3 The Heat Equation in Two Space Variables
      4. 2.4 The Weak Maximum Principle
    8. Chapter 3: Solutions of the Wave Equation
      1. 3.1 Solutions on Bounded Intervals
      2. 3.2 The Cauchy Problem
      3. 3.3 The Wave Equation in Higher Dimensions
    9. Chapter 4: Dirichlet and Neumann Problems
      1. 4.1 Laplace’s Equation and Harmonic Functions
      2. 4.2 The Dirichlet Problem for a Rectangle
      3. 4.3 The Dirichlet Problem for a Disk
      4. 4.4 Properties of Harmonic Functions
      5. 4.5 The Neumann Problem
      6. 4.6 Poisson’s Equation
      7. 4.7 Existence Theorem for a Dirichlet Problem
    10. Chapter 5: Fourier Integral Methods of Solution
      1. 5.1 The Fourier Integral of a Function
      2. 5.2 The Heat Equation on the Real Line
      3. 5.3 The Debate over the Age of the Earth
      4. 5.4 Burgers’ Equation
      5. 5.5 The Cauchy Problem for the Wave Equation
      6. 5.6 Laplace’s Equation on Unbounded Domains
    11. Chapter 6: Solutions Using Eigenfunction Expansions
      1. 6.1 A Theory of Eigenfunction Expansions
      2. 6.2 Bessel Functions
      3. 6.3 Applications of Bessel Functions
      4. 6.4 Legendre Polynomials and Applications
    12. Chapter 7: Integral Transform Methods of Solution
      1. 7.1 The Fourier Transform
      2. 7.2 Heat and Wave Equations
      3. 7.3 The Telegraph Equation
      4. 7.4 The Laplace Transform
    13. Chapter 8: First-Order Equations
      1. 8.1 Linear First-Order Equations
      2. 8.2 The Significance of Characteristics
      3. 8.3 The Quasi-Linear Equation
    14. Chapter 9: End Materials
      1. 9.1 Notation
      2. 9.2 Use of MAPLE
      3. 9.3 Answers to Selected Problems
    15. Index