Transform Methods for Solving Partial Differential Equations, Second Edition

Transform Methods for Solving Partial Differential Equations, Second Edition

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Features

  • Provides a broad perspective on the applicability and use of transform methods
  • Achieves a balance between the details a student wants and the researchers'  need for fast answers  to specific questions
  • Includes several hundred well-crafted  exercises
  • Provides complete solutions to most of the exercises and intermediate results for the remainder
  • Summary

    Transform methods provide a bridge between the commonly used method of separation of variables and numerical techniques for solving linear partial differential equations. While in some ways similar to separation of variables, transform methods can be effective for a wider class of problems. Even when the inverse of the transform cannot be found analytically, numeric and asymptotic techniques now exist for their inversion, and because the problem retains some of its analytic aspect, one can gain greater physical insight than typically obtained from a purely numerical approach.

    Transform Methods for Solving Partial Differential Equations, Second Edition illustrates the use of Laplace, Fourier, and Hankel transforms to solve partial differential equations encountered in science and engineering. The author has expanded the second edition to provide a broader perspective on the applicability and use of transform methods and incorporated a number of significant refinements:

    New in the Second Edition:

    ·         Expanded scope that includes numerical methods and asymptotic techniques for inverting particularly complicated transforms

    ·         Discussions throughout the book that compare and contrast transform methods with separation of variables, asymptotic methods, and numerical techniques

    ·         Many added examples and exercises taken from a wide variety of scientific and engineering sources

    ·         Nearly 300 illustrations--many added to the problem sections to help readers visualize the physical problems

    ·         A revised format that makes the book easier to use as a reference: problems are classified according to type of region, type of coordinate system, and type of partial differential equation

    ·         Updated references, now arranged by subject instead of listed all together

    As reflected by the book's organization, content, and many examples, the author's focus remains firmly on applications. While the subject matter is classical, this book gives it a fresh, modern treatment that is exceptionally practical, eminently readable, and especially valuable to anyone solving problems in engineering and the applied sciences.

    Table of Contents

    THE FUNDAMENTALS
    Fourier Transforms
    Laplace Transforms
    Linear Ordinary Differential Equations
    Complex Variables
    Multivalued Functions, Branch Points, Branch Cuts, and Riemann Surfaces
    Some Examples of Integration which Involve Multivalued Functions
    Bessel Functions
    What are Transform Methods?
    METHODS INVOLVING SINGLE-VALUED LAPLACE TRANSFORMS
    Inversion of Laplace Transforms by Contour Integration
    The Heat Equation
    The Wave Equation
    Laplace's and Poisson's Equations
    Papers Using Laplace Transforms to Solve Partial Differential Equations
    METHODS INVOLVING SINGLE-VALUED FOURIER AND HANKEL TRANSFORMS
    Inversion of Fourier Transforms by Contour Integration
    The Wave Equation
    The Heat Equation
    Laplace's Equation
    The Solution of Partial Differential Equations by Hankel Transforms
    Numerical Inversion of Hankel Transforms
    Papers Using Fourier Transforms to Solve Partial Differential Equations
    Papers Using Hankel Transforms to Solve Partial Differential Equations
    METHODS INVOLVING MULTIVALUED LAPLACE TRANSFORMS
    Inversion of Laplace Transforms by Contour Integration
    Numerical Inversion of Laplace Transforms
    The Wave Equation
    The Heat Equation
    Papers Using Laplace Transforms to Solve Partial Differential Equations
    METHODS INVOLVING MULTIVALUED FOURIER TRANSFORMS
    Inversion of Fourier Transforms by Contour Integration
    Numerical Inversion of Fourier Transforms
    The Solution of Partial Differential Equations by Fourier Transforms
    Papers Using Fourier Transforms to Solve Partial Differential Equations
    THE JOINT TRANSFORM METHOD
    The Wave Equation
    The Heat and Other Partial Differential Equations
    Inversion of the Joint Transform by Cagniard's Method
    The Modification of Cagniard's Method by De Hoop
    Papers Using the Joint Transform Technique
    Papers Using the Cagniard Technique
    Papers Using the Cagniard-De Hoop Technique
    THE WIENER-HOPF TECHNIQUE
    The Wiener-Hopf Technique When the Factorization Contains No Branch Points
    The Wiener-Hopf Technique when the Factorization Contains Branch Points
    Papers Using the Wiener-Hopf Technique
    WORKED SOLUTIONS TO SOME OF THE PROBLEMS
    INDEX

    Editorial Reviews

    “This book provides a detailed account of the application of transform methods to solving linear PDEs in science and engineering. The main techniques studied are the Fourier and Laplace transform, Hankel transforms, and the Wiener-Hopf technique. In this second edition the references have been extended and numerical and asymptotic techniques are included. It provides a valuable source of concrete examples from diverse fields of applications with detailed solutions.”
      — M. Kunzinger

    “The discussed solutions typically involve quite difficult algebra, while non-trivial mathematical steps, such as a change of order of integration or expansion into infinite series (product) are not justified. This gives the book more of an ‘engineering’ flavour; nonetheless it is certainly of interest to a wider interest.”
    EMS Newsletter, Sept. 2005

    “The book contains many examples and exercises taken from a wide variety of scientific and engineering sources. Also numerical examples are presented using Matlab to help readers for better understanding of the physical problems.”
    —Studia Univ. “Babes-Bolyai” Mathematica, Sept. 2005

     
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