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Transform Methods for Solving Partial Differential Equations




ISBN 9781584884514
Published July 15, 2004 by Chapman and Hall/CRC
728 Pages 294 B/W Illustrations

 
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Book Description

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

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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