Transform Methods for Solving Partial Differential Equations (Symbolic & Numeric Computation)

✍ Scribed by Dean G. Duffy

• Publisher: Chapman and Hall/CRC
• Year: 2004
• Tongue: English
• Leaves: 713
• Edition: 2
• Category: Library

✦ Synopsis

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

TRANSFORM METHODS FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS, SECOND EDITON
Acknowledgments
Introduction
List of Definitions
Contents
Worked Solutions to Some of the Problems
Chapter 1: The Fundamentals
1.1 FOURIER TRANSFORMS
Example 1.1.1
Example 1.1.2
Example 1.1.3
Example 1.1.4
1.2 LAPLACE TRANSFORMS
Example 1.2.1
Example 1.2.2
Example 1.2.3
Example 1.2.4
Example 1.2.5
Example 1.2.6
Example 1.2.7
1.3 LINEAR ORDINARY DIFFERENTIAL EQUATIONS
Example 1.3.1
Example 1.3.2
Example 1.3.3
Problems
1.4 COMPLEX VARIABLES
Example 1.4.1
Example 1.4.2
Example 1.4.3
1.5 MULTIVALUED FUNCTIONS, BRANCH POINTS, BRANCH CUTS AND RIEMANN SURFACES
1.6 SOME EXAMPLES OF INTEGRATION THAT INVOLVE MULTIVALUED FUNCTIONS
Example 1.6.1
Example 1.6.2
Example 1.6.3
Example 1.6.4
Example 1.6.6
Problems
1.7 BESSEL FUNCTIONS
Example 1.7.1
Problems
1.8 WHAT ARE TRANSFORM METHODS?
Worked Solutions to Some of the Problems
Chapter 2: Methods Involving Single-Valued Laplace Transforms
2.1 INVERSION OF LAPLACE TRANSFORMS BY CONTOUR INTEGRATION
Example 2.1.1
Example 2.1.2
Example 2.1.3
Example 2.1.4
Example 2.1.5
Example 2.1.6
Problems
2.2 THE HEAT EQUATION
Example 2.2.1
Example 2.2.2
Example 2.2.3
Example 2.2.4
Example 2.2.5
Example 2.2.6: Moving boundary
Example 2.2.7
Example 2.2.8: Heat dissipation in disc brakes
Example 2.2.9: Asymptotic Analysis
Problems
Homogeneous Heat Equation on a Semi-Infinite, Cartesian Domain
Homogeneous Heat Equation on a Finite, Cartesian Domain
Homogeneous Heat Equation on a Cylindrical Domain
Homogeneous Heat Equation on a Spherical Domain
Nonhomogeneous Heat Equation on a Semi-Infinite Cartesian Domain
Nonhomogeneous Heat Equation on a Finite Cartesian Domain
Nonhomogeneous Heat Equation on a Cylindrical Domain
Systems of Heat Equations
Applications
2.3 THE WAVE EQUATION
Example 2.3.1
Example 2.3.2
Example 2.3.3
Example 2.3.4: Ray Expansion
Problems
Homogeneous Wave Equation on a Semi-Infinite or Infinite Cartesian Domain
Homogeneous Wave Equation on a Finite Cartesian Domain
Homogeneous Wave Equation on a Cylindrical Domain
Homogeneous Wave Equation on a Spherical Domain
Systems of Wave Equations
Applications
2.4 LAPLACE’S AND POISSON’S EQUATIONS
Example 2.4.1
Example 2.4.2
Problems
Papers Using Laplace Transforms to Solve Partial Differential Equations
Aquifers, Reservoirs and Porous Media
Diffusion
Electromagnetism
Fluid Dynamics
General
Geophysical Sciences
Heat Conduction
Magnetohydrodynamics
Other
Solid Mechanics
Thermoelasticity and Magnetoelasticity
Wave Equation
Worked Solutions to Some of the Problems
Chapter 3: Methods Involving Single-Valued Fourier and Hankel Transforms
3.1 INVERSION OF FOURIER TRANSFORMS BY CONTOUR INTEGRATION
Example 3.1.1
Example 3.1.2
Example 3.1.3
Example 3.1.4
Example 3.1.5
Problems
3.2 THE WAVE EQUATION
Example 3.2.1
Example 3.2.2
Example 3.2.3
Problems
3.3 THE HEAT EQUATION
Example 3.3.1
Example 3.3.2
Problems
3.4 LAPLACE’S EQUATION
Example 3.4.1
Example 3.4.2
Example 3.4.3
Example 3.4.4
Example 3.4.5: Mixed Boundary-Value Problem
Problems
3.5 THE SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS BY HANKEL TRANSFORMS
Example 3.5.1: Elastic Wave Equation
Example 3.5.2: Heat Equation
Example 3.5.3: Laplace’s Equation
Example 3.5.4: Poisson’s Equation
Example 3.5.5: Water Waves
Example 3.5.6: Mixed Boundary-Value Problems
Example 3.5.7: Heat Conduction Outside of a Cylinder
Problems
3.6 NUMERICAL INVERSION OF HANKEL TRANSFORMS
Papers Using Fourier Transforms to Solve Partial Differential Equations
Diffusion
Electromagnetism
Fluid Dynamics
General
Geophysical Sciences
Heat Conduction
Magnetohydrodynamics
Solid Mechanics
Thermoelasticity
Wave Equation
Papers Using Hankel Transforms to Solve Partial Differential Equations
Aquifers, Reservoirs and Porous Media
Electromagnetism
Fluid Dynamics
General
Geophysical Sciences
Heat Conduction
Solid Mechanics
Thermoelasticity
Worked Solutions to Some of the Problems
Chapter 4: Methods Involving Multivalued Laplace Transforms
4.1 INVERSION OF LAPLACE TRANSFORMS BY CONTOUR INTEGRATION
Example 4.1.1
Example 4.1.2
Example 4.1.3
Example 4.1.4
Example 4.1.5
Example 4.1.6
Example 4.1.7
Example 4.1.8
Example 4.1.9
Example 4.1.10
Example 4.1.11
Example 4.1.12
Example 4.1.13
Example 4.1.14: Asymptotic Expansions
Problems
4.2 NUMERICAL INVERSION OF LAPLACE TRANSFORMS
4.3 THE WAVE EQUATION
Example 4.3.1
Problems
4.4 THE HEAT EQUATION
Example 4.4.1
Example 4.4.2
Example 4.4.3
Problems
Papers Using Laplace Transforms to Solve Partial Differential Equations
Aquifers, Reservoirs and Porous Media
Diffusion
Electromagnetism
Fluid Dynamics
General
Geophysical Sciences
Heat Conduction
Magnetohydrodynamics
Other
Solid Mechanics
Thermoelasticity and Magnetoelasticity
Wave Equation
Worked Solutions to Some of the Problems
Chapter 5: Methods Involving Multivalued Fourier Transforms
5.1 INVERSION OF FOURIER TRANSFORMS BY CONTOUR INTEGRATION
Problems
5.2 NUMERICAL INVERSION OF FOURIER TRANSFORMS
5.3 SOLUTION OF PARTIAL DIFFERENTIAL EQUATIONS
Problems
Papers Using Fourier Transforms to Solve Partial Differential Equations
Electromagnetism
Fluid Dynamics
Geophysical Sciences
Magnetohydrodynamics
Other
Solid Mechanics
Thermoelasticity
Wave Equation
Worked Solutions to Some of the Problems
Chapter 6: The Joint Transform Method
6.1 THE WAVE EQUATION
Example 6.1.1
Example 6.1.2
Example 6.1.3
Example 6.1.4
Problems
6.2 THE HEAT AND OTHER PARTIAL DIFFERENTIAL EQUATIONS
Example 6.2.1
Example 6.2.2
Problems
6.3 INVERSION OF THE JOINT TRANSFORM BY CAGNIARD’S METHOD
Example 6.3.1
Example 6.3.2
Example 6.3.3
Problems
6.4 THE MODIFICATION OF CAGNIARD’S METHOD BY De HOOP
Example 6.4.1
Example 6.4.2
(a) Direct Wave
(b) Reflected Wave
(c) Transmitted Wave
Example 6.4.3
(a) Free-Space Solution
(b) Reflection
(c) Transmission
Example 6.4.4
Problems
Papers Using the Joint Transform Techniques
Aquifers, Reservoirs and Porous Media
Diffusion
Electromagnetism
Fluid Dynamics
General
Geophysical Sciences
Heat Conduction
Magnetohydrodynamics
Other
Solid Mechanics
Thermoelasticity and Magnetoelasticity
Wave Equation
Papers Using the Cagniard Technique
Electromagnetism
Fluid Dynamics
Geophysical Sciences
Magnetoelasticity
Solid Mechanics
Wave Equation
Papers Using the Cagniard—De Hoop Technique
Aquifers, Reservoirs and Porous Media
Electromagnetism
General
Geophysical Sciences
Heat Conduction
Solid Mechanics
Wave Motion
Worked Solutions to Some of the Problems
Chapter 7: The Wiener-Hopf Technique
Problems
7.1 THE WIENER-HOPF TECHNIQUE WHEN THE FACTORIZATION CONTAINS NO BRANCH POINTS
7.2 THE WIENER-HOPF TECHNIQUE WHEN THE FACTORIZATION CONTAINS BRANCH POINTS
Example 7.2.1
Problems
Example 7.2.2
Problems
Papers Using the Wiener—Hopf Technique
Aquifers, Reservoirs and Porous Media
Diffusion
Electromagnetism
Fluid Dynamics
General
Geophysical Sciences
Heat Conduction
Magnetohydrodynamics
Other
Solid Mechanics
Thermoelasticity
Wave Equation
Worked Solutions to Some of the Problems
Worked Solutions to Some of the Problems
Section 1.3
Section 1.6
Section 1.7
Section 2.1
Section 3.1
Section 4.1
Section 5.1