Partial Differential Equations

Topics in Fourier Analysis

By M.W. Wong

© 2013 – Chapman and Hall/CRC

184 pages | 5 B/W Illus.

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Hardback: 9781466584013
pub: 2013-06-02
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About the Book

Partial Differential Equations: Topics in Fourier Analysis explains how to use the Fourier transform and heuristic methods to obtain significant insight into the solutions of standard PDE models. It shows how this powerful approach is valuable in getting plausible answers that can then be justified by modern analysis.

Using Fourier analysis, the text constructs explicit formulas for solving PDEs governed by canonical operators related to the Laplacian on the Euclidean space. After presenting background material, it focuses on:

  • Second-order equations governed by the Laplacian on Rn
  • The Hermite operator and corresponding equation
  • The sub-Laplacian on the Heisenberg group

Designed for a one-semester course, this text provides a bridge between the standard PDE course for undergraduate students in science and engineering and the PDE course for graduate students in mathematics who are pursuing a research career in analysis. Through its coverage of fundamental examples of PDEs, the book prepares students for studying more advanced topics such as pseudo-differential operators. It also helps them appreciate PDEs as beautiful structures in analysis, rather than a bunch of isolated ad-hoc techniques.

Reviews

"… nicely presented material … includes theorem with bibliographical references, exercises, and historical notes. … a useful complement to the existing literature for those who study partial differential equations."

Zentralblatt MATH 1286

Table of Contents

The Multi-Index Notation

The Gamma Function

Convolutions

Fourier Transforms

Tempered Distributions

The Heat Kernel

The Free Propagator

The Newtonian Potential

The Bessel Potential

Global Hypoellipticity in the Schwartz Space

The Poisson Kernel

The Bessel-Poisson Kernel

Wave Kernels

The Heat Kernel of the Hermite Operator

The Green Function of the Hermite Operator

Global Regularity of the Hermite Operator

The Heisenberg Group

The Sub-Laplacian and Twisted Laplacians

Convolutions on the Heisenberg Group

Wigner Transforms and Weyl Transforms

Spectral Analysis of Twisted Laplacians

Heat Kernels Related to the Heisenberg Group

Green Functions Related to the Heisenberg Group

Bibliography

Index

About the Author

M.W. Wong is a professor in and former chair of the Department of Mathematics and Statistics at York University in Toronto, Canada. From 2005 to 2009, he was president of the International Society for Analysis, its Applications and Computations (ISAAC).

Subject Categories

BISAC Subject Codes/Headings:
MAT003000
MATHEMATICS / Applied
MAT007000
MATHEMATICS / Differential Equations
MAT037000
MATHEMATICS / Functional Analysis