ABSTRACT

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 tex

chapter 1|6 pages

The Multi-Index Notation

chapter 2|4 pages

The Gamma Function

chapter 3|10 pages

Convolutions

chapter 4|12 pages

Fourier Transforms

chapter 5|10 pages

Tempered Distributions

chapter 6|10 pages

The Heat Kernel

chapter 7|10 pages

The Free Propagator

chapter 8|6 pages

The Newtonian Potential

chapter 9|4 pages

The Bessel Potential

chapter 11|6 pages

The Poisson Kernel

chapter 12|6 pages

The Bessel–Poisson Kernel

chapter 13|10 pages

Wave Kernels

chapter 14|8 pages

The Heat Kernel of the Hermite Operator

chapter 15|10 pages

The Green Function of the Hermite Operator

chapter 16|6 pages

Global Regularity of the Hermite Operator

chapter 17|10 pages

The Heisenberg Group

chapter 19|4 pages

Convolutions on the Heisenberg Group

chapter 20|6 pages

Wigner Transforms and Weyl Transforms

chapter 21|6 pages

Spectral Analysis of Twisted Laplacians