Partial Differential Equations: Topics in Fourier Analysis, Second
Edition 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; and 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.
New to the Second Edition
- Three brand new chapters covering several topics in analysis not
explored in the first edition
- Complete revision of the text to correct errors, remove redundancies,
and update outdated material
- Expanded references and bibliography
- New and revised exercises.
