This volume provides an introduction to the analytical and numerical
aspects of partial differential equations (PDEs). It unifies an
analytical and computational approach for these; the qualitative
behaviour of solutions being established using classical concepts:
maximum principles and energy methods. Notable inclusions are the
treatment of irregularly shaped boundaries, polar coordinates and the
use of flux-limiters when approximating hyperbolic conservation laws.
The numerical analysis of difference schemes is rigorously developed
using discrete maximum principles and discrete Fourier analysis. A novel
feature is the inclusion of a chapter containing projects, intended for
either individual or group study, that cover a range of topics such as
parabolic smoothing, travelling waves, isospectral matrices, and the
approximation of multidimensional advection-diffusion problems.
The underlying theory is illustrated by numerous examples and there are
around 300 exercises, designed to promote and test understanding. They
are starred according to level of difficulty. Solutions to odd-numbered
exercises are available to all readers while even-numbered solutions are
available to authorised instructors.
Written in an informal yet rigorous style, Essential Partial
Differential Equations is designed for mathematics undergraduates in
their final or penultimate year of university study, but will be equally
useful for students following other scientific and engineering
disciplines in which PDEs are of practical importance. The only
prerequisite is a familiarity with the basic concepts of calculus and
linear algebra.
