Introduction to Partial Differential Equations

This book is intended as a Partial Differential Equations (PDEs) reference for individuals who already posses a firm understanding of ordinary differential equations and at least have a basic idea of what a partial derivative is.

This book is meant to be easily readable to engineers and scientists while still being (almost) interesting enough for mathematics students. Be advised that in depth proofs of such matters as series convergence, uniqueness, and existence will not be given; this fact will appall some and elate others. This book is meant more toward solving or at the very least extracting information out of problems involving partial differential equations.

The first few chapters are built to be especially simple to understand so that, say, the interested engineering undergraduate can benefit; however later on important and more mathematic topics such as vector spaces will be introduced and used. What follows is a quick intro for the uninitiated, with analogies to ordinary differential equations.

What is a Partial Differential Equation?

Ordinary differential equations (ODEs) arise naturally whenever a rate of change of some entity is known. This may be the rate of increase of a population, the rate of change of velocity, or maybe even the rate at which soldiers die on a battlefield. ODEs describe such changes of discrete entities. Respectively, this may be the capita of a population, the velocity of a particle, or the size of a military force. More than one entity may be described with more than one ODE. For example, cloth is very often simulated in computer graphics as a grid of particles interconnected by springs, with Newton's law (an ODE) applied to each "cloth particle". In three dimensions, this would result in 3 second order ODEs written and solved for each particle.

Partial differential equations (PDEs) are analogous to ODEs in that they involve rates of change; however, they differ in that they treat continuous media. For example, the cloth could just as well be considered to be some kind of continuous sheet. This approach would most likely lead to only 3 (maybe 4) partial differential equations, which would represent the entire continuous sheet, instead of a set of ODEs for each particle.

This continuum approach is a very different way of looking at things. It may or may not be favorable: in the case of cloth, the resulting PDE system would be too difficult to solve, and so the computer graphics industry goes with a particle based approach (but a prime counterexample is a fluid, which would be represented by a PDE system most of the time).