Partial Eifferential Equations
| Date: 21 April 2011, 10:38
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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).
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