Advanced Mathematics for Engineers and Scientists/Introduction to Partial Differential Equations

This book is intended as a Partial Differential Equations (PDEs) reference for individuals who already possess a firm understanding of ordinary differential equations and have at least 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 mathematical 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).
While PDEs may not be straightforward to solve on a computer, they have a major advantage over ODEs when applicable: it is nearly impossible to gain any analytical insight from a huge system of particles, while a relatively small PDE system can reveal much insight, even if it won't yield an analytic solution.

But PDEs don't strictly describe continuum mechanics. As with anything mathematical, they are what you make of them.

The Character of Partial Differential Equations

The solution of an ODE can be represented as a function of one variable. For example, the position of the Earth may be represented by coordinates with respect to, say, the sun, and each of these coordinates would be functions of time. Note that the effects of other celestial bodies would certainly affect the solution, but it would still be expressible strictly as a function of time.

The solution of a PDE will, in general, depend on more than one variable. An example is a vibrating string: the deflection of the string will depend both on time and which part of the string you're looking at.

The solution of an ODE is called a trajectory. It may be represented graphically by one or more curves. The solution of a PDE, however, could be a surface, a volume, or something else, depending on how many variables are involved and how they're interpreted.

In general, PDEs are complicated to solve. Concepts such as separation of variables or integral transformations tend to work very differently. One significant difficulty is that the solution of a PDE depends very strongly on the initial/boundary conditions (ICs/BCs). An ODE typically yields a general solution, which involves one or more constants which may be determined from one or more ICs/BCs. PDEs, however, do not easily yield such general solutions. A solution method that works for one initial boundary value problem (IBVP) may be useless for a different IBVP.

PDEs tend to be more difficult to solve numerically as well. Most of the time, an ODE can be expressed in terms of its highest order derivative, and can be solved on a computer very easily with knowledge of the ICs (boundary value problems are a little more complicated), using well established and more or less generally applicable methods, such as Runge Kutta (RK). With this in mind, an ODE may be solved quickly by entering the equation and its ICs/BCs into the right application and pressing the solve button. An IBVP for a PDE, however, will typically require its own specialized solution, and it may take much effort to make the solution more than, say, second order accurate.

An Early Example

Many of the concepts of the previous section may be summarized in this example. We won't deal with the PDE just yet.

Consider heat flow along a laterally insulated rod. In other words, the heat is only flowing along the rod but not into the surrounding air. Let's call the temperature of the rod $u$, and let $u=u(x,t)$, where $t$ is time and $x$ represents the position along the rod. As the temperature depends both on time and position along the rod, this is exactly what $u=u(x,t)$ says. It is the change of the heat distribution over time. See the graphic below to get an idea.
Let's say that the rod has unitless length $1$, and that its initial temperature (again unitless) is known to be $u(x,0)=\sin(\pi x)$. This states the initial condition, which depends on $x$. The function is a simple hump between 0 and 1. Check for yourself with maxima (http://maxima.sourceforge.net or on android): plot2d(sin(x*%pi),[x,0,1])
Let's also say that the temperature is somehow fixed to $0$ at both ends of the rod, i.e. at $x=0$ and at $x=1$. This would result in $u(0,t)=u(1,t)=0$, which specifies boundary conditions. The BCs state that for all t, $u=0$ at $x=0$ and $x=1$.

A PDE can be written to describe the situation. This and the IC/BCs form an initial boundary value problem (IBVP). The solution to this IBVP is (with a physical constant taken to be $1$):

$u(x,t)=e^{-\pi ^{2}t}\sin(\pi x)\,$

Note that:

$u(x,0)=e^{-\pi ^{2}0}\sin(\pi x)=\sin(\pi x)\qquad {\mbox{(it satisfies the IC)}}\,$

$u(0,t)=e^{-\pi ^{2}t}\sin(\pi \cdot 0)=0\,$

$u(1,t)=e^{-\pi ^{2}t}\sin(\pi \cdot 1)=0\qquad {\mbox{(it satisfies the BCs)}}\,$

It also satisfies the PDE, but (again) that'll come later.

This solution may be interpreted as a surface, it's shown in the figure below with $x$ going from $0$ to $1$, and $t$ going from $0$ to $0.5$. That is, the distribution of heat is changing over time as the heat flows and dissipates.

$u(x,t)$ from $t=0$ to $t=0.5$ and $x=0$ to $x=1$.

Surfaces may or may not be the best way to convey information, and in this case a possibly better way to draw the picture would be to graph $u(x,t)$ as a curve at several different choices of $t$, this is portrayed below.

$u(x,t)$ in the domain of interest for various interesting values of $t$.

PDEs are extremely diverse, and their ICs and BCs can radically affect their solution method. As a result, the best (read: easiest) way to learn is by looking at many different problems and how they're solved.