Archive for March, 2008

PDEs: Some Solutions of the Wave Equation

In this post we derived the wave equation for a one-dimensional object moving in two-dimensional space (namely, a string vibrating up and down). This time we’re going to derive a family of solutions for this one-dimensional wave equation.

PDEs: Even More on Characteristics

So far, we’ve found the general solutions for two types of linear, homogeneous, first order PDEs: those with constant coefficients, and those with varying coefficients. It turns out that you can also extend the method of characteristics for linear, inhomogeneous, first order PDEs. In fact, the procedure is almost exactly the same.

PDEs: The Wave Equation (Long Post..)

So far we’ve derived and found solutions for a simple, first order, linear PDE describing the transport of a chemical through a fluid. Today we’re going to derive our first second-order equation, namely the wave equation. This equation is actually rather famous, and applies to a crap-ton of different physical phenomena. You’ve no doubt heard of Schroedinger’s wave equation, which is a fundamental result of quantum mechanics. Well, that’s really just applying quantum mechanical principles to the classical wave equation (see here, for instance). The wave equation can also describe electromagnetic waves, sonic tremors, tidal waves, guitar strings and almost anything “wave-like”, given the proper treatment.

PDEs: Extending the Method of Characteristics

Last time we derived the general solution of a linear, homogeneous PDE with constant coefficients: $c_2 u_x + c_1 u_t = 0$. Physically, we can interpret such an equation as describing the flow of some chemical through water moving at a constant speed $\frac{c_2}{c_1}$. Suppose, however, that the water does not move at a constant speed. Suppose that $c_1$ and $c_2$ are themselves functions. In other words, suppose our PDE looks like this:

$c_2 (x,t) u_x + c_1 (x,t) u_t = 0$

It turns out that it’s not very difficult to extend the method of characteristics for this situation. Let’s graph it, though, to get a better idea of what we’re doing:

PDEs: Solving Simple PDEs Using Characteristics

Last time we derived a simple PDE describing the process of non-diffusive advection in a constantly flowing stream of water. That PDE looked like this: $v\cdot u_x + u_t = 0$, where $v$ was the constant speed of the water. The solution of this PDE is a function of the form $u(x,t)$.

This PDE follows a more general form that looks like this: $c_2\cdot u_x + c_1\cdot u_t = 0$, where $c_1$ and $c_2$ are constants. If you’ve studied multivariable calculus, you might recognize this kind of equation. It is, in fact, a directional derivative.

We can rewrite $c_2 u_x + c_1 u_t = \nabla u \cdot \textbf{s} = 0$, where $\textbf{s} = (c_1, c_2)$. We can read this equation as “the gradient (or derivative) of $u(x,t)$ in the direction of $\textbf{s}$ is zero”. Which means that in the direction $\textbf{s}$, the solution $u$ attains the same constant value. It may be easier to see by graphing it:

PDEs: Deriving a Simple Advection Equation

Suppose there’s a pipe with water flowing through it at a constant rate $v$, and that a chemical of some kind is suspended in and moving with the water. This process is called advection. Suppose also that the chemical does not diffuse (i.e. spread out) as it moves. In other words, each particle of the chemical moves parallel to the pipe.

Now let $u(x,t)$ describe the concentration of this chemical at position $x$ and at time $t$. We get the following picture, where the chemical is moving in the positive $x$ direction (to see this, just note how the graph changes as time passes).

PDEs: L, H, and O

One of the math classes I’m taking this semester is on partial differential equations (or PDEs). In lieu of actually studying this troublesome topic, I think I’m going to write a few posts about it. Which will likely take more time, and which will count as studying anyway. Why am I so masochistic? Meh.

Anyways. So partial differential equations are just equations of partial derivatives of some function. As with ordinary differential equations (ODEs), the goal is to find that function. For example, you may have this equation: $\frac{{\partial u}}{{\partial x_1 }} + \frac{{\partial u}}{{\partial x_2 }} = 0$, written more concisely as $u_{x_1 } + u_{x_2 } = 0$. The goal, then, is to find the function $u(x_1,x_2)$