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Section 9.3 Partial Differential Equations

Subsection 9.3.1

Figure 9.3.1. Concavity and Heat Diffusion

Now that I have defined partial derivatives, I can introduce what is possibly the most important setting for their use: partial differential equations. I will start with two classic examples. The first is diffusion of heat (though the same principle apply to any kind of diffusion model).

Assume there is a 1-dimensional rod where length is measured with the variable \(x\text{.}\) Heat can vary along the rod, so I measure it by a function \(u(x)\text{.}\) However, this distribution of heat can also change over time. Therefore, I should measure the heat distribution both in terms of position \(x\) along the rod and time \(t\text{,}\) \(u(x,t)\text{.}\)

I need to consider several aspects of the situation to give a full account of how heat will diffuse. First, I will look at the mechanics of heat. I make the assumption that heat wants to equalize; with the absence of external addition of heat, it will diffuse until it equals out everywhere. If addition, I assume that the greater the variance in heat between two adjacent points on the rod, the faster the heat will diffuse. How do I translate this assumptions into mathematics? I need to get a measure of this variance in heat. The measure is local, since heat only diffuses to points adjacent. So how do I measure the local variance in heat?

Consider the heat picture at some fixed time \(t_0\text{:}\) \(u(x,t_0)\text{.}\) It is not the value of the heat that determines diffusion, since nearby values can be higher or lower. It is also not the slope of this graph in \(x\) that determines the diffusion, since a straight line slope represents and even flow of heat from one end to the other. Instead, heat wants to return to this even flow, this straight line: so it is the curvature of the graph that disrupts the straight line. Curvature or concavity is measured by the second derivative. Therefore \(u_{xx}(x,t_0)\) measures the tendency for the heat to diffuse. Figure 9.3.1 illustrates how concavity causes heat diffusion.

Diffusion creates change: heat will leave or enter the point. Change is measured by the time derivative \(u_t(x,t)\text{.}\) So concavity, the second space deriavitve, must be related to the first time derivative. What is the relationship between these? The simpliest relationship is a linear relationship. That means there is a constant \(\alpha\) such that

\begin{equation*} \frac{\del u}{\del t} = \alpha \frac{\del^2 u}{\del x^2}\text{.} \end{equation*}

This equation is called the Heat Equation. Though relatively simple, it is one of the most important partial differential equations.

The equation, however, isn't enough to solve the problem. I also need boundary conditions and initial conditions. The boundary conditions determine what happens at the end of the rod: \(u(0,t)\) and \(u(l,t)\text{.}\) In principle, these could be anything functions of \(t\text{.}\) For now, I will assume they are constant \(u(0,t) = a\) and \(u(l,t) = b\text{.}\)

I also need intial conditions. Initial conditions determine the starting heat distribution at at particular time, say \(t=0\text{,}\) That is \(u(x,0) = f(x)\text{,}\) a single variable function that shows the original situation.

All together, this information determines the physical system. I can then try to find a function \(u(x,t)\) which matches the equation, the boundary conditions and the initial conditions. Such a task is often very difficult to do. However, if both boundary conditions are constant \(0\) and the initial condition is \(f(x) = \sin \left( \frac{\pi x}{l} \right)\) then the function

\begin{equation*} u(x,t) = e^{-\frac{\alpha \pi^2 t}{l^2}} \sin \left( \frac{\pi x}{l} \right) \end{equation*}

solves the partial differential equation. This is an ideal case: in general the solutions become much more complicated. This initial heat distribution is half a period of a sine wave, and the time dependence is a exponential decay of the amplitude of that sine wave back towards a stable 0-level heat distribution.

This example is archtypical of many partial differential equations. There will almost always be a function which depends on time as well as some other quantities. In physics, these other quantities are usually position. Then the equation is usually organized by taking a time derivative on one side and a position derivative on the other. Then the equation encodes a relationship between the two two derivatives. In addition, boundary conditions determine what happens at the edges of our environment and initial conditions describe the situation at a fixed moment in time. Finally, I try to find a multi-variable function that fits all the information.

Subsection 9.3.2 The Wave Equation

Another very familiar situation is a wave moving through an elastic medium. If there is a 1-dimensional elastic medium (think a wire or string), then \(u(x,t)\) measures the displacement of the medium at position \(x\) and time \(t\text{.}\) The physical motivation is similar to the heat equation: the concavity measure the offset of the stiuation from a stable straight line. However, this concavity, instead of causing heat diffusion, causes acceleration on the adjacent points of the elastic medium. The elastic medium doesn't diffuse back to equilibrium, it accelerates, like a spring, back to equilibrium. Acceleration is a second time derivative, so the Wave Equation is

\begin{equation*} \frac{\del^2 u}{\del t^2} = \alpha \frac{\del^2 u}{\del x^2}\text{.} \end{equation*}

Again, there are boundary conditions and initial condition and, in general, the problem is very difficult to solve. However, if I choose the same simplistic options as I did with the heat equation, with constant zero boundary conditions at \(x=0\) and \(x=l\) and initial wave profile \(f(x) = \sin \left( \frac{ \pi x}{l} \right)\text{,}\) then the solution is

\begin{equation*} u(x,t) = \sin \left( \frac{\sqrt{\alpha} \pi t}{l} \right) \sin \left( \frac{\pi x}{l} \right)\text{.} \end{equation*}

So, instead of decay to equilibrium, there is an oscillating amplitude, resulting is a very simple standing wave on the wire or string. This is a very simple version: there are no higher harmonics and there is no friction which causes decay over time.

Without friction, the oscillations of the wave equation continue forever while the heat diffusion in the heat equation quickly decayed to equilibrium. The major difference between the two equations is one of active versus passive response. In the heat equation, the time derivative is a first derivative: the velocity is determined by the physical arrangement. This is a passive situation: some velocity is simply determined by the arrangement. It fades away over time. In the wave equation, the time derivative is a second derivative: the acceleration is determined by the physical arrangmeent. This is an active situation: the acceleration can increase the velociy of the displacement. The result is propogation of waves, an active process, and it will continue until otherwise damped by friction.

Subsection 9.3.3 Other PDEs

Many other famous equations are relationships between times derivatives and position derivatives. In both of the following examples, the left side is a time derivative and the right is a space derivative.

The Schrodinger equation is the centre of quantum mechanics: it measures a wave function \(\Psi(x,y,z,t)\) in three dimensions and time. The symbol \(\nabla\) is a 3-dimemsional differential operator which will be defined in Subsection 9.4.4; for now, just know that it is a combination of position derivatives. \(\hbar\) is a constant, \(\imath\) is a number with \(\imath^2 = -1\text{,}\) \(m\) is mass and \(V(x,y,z)\) is a potential energy function.

\begin{equation*} \imath \hbar \frac{\del \Psi}{\del t} = - \frac{ \hbar^2}{2m} \nabla^2 \Psi + V \Psi \end{equation*}

Another famous example is the Navier-Stokes equation, which is the fundamental equation of fluid dynamics. There are a number of versions of it, but I'll only write one. The function \(v(x,y,z,t)\) is the flow velocity of a three-dimensional fluid. Again, \(\nabla\) is a 3-dimensional position differential operator. \(\rho\) is the fluid density, \(p\) is the pressure, \(T\) is something called a stress tensor and \(f\) is an external force of the fluid. The equation is

\begin{equation*} \rho \frac{\del v}{\del t} = \rho v \cdot \nabla v - \nabla p + \nabla \cdot T + f \end{equation*}

Solving the Navier-Stokes equation for various initial and boundary conditions is the subject of a whole branch of mathematical physics called fluid dynamics.