# Diffusion Equation/Change of Variable

1. Feb 27, 2009

### LP_MAX

1. The problem statement, all variables and given/known data

Apologies if this doesn't come through properly.

The question states

Use a change of time variable to show that the equation

$$c(\tau) \frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x^2}$$

can be reduced to the diffusion equation.

3. The attempt at a solution

I've tried a couple of things, primarily setting up

$$v(\tau) = \int c(\tau)$$

in the hope that the product rule would give me something to cancel out on the left hand side, but no luck. I'm pretty certain this is going to be one of those nasty little mathematical tricks that can be described in six words or less. If anybody wants to give me a pointer as to what I should be looking for, it would be appreciated.

Thanks.

Last edited: Feb 27, 2009
2. Feb 27, 2009

### HallsofIvy

What, exactly, is the "diffusion equation" you trying to get to? I ask because I would certainly consider the equation you give a "diffusion equation".

3. Feb 27, 2009

### LP_MAX

Sorry, *the* diffusion equation in this case just means that by change of variable I need to show it can be reduced to

$$\frac{\partial U}{\partial \tau} = \frac{\partial^2 U}{\partial \tau^2}$$

for example, the previous part of this question involves

Suppose that a and b are constants. Show that the parabolic equation
$$\frac{\partial \mu}{\partial \tau } = \frac {\partial^2 \mu}{\partial x^2} + a \frac {\partial \mu} {\partial x} + b\mu$$
can always be reduced to the diffusion equation.

can be reduced by substituting in

$$v = e^{\alpha x +\beta \tau} u(x,\tau)$$

and liberal application of the product rule. I'm sure there's a really simplistic substitution I should be able to rattle off to do this, but I am not a mathematician, sadly. I'm not a physicist and "diffusion equation" has a very narrow meaning and form for the area I'm involved in, at least at the level I'm currently at (probably more a reflection of my limited understanding than anything else). Thanks for the pointer on the tags. Been knee deep in latex all day and just didn't see that.

Last edited: Feb 27, 2009
4. Feb 27, 2009

### John Creighto

Use \ for letex and / for discussion board tags. Notice your closing tex tags use the wrong slash.