Scaling Problem in Diffusion Equation

[kayahan]

Hello everyone,

I have a question that is bothering me a bit. I would be happy if you could give an idea or tell me a specific point to look at.

Lets say that we have an arbitrary function that obeys diffusion equation:

f = f(η), here η is the scaling parameter for pde which equals to x/√t
(Diffusion eq. becomes $df/d\eta=-2\eta d^{2}f/d\eta^{2}$ after scaling)

As I understood, diffusion eq. becomes an ode after scaling and it contains all the possible solutions for x and t pairs in η. Can we say that f(η) is invariant with respect to x and t? or in other words say ∂f/∂t=0 or not? What kind of physical information can we get just by lloking at f(η)?

Thanks in advance!

 

[HallsofIvy]

No, $\partial f/\partial x$ and $\partial f/\partial x$ are NOT 0 because $\eta$ is itself a function of x and t.

$$\frac{\partial f}{\partial x}= \frac{df}{d\eta}\frac{\partial \eta}{\partial x}= \frac{1}{\sqrt{t}}\frac{df}{d\eta}$$
$$\frac{\partial f}{\partial t}= \frac{df}{d\eta}\frac{\partial \eta}{\partial t}= -\frac{x}{\sqrt{t^3}}\frac{df}{d\eta}$$