How Can You Effectively Change Variables to Solve a Specific PDE?

[jimbo007]

hi,
i am having difficulty trying to find a change of variables to solve this partial differential equation
$$\frac{\partial f}{\partial t} = t^\gamma \frac{\partial ^2 f}{\partial x^2}$$
not sure how to pluck out a change of variables by looking at the equation as its definitely not obvious to the untrained eye (me).
have tried
$$f(x,t) = p(\eta)$$ (not sure if i am even on the right track doing this)
where
$$\eta = \frac{x}{t^{\gamma}}$$
but that just gets me in a giant mess with
$$-\gamma\frac{x}{t}\frac{\partial p}{\partial \eta} = \frac{\partial ^2 p}{\partial \eta ^2}$$
i may have done $$\frac{\partial ^2 f}{\partial x^2}= \frac{\partial \eta}{\partial x}\frac{\partial ^2 p}{\partial \eta ^2}\frac{\partial \eta}{\partial x}$$ wrongly.

any suggestions?
thanks for the time

 

[topsquark]

hi,
i am having difficulty trying to find a change of variables to solve this partial differential equation
$$\frac{\partial f}{\partial t} = t^\gamma \frac{\partial ^2 f}{\partial x^2}$$
not sure how to pluck out a change of variables by looking at the equation as its definitely not obvious to the untrained eye (me).
have tried
$$f(x,t) = p(\eta)$$ (not sure if i am even on the right track doing this)
where
$$\eta = \frac{x}{t^{\gamma}}$$
but that just gets me in a giant mess with
$$-\gamma\frac{x}{t}\frac{\partial p}{\partial \eta} = \frac{\partial ^2 p}{\partial \eta ^2}$$
i may have done $$\frac{\partial ^2 f}{\partial x^2}= \frac{\partial \eta}{\partial x}\frac{\partial ^2 p}{\partial \eta ^2}\frac{\partial \eta}{\partial x}$$ wrongly.

any suggestions?
thanks for the time

I don't know about a substitution, but if you're just after a solution you can set f(x,t) = X(x)T(t) and the equation becomes separable.

-Dan

 

[HallsofIvy]

There is no need to get "x" involved in your change of variable. You want to go from
$$\frac{\partial f}{\partial t} = t^\gamma \frac{\partial ^2 f}{\partial x^2}$$
or, similarly,
$$t^{-\gamma}\frac{\partial f}{\partial t} = \frac{\partial ^2 f}{\partial x^2}$$
to something like
$$\frac{\partial f}{\partial \eta} = \frac{\partial ^2 f}{\partial x^2}$$
which means you want
$$\frac{\partial f}{\partial \eta}= \frac{\partial f}{\partial t}\frac{dt}{d \eta}= t^{-\gamma}\frac{\partial f}{\partial t}$$
So you must have
$$\frac{dt}{d\eta}= t^{-\gamma}$$
or
$$\frac{d\eta}{dt}= t^{\gamma}$$