I wish to solve the following PDE:
$$
u_t – (u^{2})_{xx}=0
$$
with some boundary conditions which right now is not of much relevance apart from the fact that $u(t,x)$ is a piecewise continuous function and $u_x(t,0)=0$.
I have used the following substitution: $u(t,x)=f(\eta)t^{-\frac{1}{3}}$ and $\eta=xt^{-\frac{1}{3}}$ which reduces the above equation to a single variable of the form:
$$
\partial_{\eta}\left ( 2ff_\eta + \frac{\eta f}{3} \right )=0
$$
Now, I am very confused about how to proceed with this reduced form to solve the above equation and find the analytical solution of $f(\eta)$ thereby finding the analytical solution of $u(t,x)$.
Help of any sort is deeply appreciated. Thanks in advance.
Best Answer
Actually, there was a key point which we were missing from the question which would make the solution very easy (it clicked to me after spending 3-4 hours on this question) which is $u_x(t,0)=0$
Now, using the chain rule we can easily derive the following:
$$ u_x(t,x) = t^{-\frac{2}{3}}f_{\eta}(\eta)=0 $$
Also, from the very definition of $\eta$, $x=0$ implies $\eta=0$ and thus
$$ u_x(t,0) = t^{-\frac{2}{3}}f_{\eta}(0)=0 $$
or simply $f_{\eta}(0)=0$
Now, we can write
$$ \partial_{\eta}\left ( 2ff_\eta + \frac{\eta f}{3} \right )=0 $$
as (for a constant $K$)
$$ 2ff_\eta + \frac{\eta f}{3} = K $$
When $\eta=0$, $f_{\eta}(0)=0$ and thus substituting them in the above equation gives $K=0$ which makes our equation
$$ 2ff_\eta + \frac{\eta f}{3} = 0 $$
Now, since we are not looking for the trivial solution, hence $f(\eta) \neq 0$ which means
$$ 2f_\eta + \frac{\eta}{3} = 0 $$
or
$$ f(\eta) = -\frac{\eta^2}{12} + C $$
for a constant $C$, which can be found from the other boundary conditions.