Nonlinear PDE Help: Solving \alpha(uu_x)_x = u_t | Initial Value Problem Tips"

[maka89]

Hello. I was wondering if anyone here had come across an equation similar to this one:
$\alpha(uu_x)_x= u_t$

Any info regarding this equation or tips on how to solve this would be appreciated :)

I came across these solutions: http://eqworld.ipmnet.ru/en/solutions/npde/npde1201.pdf, but how do I choose which one to use? I am looking at a initial value problem. And u > 0.

 

[Mark44]

Hello. I was wondering if anyone here had come across an equation similar to this one:
$\alpha(uu_x)_x= u_t$

Any info regarding this equation or tips on how to solve this would be appreciated :)

I came across these solutions: http://eqworld.ipmnet.ru/en/solutions/npde/npde1201.pdf, but how do I choose which one to use? I am looking at a initial value problem. And u > 0.

I don't think the equation given in the link you posted will be helpful. The equation there is equivalent to $a(w^m w_x)_x = w_t$.
The only thing that comes to mind in your equation is to take the partial w.r.t x of the left side (using the product rule). That would leave you with $\alpha[(u_x)^2 + uu_{xx}] = u_t$, although I'm not sure that gets you anywhere.

 

[pasmith]

Hello. I was wondering if anyone here had come across an equation similar to this one:
$\alpha(uu_x)_x= u_t$

Any info regarding this equation or tips on how to solve this would be appreciated :)

For $\alpha > 0$ this is a one-dimensional diffusion equation $$u_t - (Du_x)_x = 0$$ where the diffusivity $D$ is not constant, but is instead proportional to the density $u$ of the diffused quantity: $D = \alpha u$.

I came across these solutions: http://eqworld.ipmnet.ru/en/solutions/npde/npde1201.pdf, but how do I choose which one to use? I am looking at a initial value problem. And u > 0.

You seem to be dealing with the case $m = 1$. I don't think the given analytic solutions will help you, except for particular special cases of the initial condition. For generic initial conditions you must fall back on numerical methods.