Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

$$K\frac{\partial }{{\partial x}}(h\frac{{\partial h}}{{\partial x}}) = \mu \frac{{\partial h}}{{\partial t}}$$ K and u are constants. If no,how to get a asymptotic solution?ie,linearize

share|improve this question

closed as too localized by Ricky Demer, Qiaochu Yuan, Andy Putman, Michael Renardy, Pietro Majer Nov 19 '12 at 17:38

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

1 Answer 1

We may assume WLOG that $K=1$. One family of solutions is $$h(x,t) = \frac{v(x)}{a+bt}$$ for arbitrary constants $a,b$, where $v(x)$ is a solution of the ordinary differential equation $$v v'' + (v')^2 + a \mu v = 0$$

Another family of solutions is $$h(x,t) = a \left(W\left(b e^{(x+ct) \mu c/a}\right)+1\right)$$ for arbitrary constants $a,b,c$, where $W$ is the Lambert W function.

share|improve this answer
    
"We may assume WLOG that $\: K=1 \:$" $\;\;$ except when $\: K=0 \:$. $\;\;\;\;$ –  Ricky Demer Nov 18 '12 at 5:44
2  
OK, but the case $K=0$ is left as an exercise. –  Robert Israel Nov 18 '12 at 8:09

Not the answer you're looking for? Browse other questions tagged or ask your own question.