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**Misc Math / Bonus Web Problem 1**

« **on:**October 09, 2012, 03:49:08 PM »

This is not very difficult problem but it contains one tricky point

Consider heat equation with thermo-conductivity depending on the temperature:

\begin{equation}

u_t= (u^m u_x)_x

\label{eq-1}

\end{equation}

with $m>0$ and find solution(s) $u$ which are self-similar

\begin{equation}

u_\lambda:= \lambda u(\lambda x, \lambda^\gamma t )=u(x,t)\qquad \forall \lambda>0

\label{eq-2}

\end{equation}

and

\begin{equation}

u(x,t)\to 0 \qquad\text{as } x\to \pm \infty.

\label{eq-3}

\end{equation}

Hint: first find $\gamma$ and then plugging $\lambda =t^{1/\gamma}$ reduce $u$ to a function of one variable and PDE (\ref{eq-1}) to ODE (Follow lecture 9 with the necessary modifications)

Consider heat equation with thermo-conductivity depending on the temperature:

\begin{equation}

u_t= (u^m u_x)_x

\label{eq-1}

\end{equation}

with $m>0$ and find solution(s) $u$ which are self-similar

\begin{equation}

u_\lambda:= \lambda u(\lambda x, \lambda^\gamma t )=u(x,t)\qquad \forall \lambda>0

\label{eq-2}

\end{equation}

and

\begin{equation}

u(x,t)\to 0 \qquad\text{as } x\to \pm \infty.

\label{eq-3}

\end{equation}

Hint: first find $\gamma$ and then plugging $\lambda =t^{1/\gamma}$ reduce $u$ to a function of one variable and PDE (\ref{eq-1}) to ODE (Follow lecture 9 with the necessary modifications)