Toronto Math Forum
APM3462018S => APM346––Home Assignments => Topic started by: Andrew Hardy on April 07, 2018, 05:38:41 PM

Dr. Ivrii,
Jingxuan and I are studying together over the old term tests. We're confused by your motivation for the initial guess for Fall 2016 Q2, the heat equation?
$$ v(x,t)=t^{\frac{1}{2}}e^{\frac{x^2}{t}} $$
It ends up being quite quick, but I don't know how I could conjure up the $ t^{\frac{1}{2}} $ term on our own. Is there a method?

You mean "get the same result fast" in
http://forum.math.toronto.edu/index.php?topic=862.0 (http://forum.math.toronto.edu/index.php?topic=862.0)
We do not guess, we know that $t^{1/2}e^{x^2/4kt}$ satisfies $u_tku_{xx}=0$. We found it in Chapter 3

So what if IC is not as in that question? $x^2e^{x^3}$, say? Can we still use this or similar result?

So what if IC is not as in that question? $x^2e^{x^3}$, say? Can we still use this or similar result?
Sometimes... but usually not. F.e. solving Cauchy problem for
\begin{align}
&u_t ku_{xx}=0,\tag{*}\\
&u_{t=0}=x^2e^{ax^2}\tag{**}
\end{align}
the same way would work: again, we know that $v(x,t)=t^{1/2}e^{x^2/4kt}$ satisfies (*), and for $t=t_0$ (find it) gives us $Ce^{ax^2}$....
You need to know the regular way, cutting corners works sometimes ... but usually does not