Author Topic: Clever Guesses with Heat Equation  (Read 744 times)

Andrew Hardy

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Clever Guesses with Heat Equation
« 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?

Victor Ivrii

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Re: Clever Guesses with Heat Equation
« Reply #1 on: April 07, 2018, 08:16:07 PM »
You mean "get the same result fast" in
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  $u_t-ku_{xx}=0$. We found it in Chapter 3
« Last Edit: April 07, 2018, 08:18:25 PM by Victor Ivrii »

Jingxuan Zhang

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Re: Clever Guesses with Heat Equation
« Reply #2 on: April 07, 2018, 08:37:24 PM »
So what if IC is not as in that question? $x^2e^{-x^3}$, say? Can we still use this or similar result?

Victor Ivrii

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Re: Clever Guesses with Heat Equation
« Reply #3 on: April 07, 2018, 09:07:32 PM »
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