# Toronto Math Forum

## APM346-2012 => APM346 Math => Home Assignment 3 => Topic started by: Kun Guo on October 04, 2012, 05:47:45 PM

Title: Problem 1
Post by: Kun Guo on October 04, 2012, 05:47:45 PM
I noticed the given error function(Erf) in the problem set sheet is different with what's given in WolframAlpha. Which one should I use? Also, are we expected to write our final answers in form of Erf?
Title: Re: Problem 1
Post by: Victor Ivrii on October 04, 2012, 05:52:05 PM
I noticed the given error function(Erf) in the problem set sheet is different with what's given in WolframAlpha. Which one should I use? Also, are we expected to write our final answers in form of Erf?

It was a misprint in HA3, which I just fixed
Title: Re: Problem 1
Post by: Kun Guo on October 05, 2012, 12:22:45 AM
I think there is still problem 2/ sqrt(pi) instead of sqrt(2/pi), is this true?
Title: Re: Problem 1
Post by: Victor Ivrii on October 05, 2012, 07:03:41 AM
I think there is still problem 2/ sqrt(pi) instead of sqrt(2/pi), is this true?

$\sqrt{\frac{2}{\pi}}$ as it should be
Title: Re: Problem 1
Post by: Aida Razi on October 07, 2012, 03:50:06 PM
I was wondering if first we need to find IVP solution to heat equation by separation of variable and then apply IBVP OR
we can use IVP solution from lecture note and just apply IBVP?

Thank you,
Title: Re: Problem 1
Post by: Victor Ivrii on October 07, 2012, 04:47:44 PM
I was wondering if first we need to find IVP solution to heat equation by separation of variable and then apply IBVP OR
we can use IVP solution from lecture note and just apply IBVP?

Thank you,

So far we have not studied separation of variables for heat equation (at least not on infinite or semiinfinite interval).
Title: Re: Problem 1
Post by: Jinlong Fu on October 09, 2012, 10:59:24 PM
The two forms are actually the same,

\begin{equation*}
erf(z)=\sqrt{\frac{2}{\pi}}\int_0^ze^{-y^2/2}\,dy
\tag{Erf}\label{eq-Erf}
\end{equation*}

let $\frac{y}{\sqrt{2}}= t$, you can transform the above formula into the form in Wolfram

\begin{equation*}
erf(z)=\frac{2}{\sqrt{\pi}}\int_0^ze^{-t^2}\,dt
\end{equation*}
Title: Re: Problem 1
Post by: Zarak Mahmud on October 10, 2012, 09:31:36 PM
(a)

\begin{equation}
G(x,y,t) = \frac{1}{2 \sqrt{k \pi t}} e^{\frac{-(x-y)^2}{4kt}}\
\end{equation}

\begin{equation*}
u(x,t) = \int_{-\infty}^{\infty} G(x,y,t)g(y)dy \\
= \int_{-\infty}^{0} G(x,y,t)g(y)dy + \int_{0}^{\infty} G(x,y,t)g(y)dy \\
\end{equation*}
Let $-y = z$,
$$= \int_{\infty}^{0} G(x,-z,t)g(-z)(-dz) + \int_{0}^{\infty} G(x,y,t)g(y)dy$$

We have Dirichlet boundary condition so g(z) is odd;$g(-z) = -g(z).$

\begin{equation*}
= -\int_{\infty}^{0} G(x,-z,t)g(z)dz + \int_{0}^{\infty} G(x,y,t)g(y)dy\\
= -\int_{\infty}^{0} G(x,-y,t)g(y)dy + \int_{0}^{\infty} G(x,y,t)g(y)dy\\
= \int_{0}^{\infty}\big[G(x,y,t)g(y) - G(x,-y,t)g(y)\big]dy\\
= \int_{0}^{\infty} \frac{1}{2 \sqrt{k \pi t}} \left[ e^{\frac{-(x-y)^2}{4kt}} - e^{\frac{-(x+y)^2}{4kt}} \right] g(y)\,dy.\\
\end{equation*}

(b)
Neumann boundary conditions: $g(y)$ is even $\implies g(x) = g(-x)$
Making the same substitution as in (a), we have:

\begin{equation*}
u(x,t) = \int_{0}^{\infty} \frac{1}{2 \sqrt{k \pi t}} \left[ e^{\frac{-(x-y)^2}{4kt}} + e^{\frac{-(x+y)^2}{4kt}} \right] g(y)\,dy.
\end{equation*}
Title: Re: Problem 1
Post by: Aida Razi on October 13, 2012, 02:17:05 PM
There is a missing g(y) in the final solution for both conditions.
Title: Re: Problem 1
Post by: Victor Ivrii on October 13, 2012, 02:47:52 PM
There is a missing g(y) in the final solution for both conditions.

Right, I fixed them. Thanks!
Title: Re: Problem 1
Post by: Victor Ivrii on October 13, 2012, 02:51:37 PM
Zarak,

You don't need to surround
Code: [Select]
 \begin{equation} ....\end{equation}by double dollars as equation, align, gather (and their * versions) are LaTeX environments, and multline  (and its * version) is AMS-LaTeX environment, all recognizable by MJ. Basically DD are  deprecated in LaTeX.
Title: Re: Problem 1
Post by: Zarak Mahmud on October 14, 2012, 08:15:42 AM
Zarak,

You don't need to surround
Code: [Select]
 \begin{equation} ....\end{equation}by double dollars as equation, align, gather (and their * versions) are LaTeX environments, and multline  (and its * version) is AMS-LaTeX environment, all recognizable by MJ. Basically DD are  deprecated in LaTeX.

Oh I see. I edited them out of the code for this post and will dispense with them in future posts. Now that I think about it, the double dollars do seem redundant when using equation, align, gather etc.