Toronto Math Forum
APM3462015F => APM346Misc => Textbook errors => Topic started by: Bruce Wu on October 07, 2015, 07:01:08 PM

http://www.math.toronto.edu/courses/apm346h1/20159/PDEtextbook/Chapter3/S3.1.html#mjxeqneq3.1.9
The $kt$ in the denominator should also be under the square root. Similar error in the line below as a part of remark 2.

I mentioned about this error during lecture today
(9) Should be:
\begin{equation} u(x,t) = \frac{1}{2\sqrt{\pi kt}}e^{\frac{x^2}{4kt}} \end{equation}
Indeed
Error in change of variable
\begin{equation} z = x/ \sqrt{2kt} \end{equation}
(12) Shouldn't upper limit be
\begin{equation} \frac{x}{ \sqrt{4kt}}\end{equation}
You can change variables in a different way and get different expressions; it looks like with $e^{z^2}$ rather than $e^{z^2/2}$ it became more standard; so I change it (but it will take time to deal with all instances in forthcoming sections, so be vigilant)
Error function shouldn't have variables in the limits
\begin{equation}erf(x) = \sqrt{ \frac{2}{\pi} } \int_{0}^{x} \ e^{ z^{2}/2 }dz\end{equation}
It can have variable limits or we can get constant limit but then we need integrand depending on $x$
It's actually equivalent to the wikipedia version:
\begin{equation}erf(x) = \frac{2}{ \sqrt{\pi} } \int_{0}^{x} \ e^{ t^{2} } dt \end{equation}