Toronto Math Forum

APM346-2012 => APM346 Math => Misc Math => Topic started by: Thomas Nutz on October 05, 2012, 08:59:09 AM

Title: inhomogeneous b.c.
Post by: Thomas Nutz on October 05, 2012, 08:59:09 AM

In the 10th lecture we are asked to consider
$$
0=\int_{II}G(x,y,t-\tau)(-u_{\tau}(y,\tau))+ku_{yy}(y,\tau)d\tau'dy
$$

1. question: What are we integrating over here? Is $\tau'=t-\tau$?

2. Where is this expression coming from? Is it a trial solution that I simply have to take as given, or does it follwow from any other expression?

Thanks!

Title: Re: inhomogeneous b.c.
Post by: Victor Ivrii on October 05, 2012, 10:42:42 AM

Quote from: Thomas Nutz on October 05, 2012, 08:59:09 AM

In the 10th lecture we are asked to consider
$$
0=\int_{II}G(x,y,t-\tau)(-u_{\tau}(y,\tau))+ku_{yy}(y,\tau)d\tau'dy
$$

1. question: What are we integrating over here? Is $\tau'=t-\tau$?

2. Where is this expression coming from? Is it a trial solution that I simply have to take as given, or does it follwow from any other expression?

Thanks!

$'$ is an artefact (removed). BTW domain is $\Pi$, not $II$

$\bigl(-u_{\tau}(y,\tau)+ku_{yy}(y,\tau)\bigr)=0$ due to equation $u_t-ku_{xx}=0$ (if $u_t-ku_{xx}=f$) we would get an extra term (17) in the lecture 10 (http://www.math.toronto.edu/courses/apm346h1/20129/L10.html)