how can we compute the integral of PDE if both the PDE and integral is with respect to $x$?
For example:
$\int w(x) \, \partial_{x} \, [\mu(x)\,\partial_{x}\,u(x,t)]dx$
I know we have to do integration by parts, but how do we do integration of:
$\int \partial_{x} \, [\mu(x)\,\partial_{x}\,u(x,t)]dx$
The book says that the result is:
$\int \mu(x) \, \partial_{x} \,w(x)\,\partial_{x}\,u(x,t)dx$
I cannot get that result.
Thanks
You are integrating only in $x$. As such you should treat $y$ constant. Let $y$ be fixed, and set $U(x) := u(x,y)$. Then we have $\partial_x u = U'$, and the integral is $$ \int_0^L w (\mu u')' dx$$
Integration by parts: $$ \int_0^L w (\mu U')' dx = w \mu U'\Big|_0^L - \int_0^L w' (\mu U') dx$$ the boundary term disappears using the Neumann boundary condition.
PS note that $w:G\to\mathbb R$ is a one-variable function so some people would say you cannot write $\partial_x w$, only $w'$. The person who wrote your notes is not this kind of person, so you should get used to treating $\partial_x$ as a synonym for $d/dx$ if you want to continue reading.