The question I am working on is:
Suppose $\displaystyle \frac{\partial u}{\partial t}= \frac{\partial ^2 u}{\partial x^2}+x,u(x,0)=f(x),\frac{\partial u}{\partial x}(0,t)=\beta,\frac{\partial u}{\partial x}(L,t)=7$.
Calculate the total thermal energy in the one-dimensional rod (as a function of time).
What I tried:
We got a hint from the professor that we should try to calculate the integral:
$\displaystyle \int_0^L \frac{\partial u}{\partial t}\ dx= \int_0^L \frac{\partial ^2 u}{\partial x^2}+x\ dx $
I tried looking up resources on how to integrate a PDE and tried wolfram, but cant figure out where to start on this integral.