# Total thermal energy from heat equation

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1. Sep 28, 2016

### Conservation

1. The problem statement, all variables and given/known data

2. Relevant equations
Heat equation

3. The attempt at a solution
I can derive E(t) to get integral of du/dt over 0 to L, which is the same as integrating the right hand side of the original equation (d2u/dx2+sin(5t); while this allows me to take care of the d2u/dx2, I don't know what to do with the sin(5t) term or proceed from there.

Thanks.

2. Sep 29, 2016

### Ssnow

Hi, mmm $\int_{0}^{L}\sin{5t}dx=L\sin{5t}$ ...

3. Sep 29, 2016

### Ssnow

Another hint, when you find $\frac{\partial}{\partial t} E(t)$ as a function of $t$ you must integrate in order to find $E(t)$, this gives you the problem to determine a constant of integration ... you can solve this calculating $E(0)=\int_{0}^{L}u(x,0)dx=\int_{0}^{L}1+100\sin{\left(\frac{2\pi x}{L}\right)}dx$ ...

Ssnow