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I'm trying to set up and solve the heat equation in 1D that conforms to conservation of energy. So intuitively in addition to the initial condition $$u(x,0) = f(x)$$ the boundary condition would be $$\int_0^L u(x,t) dx = k,\quad t\in (0,\infty)$$

But I'm not very familiar with multivariate calculus and PDEs in general so I'm not sure how to proceed from here. I'm not really looking for an analytical solution as I suspect it'll be very complicated, but it'd be nice if someone pointed me in the correct solution. I am hoping to solve this in MATLAB if possible. Thanks.

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  • $\begingroup$ Please tell us typically about the PDE. $\endgroup$ – doraemonpaul Apr 5 '13 at 16:44
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Energy conservation holds, for instance, if you solve the heat equation with the homogeneous Neumann or periodic boundary conditions.

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