We now consider an inhomogeneous heat equation given by

\[\pd{T}{t}=D \nabla^2 T+f(x,y),\] \[f(x,y) = D\pi^2\left(\frac{n^2}{L_x^2} + \frac{m^2}{L_y^2}\right)\cos \left(\frac{n\pi x}{L_x} \right)\cos \left(\frac{m\pi y}{L_y} \right)\]

with homogeneous Neumann (aka no-flux) boundary conditions on a rectangular domain with side lengths $L_x$, $L_y$. You can use separation of variables to show that the solution at steady state looks like

\[T(x,y) = -\cos \left(\frac{n\pi x}{L_x} \right)\cos \left(\frac{m\pi y}{L_y} \right).\]

  • Load the interactive simulation.

  • You can change the values of $m$ and $n$ to observe different patterns of sources/sinks of heat in the domain.

  • You can use any function $f(x,y)$ instead of the one given above. However, if $f(x,y)$ does not satisfy the constraint that $\int_0^{L_y}\int_0^{L_x} f(x,y) \, \d x \, \d y=0$, then the solution will either grow or decrease without bound. An easy way to prove this is to multiply the equation by $T$ and integrate to find, after applying the Neumann boundary conditions,

\[\frac{1}{2}\pd{}{t}\int_0^{L_y} \int_0^{L_x} T^2 \, \d x \, \d y = \int_0^{L_y}\int_0^{L_x} f(x,y) \, \d x \, \d y.\]

Inhomogeneous transport

We can also consider a diffusion coefficient which varies in space by studying

\[\pd{T}{t}= \vnabla\cdot(g(x,y)\vnabla T),\]

where we need $g(x,y)>0$ for all $x,y$ in the domain. As a simple (though complicated-looking) example, we take,

\[g(x,y) = D\left[1+E\cos\left(\frac{n \pi}{L_xL_y}\sqrt{(x-L_x/2)^2+(y-L_y/2)^2}\right)\right],\]

where $D>0$, $n>0$, and $\lvert E\rvert <1$ are constants. This represents radially-oscillating regions of high and low diffusion. Setting an initial condition of \(u(x,y,0)=1\) and Dirichlet boundary conditions, we can observe an immediate partitioning of the initial heat into regions bounded by the maxima of the cosine function. Click here to see this, and play around with the values of $n$, $E$ and $D$.