Say we have Laplace's equation:
$$\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} = 0$$
These are the boundary conditions.
Now, to solve this by separation of variables, we need the boundary conditions to be homogeneous.
We can apparently (according to my mark-scheme) do so by defining a new variable $u = T-T_0$.
Then, the boundary conditions become $u=0$ for $x=0, x=1, y=1$ and $T_0 \sin (2\pi x)$ for $y = 0$.
Now obviously, the first three of those are homogeneous as they are equal to 0, but the fourth BC is what I'm not so sure about. Is $u(x,0) = T_0 \sin (2\pi x)$ a homogeneous boundary condition? My understanding of a PDE having homogeneous BCs was that the function (in this case $u$) is zero at the boundaries, which isn't completely fulfilled by the fourth boundary condition.
However, the mark-scheme proceeds from here to solve by separation of variables, which leads me to wonder if I'm misunderstanding what a homogeneous boundary condition is?
Best Answer
It is enough for three of the four boundary conditions to be "homogeneous" (i.e. equal to $0$) so that separation of variables is analytically tractable for the 2D Laplace equation on a square domain. Cf. for example https://tutorial.math.lamar.edu/classes/de/laplaceseqn.aspx for more details.