# Boundary value problem without separation of variables

It is well-known that the separation of variables method is useful for boundary value PDE problems. For example, it is usual to use separation of method for this problem $$V:\mathbb{R}^2\to \mathbb{R}$$ $$\nabla^2V=0$$ $$V(0,y)=V_0(y),V(x,0)=V(x,a)=0,\lim_{x\to\infty}V(x,y)=0$$ by letting $$V(x,y)=X(x)Y(y)$$ to get $${1\over X}{\partial^2X\over \partial x^2}=-{1\over Y}{\partial^2Y\over \partial y^2}=k^2$$ and so on.

Although this method is very useful and is general approach to various kind of PDE problems, I feel somewhat awkward with it because it feels like the solution comes out of the blue, just as if the solver already knows that this separation will turn out to be valid.

Or from the other view, as I'm a physics student, I think that if the problem is not that mathematically complicated, then there must be a corresponding physics problem for each PDE problem (like the one I suggested), then the solution of the problem must have its physical meaning. So if one can get the solution with one mathematical method, then I think there must be some other way of deduction to get the answer. (As you can interpret one phenomenon in various ways)

So, my question is, is there any method that can substitute the separation of variables method for simple PDE problems (with adequate generality)?

Separation of variables does not always work, it does for homogenous PDEs with these boundary conditions (linear PDEs only, thank you Dmoreno). In general there needs to be a symmetry between the variables in order for the equation to be separable.

Without knowing initial conditions such as $\frac{\partial V}{\partial x}(0, y)$ or $\frac{\partial V}{\partial y}(x, 0)$ we can't use a Laplace Transform in one variable, or a Fourier Transform. As a note, I notice you are using $x$ and $y$ for your variables, which from a physics perspective imply spatial coordinates. Laplace and Fourier transforms are generally used on coordinates in the time domain, if only because these usually have initial conditions.

The Method of Characteristics is usually better suited for first order equations.

You can look for a change of variables for any PDE, the accepted change of variables in this case is to separate the variables.

I agree that there is a (sometimes false!) assumption that separation of variables will/should work. I would recommend taking a good course in PDEs through the math department, you will get a very strong background in the theory of why certain techniques work in some cases and fail in others, I found this very helpful for upper level courses in physics.

The method of SV is almost always useful and suitable for solving such PDEs when they are linear and homogeneous with, in general, homogeneous boundary conditions.

In case the boundary conditions are not homogeneous, you may use the superposition principle, that is $V = u+w$ (where $u$ satisfies homogeneous boundary conditions and $w$ absorbs the non-homogeneous ones) and solve for $u$. In case the PDE for $u$ results non homogeneous you may use Fredholm's alternative and Sturm-Liouville theory to achieve the solution $u$ by considering the problem:

$$\nabla^2 \theta = 0,$$

with homogeneous boundary conditions. This problem is linear and homogeneous and suitable for separation of variables. Then you expand $u$ as a Fourier expansion of the eigenfunctions (which depend on every problem).

I hope this is useful to you.

Cheers!