Separation of variables is a powerful method which comes to our help for finding a closed form solution for a linear partial differential equation (PDE). For example, we all know that how the method works for the two dimensional Laplace equation in Cartesian Coordinates
$$\nabla^2 \phi(x,y) = \partial^{2}_{x}\phi(x,y) + \partial^{2}_{y}\phi(x,y) = 0 \tag{1}$$
The steps are
$1$. Consider a solution of the form $\phi(x,y)=X(x)Y(y) \tag{2}$
$2$. Put it into the equation, assuming $X(x) \ne 0,$ and $Y(y) \ne 0$, to get
$$\dfrac{\text{D}^2 X}{X}+\dfrac{\text{D}^2 Y}{Y}=0 \tag{3}$$
where $\text{D}$ is the differential operator.
$3$. Observe that the above functional equation $(3)$ is possible if and only if
$$\dfrac{\text{D}^2 X}{X} = -\dfrac{\text{D}^2 Y}{Y} = \lambda \tag{4}$$
where $\lambda$ is some constant.
$4$. Conclude that $X$ and $Y$ should satisfy the following ordinary differential equations (ODEs)
$$\begin{align}
\left[ \text{D}^2 – \lambda \right]X &= 0 \\
\left[ \text{D}^2 + \lambda \right]Y &= 0
\end{align} \tag{5}$$
However, I always had some questions about this method which is not addressed in the elementary books. Here are my questions
$1$. What are the restrictions of this method? I mean when it is not going to work! or equivalently When a linear PDE cannot have a separable solution of the form $(2)$?
$2$. Has the method been used for system of linear PDEs? If Yes, would you please give an example.
Best Answer
For linear equations, the technique of separation of variables is used to find all separated solutions of the form $X_1(x_1)X_2(x_2)\cdots X_n(x_n)$. You find them all if your equation can be separated.
If you make a change of variables, then you will generally find a different set of separated solutions. For example, you might separate $X(x)Y(y)$, or instead $R(r)\Theta(\theta)$ where $r=\sqrt{x^2+y^2}$, etc. The separated solutions you end up with using a different set of coordinates will be different because the separated solutions $u(x,y)=X(x)Y(y)$ that vanish at some $x$ vanish for all $y$ at that $x$; so the zero sets are coordinate-aligned. The solutions $u(r,\theta)=R(r)\Theta(\theta)$ will vanish on circles and or rays starting at the origin. Note: just because you can separate in one set of coordinates does not mean you can separate in another.
The Laplacian separates in Cartesian coordinates, and one contributing factor is that there are no mixed derivatives. Coordinate changes that introduce mixed terms most often result in equations that are no longer separable. For example, try $u=X(x)Y(y)$ in the following $$ u_{xx}+ 2u_{xy} +u_{yy} = 0 \\ X''Y + 2 X'Y' + XY'' = 0 \\ \frac{X''}{X}+2\frac{X'}{X}\frac{Y'}{Y}+\frac{Y''}{Y}=0. $$ For the Laplacian, avoiding mixed terms essentially requires that you use orthogonal coordinate systems to separate variables, but that's not always enough to be able to separate. The spherical coordinate system and the cylindrical coordinate system are orthogonal because the coordinate surfaces $r=A$, $\theta=B$, $\phi=C$ are mutually orthogonal where they intersect. So, when you separate variables you don't end up with mixed derivative terms. As I recall, there are 24 orthogonal coordinate systems where the Laplacian separates. This limits you, because you can only impose conditions on surfaces where a coordinate is constant, but there are enough systems that significant problems could be solved exactly a century ago, enough to develop Quantum Mechanics, for example.
If you're going to use the technique on a system, then all of the equations need to separate, and maybe in the same coordinate system.
The basic reason you have enough separated solutions to build up a full solution for classical operators is that they are formally selfadjoint. Symmetry is a big part of what makes separation of variables work to give you a full range of solutions. For example, $$ \int_{\Omega} \nabla^2 f g dx = \int_{\Omega} f \nabla^2 g dx + \mbox{eval terms} $$ Fortunately, symmetry and Physics combine in a powerful way; and symmetry and math combine in a powerful way. It's difficult to do much in either field without symmetry or something close to it.