Separation of Variables and How to Solve Partial Differential Equations

by Alexander FufaeV

Separation approach is used to convert partial DEQ of arbitrary order into ordinary DEQ.

Let's look at the last solving method, the Separation of Variables method (also called separation ansatz). This method is sometimes called the product ansatz for reasons you will soon understand. separation of variables is suitable:

for partial differential equations (PDE's)
of arbitrary order.

This solution method is mainly used only to transform a PDE into several ordinary differential equations and to solve these then with other methods (e.g. with the exponential ansatz).

Let us illustrate this method directly with an example. The one-dimensional wave equation for an electric field is best suited for this:

$$ \begin{align} \frac{\partial^2 E}{\partial x^2} ~=~ \frac{1}{c^2} \, \frac{\partial^2 E}{\partial t^2} \end{align} $$

An electromagnetic wave whose E-field and B-field components oscillate.

The function we are looking for is the electric field $E(t,x)$. This depends on $x$ and on $t$. Since the function depends on two variables and their derivatives appear in the equation, it is a partial differential equation.

Here we make a product ansatz for the searched solution $E$:

Product ansatz for searched function

$$ \begin{align} E(x,t) ~=~ R(x) \, U(t) \end{align} $$

We assume that the solution $E$ can be split into a product of two functions that we call $R(x)$ and $U(t)$. One function, $R$, depends only on $x$. And the other function, $U$, depends only on $t$. In the wave equation, we have a second derivative of $E$ with respect to location $x$ and a second derivative with respect to time $t$. So we must first differentiate our product ansatz before we insert it into the wave equation. Differentiating the product ansatz with respect to $x$ yields $U(t)\,\frac{\partial^2 R(x)}{\partial x^2}$, since $U$ is independent of $x$ and thus acts like a constant. In contrast, in the derivative of the product with respect to time $t$, the function $R$ acts like a constant because it does not depend on $t$:

$$ \begin{align} U(t)\,\frac{\partial^2 R(x)}{\partial x^2} ~=~ \frac{1}{c^2} \, R(x)\, \frac{\partial^2 U(t)}{\partial t^2} \end{align} $$

The goal now is to separate everything that depends on $x$ from what depends on $t$. For this we divide this equation by $R(x) \, U(t)$:

$$ \begin{align} \frac{1}{R(x)}\,\frac{\partial^2 R(x)}{\partial x^2} ~=~ \frac{1}{c^2} \, \frac{1}{U(t)}\, \frac{\partial^2 U(t)}{\partial t^2} \end{align} $$

Thus we have achieved that everything that depends on $x$ is on the left side and everything that depends on $t$ is on the right side. If you manage to separate a partial differential equation this way, then the product ansatz was successful.

Now we can vary $x$ on the left side without changing the right side, because there is no $x$ on the right side. The same is true for the time $t$. If we vary the time on the right side, the left side remains unchanged, because there is no time $t$. Thus both sides must be constant. So let's set the left and the right side equal to a constant $K$:

$$ \begin{align} \frac{1}{R(x)}\,\frac{\partial^2 R(x)}{\partial x^2} &~=~ K\\\\
\frac{1}{c^2} \, \frac{1}{U(t)}\, \frac{\partial^2 U(t)}{\partial t^2} &~=~ K \end{align} $$

Thus we have transformed a partial differential equation into two ordinary differential equations. And the good thing is that the two differential equations are not coupled. That is, you can solve them independently and then multiply the solutions like in the product ansatz 2 to get the general solution for the partial differential equation.

You can solve the two ordinary differential equations with the exponantial ansatz.