# Perturbation Theory

There exists a certain class of “hard” problems that can’t be solved with exact form. Examples include solutions to certain differential equation or higher order polynomials like quintics which can’t be solved with a simple cubic formula or quadratic formula. Perturbation theory is a tool commonly used in mathematical physics and can easily provide solutions to seemingly impossible problems. Let’s look at how perturbation theory works and is applied.

Look at the following differential equation and initial conditions with no closed form solution.

$y''(x)=\textup{ln}(x)y(x)\rightarrow&space;\begin{matrix}&space;y(0)=0\\&space;y'(0)=1&space;\end{matrix}$

Assume one were to insert an epsilon in the equation.

$y''(x)= \textup{ln}(x)(\epsilon y(x))$

Now we can say the solution to the differential equation is dependent on the value of epsilon. Furthermore, we can propose that this solution takes some form similar to a Taylor series as shown below.

$y(\epsilon,x)=\sum_{n=0}^{\infty}\epsilon^ny_n(x)=y_0(x)+\epsilon&space;y_1(x)+\epsilon^2y_2(x)\textup{&space;}...$

One by one, we can solve for values of this taylor series. Setting $\inline&space;\epsilon=0$ and substituting back into the original differential equation allows us to solve for the first unknown. This is called the “unperturbed” case.

$y''(\epsilon,x)= \textup{ln}(x)(\epsilon y(\epsilon,x))$

$y''_0(x)=0$

$y_0(x)=ax+b$

This equation can be further simplified using the initial conditions shown earlier.

$y_0(x)=x$

This allows for the solution to become more defined and can actually itself be used as an approximation on a small enough range.

$y(\epsilon,x)=x+\epsilon&space;y_1(x)+\epsilon^2&space;y_2(x)\textup{&space;}...$

The rest can be solved for by substituting this solution back into the original differential.

$\left&space;(&space;\frac{\mathrm{d}&space;}{\mathrm{d}&space;x}&space;\right&space;)^2&space;(x+\epsilon&space;y_1(x)+\epsilon^2y_2(x))=\epsilon&space;\textup{ln}(x)&space;(x+\epsilon&space;y_1(x)+\epsilon^2y_2(x))$
$\epsilon&space;y_1''(x)+\epsilon^2y_2''(x))=\epsilon&space;\textup{ln}(x)x+\epsilon^2&space;\textup{ln}(x)y_1(x)+\epsilon^3&space;\textup{ln}(x)y_2(x)$

For now, we will only consider terms with a power of 2 or below because there are some hidden third power terms in the higher level approximations.

$\epsilon&space;y_1''(x)+\epsilon^2y_2''(x)=\epsilon&space;\textup{ln}(x)x+\epsilon^2&space;\textup{ln}(x)y_1(x)$

From here, it becomes easy to solve for the unknown functions assuming epsilon can take up any value and the original initial conditions hold (Note: the rest of these equations should have a slope of 0 at the origin because the $y_0(x)$ already accounted for the slope of 1).

$y_1''(x)=\textup{ln}(x)x$

$y_1(x)=ax+b+\frac{x^3}{6}\left&space;(&space;\textup{ln}(x)-\frac{5}{6}&space;\right&space;)=\frac{x^3}{6}\left&space;(&space;\textup{ln}(x)-\frac{5}{6}&space;\right&space;)$

$y_2''(x)=\textup{ln}(x)y_1(x)=\textup{ln}(x)\frac{x^3}{6}\left&space;(&space;\textup{ln}(x)-\frac{5}{6}&space;\right&space;)$

$y_2(x)=ax+b+\frac{x^5}{120}\left&space;(\textup{ln}^2(x)-\frac{26}{15}\textup{ln}(x)+&space;\frac{17}{25}\right&space;)=\frac{x^5}{120}\left&space;(\textup{ln}^2(x)-\frac{26}{15}\textup{ln}(x)+&space;\frac{17}{25}\right&space;)$

Subsituting back into $\inline&space;y(\epsilon,x)$ yields the following.

$y(\epsilon,x)\approx&space;x+\frac{\epsilon&space;x^3}{6}\left&space;(&space;\textup{ln}(x)-\frac{5}{6}&space;\right&space;)+\frac{\epsilon^2x^5}{120}\left&space;(\textup{ln}^2(x)-\frac{26}{15}\textup{ln}(x)+&space;\frac{17}{25}\right&space;)$

The answer we want is for the case where epsilon is equal to 1 so the answer we were looking for is simply

$y(1,x)\approx&space;x+\frac{x^3}{6}\left&space;(&space;\textup{ln}(x)-\frac{5}{6}&space;\right&space;)+\frac{x^5}{120}\left&space;(\textup{ln}^2(x)-\frac{26}{15}\textup{ln}(x)+&space;\frac{17}{25}\right&space;)$

So, using perturbation theory, we have provided a solution to a seemingly impossible problem through approximations that only used simple integration methods. Graphing the function shows that this is a good approximation.

Perturbation theory is a powerful method and can provide fantastic results to amazingly hard problems throughout physics. Although they do not give exact answers, the error of the approximations can be made arbitrarily small allowing one to accomplish the same goal. Sometimes, however, this does lead to divergent sums but such a case will be acknowledged in a later post.

If you want to know more or see where I learned this from, watch the lectures series on Mathematical Physics by Carl Bender. It is really an amazing series that is both detailed and very easy to follow.