How to Solve Complex Equations When Perturbation Theory Fails

Perturbation theory is useful to any mathematician or physicist who needs to solve a seemingly unsolvable equation such as the one below:

Equation 1 may be hard to solve, but its structure looks simple. Perturbation theory works fine in its case. Later, we will try to solve a much more complex equation, where perturbation theory ceases to be practical, and we will use an alternative method. But for now, let's explore the benefits and limitations of perturbation theory. To solve equation 1 we first reformat it as follows:

Then we expand x and x^2:

Next, we make substitutions:

Let's assume the following terms equal zero:

Using a little algebra and more substitutions we solve for x0, x1, and x2. When we add these x's up we get the value of the original x:

Not bad! But check out this monster:

OK, let's set some values for y and z and solve for x. We derive equation 19:

Should we solve it with perturbation theory? Well, let's see. We first have to expand x^5:

I don't know about you, dear reader, but I don't even want to think about expanding x^5! In our previous example we only had to expand x^2 to only the second power of epsilon. Can you imagine the can of worms we will open if we expand x^5 to two or more powers of epsilon? There's got to be an easier way ... and there is.

To solve equation 19 we use a systematic guess-and-check method. The idea is to choose a value for x that will give us an answer that is close to zero if not exactly zero. We first test -1, 0, and 1. We see that 1 is closer to zero. We try 10. If 10 is better, we try 100, and higher powers of 10 as we get closer to zero. After all, the value for x could be any real number from negative infinity to positive infinity.

Now, after checking 10, we see it's no good. So we now know x is between 1 and 10. We try 2. Whoops, 1 was better. So we try 1.1 and see an improvement. Then we try 1.2--even better. We increment by tenths to 1.3. At 1.4, we overshoot. So the answer is around 1.3, so we try 1.29 and 1.31, i.e., we increment in hundredths. Our final answer is 1.29. If we want more accuracy, we increment in thousandths, and ever smaller powers of 10. Below is a table of our results:

The chief advantage of this method is it is possible to write a computer program that will yield a quick answer.

Update: Click here for a proof of perturbation theory