Questions tagged [perturbation-theory]

Perturbation theory describes a range of tools and techniques to find approximate solutions to problems containing small parameters.

Introduction

Perturbation theory exploits small parameters to find approximate solutions to complicated equations. It can be applied to algebraic equations, difference equations, and ordinary and partial differential equations, including boundary value problems.

An Example

For example, consider the quadratic equation $$ x^2+\epsilon x-1=0, $$ where $\epsilon$ is a very small number. The solution for $\epsilon=0$ is simply $x=\pm 1$, however we assume $x$ can be expanded as an asymptotic series in powers of $\epsilon$, as $x=x_0+\epsilon x_1+\epsilon^2x_2+\ldots$. Then the equations becomes, to $O(\epsilon^2)$, $$ x_0^2+2\epsilon x_0x_1+\epsilon^2(x_1^2+2x_0x_2)+\epsilon x_0+\epsilon^2 x_1-1=0. $$ We solve the equation for each power of $\epsilon$, starting with $O(1)$ terms, $$ x_0^2-1=0\Rightarrow x_0=\pm1,$$ which is what we had for $\epsilon=0$. Then, at $O(\epsilon)$, we have $$ 2x_0x_1+x_0=0\Rightarrow \pm2x_1\pm1=0\Rightarrow x_1=-\frac{1}{2}. $$ We can continue this process indefinitely, for example the $O(\epsilon^2)$ solution is $x_2=\pm 1/8$. We can use these solutions to write $x$ as $$ x=\pm1-\frac{\epsilon}{2}\pm\frac{\epsilon^2}{8}+O(\epsilon^3), $$ and since $\epsilon$ is small, the higher-order correction terms have a very small effect.