Slight Perturbations — An Introduction to Perturbation Methods (3)

轻微的扰动——摄动法简介(3) 苏剑林 · 2013-03-07 · Mathematics differential equation approximation perturbation perturbation

Where Differential Equations Truly Shine

Although perturbation methods can show off their skill in all kinds of computational domains, their most brilliant achievements are unquestionably found in the field of differential equations, both ordinary and partial. Neptune — "the planet discovered at the tip of a pen" — is a famous triumph of perturbation theory, and the discovery of Pluto is similar. Celestial mechanicians explained the anomalous motion of known planets by positing the gravitational perturbation of a hypothetical planet, and from this they worked backward to derive the orbit of the unknown planet. As we have mentioned more than once, the general three-body problem is chaotic and has no exact analytical solution. This forces us to consider approximate methods, and the development of such methods is exactly what became perturbation theory.

Just as with solving algebraic equations, the basic idea of perturbation methods for solving differential equations containing a small (or large) parameter is to express the solution as a power series in that small (or large) parameter. This is of course the most direct approach, and reasonably easy to understand, but the resulting series solution may sometimes behave badly — for instance, the true solution might be a periodic motion, yet the series solution contains "secular terms" such as $t \sin t$, which is quite undesirable. Various techniques have therefore been developed to eliminate such terms. As we can see, perturbation theory is a widely applicable, practical theory that draws together the accumulated wisdom of many hands. Below we will illustrate this technique through some concrete examples. more

Example 1: Direct Series Expansion

Before formally discussing how perturbation methods solve differential equations, let's first look at how to directly expand the solution of a differential equation as a power series in the independent variable — this will help us understand how such series work.

$$\frac{dx}{dt}=\sin x+t,x(0)=0$$

This is a fairly simple example, but it will serve our purposes for this section well enough. We know that a function can be expressed via its Taylor series as:

$$x(t)=x(0)+x'(0)t+\frac{1}{2}x''(0)t^2+\frac{1}{6}x'''(0)t^3+...$$

So we need to find the values of the derivatives of each order $x'(0),x''(0),x'''(0),...$. From $\frac{dx}{dt}=sin x+t$, substituting in $t=0,x(0)=0$, we can find $x'(0)=0+0=0$. Then differentiating both sides of $\frac{dx}{dt}=sin x+t$, we get $x''(t)= (cos x)x'(t)+1$, and substituting $t=0,x(0)=0,x'(0)=0$ gives $x''(0)=1$. Differentiating again gives $x'''(t)=-(sin x)[x'(t)]^2+(cos x) x''(t)$, from which we find $x'''(0)=1$. Continuing in this way we can find $x^{(4)}(0)=0$... and so we obtain:

$$x(t)=\frac{1}{2}t^2+\frac{1}{6}t^3+o(t^5)$$

This is the series solution of the differential equation. Here every term is a power of the independent variable, and since the independent variable can be arbitrary, the convergence of the series solution cannot be guaranteed. In fact, this series only gives an accurate solution for very small t. Moreover, as the order of differentiation increases, the expressions for the derivatives become extraordinarily complicated, so the sheer amount of computation would also overwhelm us. This calls for a different approach.

Example 2: Expansion in a Small Parameter

Rather than expanding the solution as a power series in the independent variable, perturbation methods expand it as a power series in a small parameter, which allows certain desirable properties of the original equation's solution to be preserved. This is consistent with the idea behind using perturbation methods to solve algebraic equations. We illustrate directly with a simple example.

$$x''+x=\varepsilon x^2,x(0)=1,x'(0)=0$$

If we set $\varepsilon=0$, we obtain the unperturbed equation $x''+x=0$, which is the simplest second-order linear differential equation, and from the initial conditions we can find the solution $x_0 (t)=Cos t$.

Building on this, we express the solution of the original equation as:

$$x(t)=x_0 (t) +\varepsilon x_1 (t)+\varepsilon^2 x_2 (t)+\varepsilon^3 x_3 (t)+...$$

Substituting into the original equation gives:

$$\begin{aligned}[x_0 (t) +\varepsilon x_1 (t)+\varepsilon^2 x_2 (t)+\varepsilon^3 x_3 (t)+...]''+[x_0 (t) +\varepsilon x_1 (t)+\varepsilon^2 x_2 (t)+\varepsilon^3 x_3 (t)+...] \\ =\varepsilon [x_0 (t) +\varepsilon x_1 (t)+\varepsilon^2 x_2 (t)+\varepsilon^3 x_3 (t)+...]^2\end{aligned}$$

Expanding, and matching terms correspondingly, we have

$$[x_0 ''(t)+x_0 (t)]+\varepsilon[x_1 ''(t)+x_1 (t)-x_0^2 (t)]+\varepsilon^2 [x_2 ''(t)+x_2 (t)-2 x_0 (t) x_1 (t)] +...$$

Setting each term in parentheses equal to zero gives us:

$$\begin{aligned}x_0 ''(t)+x_0 (t)=0 \\ x_1 ''(t)+x_1 (t)-x_0^2 (t)=0 \\ x_2 ''(t)+x_2 (t)-2 x_0 (t) x_1 (t)=0\end{aligned}$$

...

The first equation is our unperturbed equation, with initial conditions $x_0 (0)=1,x_0'(0)=0$, while for the later $x_i (t)$ terms, $x_i (0)=x_i '(0)=0$. Every one of these equations is a linear differential equation — in other words, perturbation methods turn a nonlinear differential equation into infinitely many linear differential equations. Solving them one by one, we obtain:

$$\begin{aligned}x_0 (t)=Cos t \\ x_1 (t)=\frac{1}{6} (3 - 2 Cos t - Cos 2t) \\ x_2 (t)=\frac{1}{144} (-48 + 29 Cos t + 16 Cos 2t + 3 Cos 3t + 60 t \sin t)\end{aligned}$$

...

These linear differential equations can be solved quite quickly using software such as Mathematica.

We can then express the solution of the original equation as:

$$\begin{aligned}x(t)=Cos t +\frac{1}{6} \varepsilon (3 - 2 Cos t - Cos 2t) \\ + \frac{1}{144}\varepsilon^2 (-48 + 29 Cos t + 16 Cos 2t + 3 Cos 3t + 60 t \sin t)+...\end{aligned}$$

This works very well for small $\varepsilon$, and gives quite accurate predictions of the motion over reasonably short times. Some readers might ask: what if the nonlinear term isn't a simple monomial like $x^2$ that can be directly expanded, but rather a more complicated expression like $sin x$? That's easy too — since we've assumed the solution $x(t)$ is actually a function of $x(t,\varepsilon)$, we just need to keep taking partial derivatives with respect to $\varepsilon$ on both sides to find each order of the derivative with respect to $\varepsilon$. This poses no real difficulty in substance.

The real difficulty, however, lies in the appearance of the "secular term" $ t Sin t$, also called a "long-term" term. There are many reasons such a term can appear — perhaps the true solution is genuinely non-periodic, in which case we wouldn't expect a periodic series solution anyway. But more often it stems from an inherent shortcoming of the perturbation method itself, since secular terms appear even for equations of motion — like that of a pendulum — that are already known to be periodic. Many mathematicians have worked tirelessly to eliminate secular terms, proposing methods such as the PL (Poincaré–Lindstedt) method, the multiple-scales method, the PLK (Poincaré–Lighthill–Kuo) method, the method of averaging, and so on. If time permits, I will continue writing in this direction.