Shining Brightly in the Field of Differential Equations
While perturbation methods can demonstrate their talent in various computational fields, their most brilliant light undoubtedly shines in the field of differential equations, including both ordinary differential equations (ODEs) and partial differential equations (PDEs). Neptune — “the planet discovered at the tip of a pen” — is a famous achievement of perturbation methods, as was the discovery of Pluto. Celestial mechanicians used the gravitational perturbation of a hypothetical planet to explain the anomalous motion of known planets and thereby deduced 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 requires us to consider approximate methods, and the development of such methods has become perturbation theory.
Just like solving algebraic equations, the basic idea of using perturbation methods to solve differential equations with small or large parameters is to express the solution of the differential equation as a power series of the small or large parameter. Of course, this is the most direct approach and is quite easy to understand. However, the resulting series solution may exhibit poor behavior. For example, sometimes the original solution should be a periodic motion, but the series solution contains “secular terms” such as t \sin t, which is quite disadvantageous. Therefore, various techniques have been developed to eliminate these terms. It can be seen that perturbation theory is a widely applicable and practical theory that synthesizes many techniques. Below, we will illustrate this technique through some practical examples.
E.g.1 Direct Series Expansion
Before formally discussing the use of perturbation methods to solve differential equations, let us first look at how to directly expand the solution of a differential equation into a power series of the independent variable. This will help us understand how series work.
\frac{dx}{dt} = \sin x + t, \quad x(0) = 0
This is a fairly simple example, but it is sufficient for the purpose of this section. We know that a function can be represented by a Taylor series as: x(t) = x(0) + x'(0)t + \frac{1}{2}x''(0)t^2 + \frac{1}{6}x'''(0)t^3 + \dots
Therefore, we need to find the values of the derivatives x'(0), x''(0), x'''(0), \dots at the origin. Based on \frac{dx}{dt} = \sin x + t, substituting t=0, x(0)=0 into it, we find x'(0) = 0 + 0 = 0. Next, differentiating both sides of \frac{dx}{dt} = \sin x + t gives x''(t) = (\cos x)x'(t) + 1. Substituting t=0, x(0)=0, x'(0)=0, we get x''(0) = 1. Differentiating again, it becomes x'''(t) = -(\sin x)[x'(t)]^2 + (\cos x)x''(t), which yields x'''(0) = 1. Subsequently, we can find x^{(4)}(0) = 0, and so on. Thus, we have: 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, each term is a power of the independent variable. Since the independent variable is arbitrary, the convergence of the series solution cannot be guaranteed. In fact, this series only provides a relatively accurate solution for very small t. On the other hand, as the order of differentiation increases, the expression for the derivative functions will become exceptionally complex, leading to a breakdown in terms of computational effort. This requires us to consider other approaches.
E.g.2 Expansion in Small Parameters
Perturbation methods do not expand the solution into a power series of the independent variable, but rather into a power series of a small parameter. This allows some excellent properties of the original equation’s solution to be preserved. This is consistent with the idea of solving algebraic equations using perturbation methods. We will illustrate this directly with a simple example.
x'' + x = \varepsilon x^2, \quad x(0) = 1, \quad x'(0) = 0
If we let \varepsilon = 0, we obtain the unperturbed equation x'' + x = 0, which is the simplest second-order linear differential equation. Based on the initial conditions, the solution is x_0(t) = \cos t.
Using this as a basis, 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) + \dots
Substituting this into the original equation, we get: \begin{aligned} &[x_0(t) + \varepsilon x_1(t) + \varepsilon^2 x_2(t) + \varepsilon^3 x_3(t) + \dots]'' + [x_0(t) + \varepsilon x_1(t) + \varepsilon^2 x_2(t) + \varepsilon^3 x_3(t) + \dots] \\ &= \varepsilon [x_0(t) + \varepsilon x_1(t) + \varepsilon^2 x_2(t) + \varepsilon^3 x_3(t) + \dots]^2 \end{aligned}
Expanding and matching terms, 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)] + \dots
Setting each term in the brackets to zero, we obtain: \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 \\ &\dots \end{aligned}
The first equation is our unperturbed equation, with initial conditions x_0(0) = 1, x_0'(0) = 0. For the subsequent x_i(t) terms, x_i(0) = x_i'(0) = 0. Each equation is a linear differential equation; in other words, the perturbation method transforms a nonlinear differential equation into an infinite sequence of linear differential equations. We solve these sequentially to find: \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) \\ &\dots \end{aligned} The solutions to these linear differential equations can be obtained quite quickly using software like Mathematica.
Thus, we can represent 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) + \dots \end{aligned}
This is very effective for small \varepsilon, and can accurately predict the results of motion over a relatively short period of time. Some readers might ask: what if the nonlinear term is not a simple monomial like x^2 that can be expanded, but a complex expression like \sin x? That is also simple, because we assume the solution x(t) is actually a function x(t, \varepsilon). By repeatedly taking partial derivatives with respect to \varepsilon on both sides, we can find the derivatives for each order of \varepsilon. This is not a substantial difficulty.
However, the core difficulty lies in the appearance of the “secular term” t \sin t. There are many reasons for the appearance of this term; it may be that the solution itself is non-periodic, in which case we would not expect a periodic solution. But more often, it is a flaw inherent in the perturbation method itself, as secular terms appear even for differential equations like simple pendulum motion, which are confirmed to be periodic. Many mathematicians have made unremitting efforts to eliminate secular terms, proposing methods such as the Poincaré–Lindstedt (PL) method, the method of multiple scales, the Poincaré–Lighthill–Kuo (PLK) method, and the method of averaging. If I have time, I will continue writing in this direction.
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