At the beginning of this journey, I happened to come across a book in the library titled Beyond Perturbation: Introduction to the Homotopy Analysis Method. It introduced a new method for finding approximate solutions to differential equations. Crucially, the content did not seem overly difficult to understand, so I borrowed it to study with great interest. As it turns out, this is a very fascinating method and, in a sense, a very elegant one. It solved a problem I had long wanted to investigate: using Fourier series to describe the approximate solution of pendulum motion. Of course, the impact it had on me went beyond just that. To derive periodic solutions, I also studied various techniques of perturbation methods, such as the Poincaré–Lindstedt (PL) method for eliminating secular terms. This simultaneously increased my understanding of various approximate analytical methods. I have been researching these issues for nearly three weeks since the start of the semester.
My mathematical intuition tells me that the Homotopy Analysis Method (HAM) indeed possesses great potential. However, what I have found is that although the author, Shijun Liao, repeatedly emphasizes that “this method not only overcomes the limitations of perturbation methods’ dependence on small parameters but also logically contains other non-perturbation methods, such as Lyapunov’s artificial small parameter method, the Adomian decomposition method, and the \delta-expansion method,” I feel that the Homotopy Analysis Method essentially still belongs to the category of perturbation methods. It is merely a variation of the artificial small parameter approach, and its processing techniques are almost identical to those of perturbation methods. The author’s most successful work was developing this technique into a relatively systematic treatment method and proving related properties (such as convergence). Therefore, I do not entirely agree with the author’s tendency to “over-deify” this method.
Regardless, I believe the author was the first to independently propose these results, so I have no wish to criticize. On the contrary, I am very pleased to be able to see and use this method. Below, I present my simple research results on pendulum motion (accurate only up to \theta_0^5):
The equation of motion for any simple pendulum can be transformed into the following by appropriately choosing the units: \ddot{\theta} + \sin\theta = 0 with an initial angle \theta_0.
The approximate series solution is: \theta = \left( \theta_0 + \frac{\theta_0^3}{192} + \frac{17 \theta_0^5}{61440} \right) \cos(\omega t) - \left( \frac{1}{192}\theta_0^3 + \frac{1}{3072}\theta_0^5 \right) \cos(3\omega t) + \frac{\theta_0^5 \cos(5 \omega t)}{20480}
where \omega = 1 - \frac{1}{16}\theta_0^2 + \frac{1}{3072}\theta_0^4 - \dots
The period is: T = 2\pi \left( 1 + \frac{1}{16}\theta_0^2 + \frac{11}{3072}\theta_0^4 + \dots \right)
This series solution satisfies \theta = \theta_0 at t=0 and \theta = 0 at t = \frac{\pi}{2\omega} = \frac{T}{4}. However, although this solution escapes the dependence on t being small, it does not escape the dependence on \theta_0 being small. Fortunately, in realistic scenarios, \theta_0 is generally quite small (even a 90-degree right angle is only \frac{\pi}{2} \approx 1.57, which is only slightly larger than 1). Therefore, convergence can usually be guaranteed. Of course, a better approach would be to change the power series expansion of \theta_0 into a power series expansion of \sin\frac{\theta_0}{2}, just as we previously handled the pendulum period. This is still under exploration.
The exact series solution should take the following form: \theta = a_1(\theta_0) \cos(\omega t) + a_3(\theta_0) \cos(3\omega t) + a_5(\theta_0) \cos(5\omega t) + \dots
where a_1(\theta_0), a_3(\theta_0), a_5(\theta_0), \dots are all infinite series of \theta_0, and ideally, they would be infinite series of \sin\frac{\theta_0}{2} \dots
I am placing the results here for everyone to enjoy! As for the detailed derivation process and the specifics of the Homotopy Analysis Method, I will introduce them in future articles.
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