I am reading a book "Homotopy Analysis Method in Nonlinear Differential Equations" by Shijun Liao chapter 13 Applications in Finance: American Put Options.
It is stated there that
Substituting the series (13.27) and (13.28)into (13.15),(13.16)and (13.18),then equating the like-power of $q$, we have the so-called $n$th-order deformation equation ($n \geq 1$)
\begin{equation} \label{ 13.32 } -\frac{\partial V_n(S,\tau)}{\partial \tau}+S^2\frac{\partial^2 V_n(S,\tau)}{\partial S^2}+\gamma S \frac{\partial V_n(S,\tau)}{\partial S}-\gamma V_n(S,\tau)=0 \end{equation}
subject to the initial/boundary conditions
\begin{equation} \label{ 13.33 } V_n(S,0)=0, \end{equation} \begin{equation} \label{ 13.34 } \lim_{S\rightarrow \infty} V_n(S,\tau)=0, \end{equation}
where equation (13.27), (13.28), (13.15), (13.16), and (13.18) are
\begin{equation} \label{ 13.27 } \phi(S,\tau;q)=V_0(S,\tau)+\sum_{n=1}^{\infty}V_n(S,\tau)q^n, \end{equation}
\begin{equation} \label{ 13.28 } \Lambda(\tau;q)=B_0(\tau)+\sum_{n=1}^{\infty}B_n(\tau)q^n, \end{equation}
\begin{equation} \label{ 13.15 } -\frac{\partial \phi(S,\tau;q)}{\partial \tau}+S^2\frac{\partial^2 \phi(S,\tau;q)}{\partial S^2}+\gamma S \frac{\partial \phi(S,\tau;q)}{\partial S}-\gamma \phi(S,\tau;q)=0 \end{equation}
\begin{equation} \label{ 13.16 } \phi(S,0;q)=0, \end{equation}
\begin{equation} \label{ 13.18 } \lim_{S\rightarrow \infty} \phi(S,\tau;q)=0, \end{equation}
Note: series (13.27) and (13.28) are homotopy-Maclaurin series where
\begin{equation} \label{13.29} \begin{split} & V_n(S,\tau)=\frac{1}{n!}\frac{\partial^n \phi(S,\tau;q)}{\partial q^n}\bigg|_{q=0}=\mathcal{D}_n[\phi(S,\tau;q)],\\ & B_n(\tau)=\frac{1}{n!}\frac{\partial^n \Lambda(\tau;q)}{\partial q^n}\bigg|_{q=0}=\mathcal{D}_n[\Lambda(\tau;q)],\\ \end{split} \end{equation} are the $n$th-order homotopy-derivatives of $\phi (S, \tau ;q)$ and $\Lambda(q)$, respectively, and $D_n$ is the $n$th-order homotopy-derivative operator.
I don't understand what "equating the like-power of $q$" mean and thus I have no idea how to obtain the first three equations. I considered asking this question in ell.stackexchange.com but I think it's better to ask it in this site.
Can anyone tell me what does "equating the like-power of $q$" mean by explaining (step by step) how to get equation the first three equations?
OR, could anyone please tell me what is "like-power" or give any reference about it?
Thanks.
If you have power series $\sum_{n\geq0}a_nq^n$ and $\sum_{n\geq0}b_nq^n$, and you know that they are equal, then you know that $a_n=b_n$ for all $n\geq0$. That is «equating the like powers of $q$» in the two series.