# Asymptotic Analytical Solution on Lamb Waves in Functionally Graded Nano Copper Layered Wafer
## 3 Solution to the Problem
### Ordinary Differential Equations with Variable Coefficients and Power Series Solution
\begin{array}{l}\left[\sum\limits_{u=0}^{\infty}a_{n\lfloor 2\rfloor}\left(\frac{x_{ 3}}{h}\right)^{n}\right]\left[\sum\limits_{u=0}^{\infty}s_{n+2}(n+2)(n+1) \left(\frac{x_{3}}{h}\right)^{n}\right]+\left[\sum\limits_{u=0}^{\infty}a_{n+ 1\lfloor 2\rfloor}(n+1)\left(\frac{x_{3}}{h}\right)^{n}\right]\left[\sum \limits_{u=0}^{\infty}s_{n+1}(n+1)\left(\frac{x_{3}}{h}\right)^{n}\right]\\ +(kh)\left[\sum\limits_{u=0}^{\infty}a_{n+1\lfloor 2\rfloor}(n+1)\left(\frac{x_{3}}{h} \right)^{n}\right]\left[\sum\limits_{u=0}^{\infty}t_{n}\left(\frac{x_{3}}{h} \right)^{n}\right]+(kh)\left[\sum\limits_{u=0}^{\infty}\left(a_{n\lfloor 1 \rfloor}+a_{n\lfloor 2\rfloor}\right)\left(\frac{x_{3}}{h}\right)^{n} \right]\left[\sum\limits_{u=0}^{\infty}t_{n+1}(n+1)\left(\frac{x_{3}}{h} \right)^{n}\right]\\ +(kh)^{2}\left[\sum\limits_{u=0}^{\infty}\left(a_{n\lfloor 3\rfloor}c^{2}-a_{n \lfloor 1\rfloor}-2a_{n\lfloor 2\rfloor}\right)\left(\frac{x_{3}}{h} \right)^{n}\right]\left[\sum\limits_{u=0}^{\infty}s_{n}\left(\frac{x_{3}}{h} \right)^{n}\right]=0\\ \left[\sum\limits_{u=0}^{\infty}\left(a_{n\lfloor 1\rfloor}+2a_{n\lfloor 2 \rfloor}\right)\left(\frac{x_{3}}{h}\right)^{n}\right]\left[\sum\limits_{u=0 }^{\infty}t_{n+2}(n+2)(n+1)\left(\frac{x_{3}}{h}\right)^{n}\right]-(kh)\left[ \sum\limits_{u=0}^{\infty}(n+1)a_{n+1\lfloor 1\rfloor}\left(\frac{x_{3}}{h} \right)^{n}\right]\left[\sum\limits_{u=0}^{\infty}s_{n}\left(\frac{x_{3}}{h} \right)^{n}\right]\\ -(kh)\left[\sum\limits_{u=0}^{\infty}\left(a_{n\lfloor 1\rfloor}+a_{n\lfloor 1 \rfloor}\right)\left(\frac{x_{3}}{h}\right)^{n}\right]\left[\sum\limits_{u=0 }^{\infty}s_{n+1}(n+1)\left(\frac{x_{3}}{h}\right)^{n}\right]+(kh)^{2}\left[ \sum\limits_{u=0}^{\infty}\left(a_{n\lfloor 2\rfloor}c^{2}-a_{n\lfloor 2 \rfloor}\right)\left(\frac{x_{3}}{h}\right)^{n}\right]\left[\sum\limits_{u=0 }^{\infty}t_{n}\left(\frac{x_{3}}{h}\right)^{n}\right]\\ +\left[\sum\limits_{u=0}^{\infty}\left(a_{n+1\lfloor 1\rfloor}+2a_{n+1\lfloor 2 \rfloor}\right)(n+1)\left(\frac{x_{3}}{h}\right)^{n}\right]\left[\sum\limits_{u =0}^{\infty}t_{n+1}(n+1)\left(\frac{x_{3}}{h}\right)^{n}\right]=0\end{array}  \tag{19}\]  
By equating the coefficient of \(\frac{x_{3}}{h}\) to zero, the following recursive relationships are achieved. \[ \begin{array}{l}\sum\limits_{i=0}^{n}(i+2)(i+1)a_{n-i\lfloor 2\rfloor}s_{i+2}+ \sum\limits_{i=0}^{n}(n-i+1)(i+1)a_{n-i+1\lfloor 2\rfloor}s_{i+1}+(kh)\sum \limits_{i=0}^{n}(n-i+1)a_{n-i+1\lfloor 2\rfloor}t_{i}\\ +(kh)^{2}\sum\limits_{i=0}^{n}\left(a_{n-i\lfloor 3\rfloor}c^{2}-a_{n-i \lfloor 1\rfloor}-2a_{n-i\lfloor 2\rfloor}\right)s_{i}+(kh)\sum\limits_{i=0}^{n}(i+1) \left(a_{n-i\lfloor 1\rfloor}+a_{n-i\lfloor 2\rfloor}\right)t_{i+1}\\ =0\sum\limits_{i=0}^{n}(i+2)(i+1)\left(a_{n-i\lfloor 1\rfloor}+2a_{n-i \lfloor 2\rfloor}\right)t_{i+2}+(kh)^{2}\sum\limits_{i=0}^{n}\left(a_{n-i \lfloor 3\rfloor}c^{2}-a_{n-i\lfloor 2\rfloor}\right)t_{i}\\ -(kh)\sum\limits_{i=0}^{n}(n-i+1)a_{n-i+1\lfloor 1\rfloor}s_{i}-(kh)\sum \limits_{i=0}^{n}(i+1)\left(a_{n-i\lfloor 1\rfloor}+a_{n-i\lfloor 2\rfloor}\right)s_{i+1}\\ +\sum\limits_{i=0}^{n}(n-i+1)(i+1)\left(a_{n-i+1\lfloor 1\rfloor}+2a_{n-i+1 \lfloor 2\rfloor}\right)t_{i+1}=0\end{array}  \tag{20}\]  
where \(s_{0}\), \(s_{1}\), \(t_{0}\), and \(t_{1}\) are undetermined coefficients. For \(i\geq 2\), all \(s_{i}\) and \(t_{i}\) values are linear functions of \(s_{0}\), \(s_{1}\), \(t_{0}\), and \(t_{1}\). To decouple the undetermined coefficients, the following matrix is constructed. \[\left( \begin{array}{cccc}s_{0j},&s_{1j},&t_{0j},&t_{1j}\end{array} \right)= \textbf{{I}} \tag{21}\]  
where \(j=1-4\) and \(\textbf{{I}}\) is a \(4\times 4\) unity matrix. The equivalent form of Equation (18) is written as:  
\[U_{1}=\sum\
olimits_{j=1}^{4}C_{j}\sum\
olimits_{n=0}^{\infty}s_{nj}\Big{(} \frac{x_{3}}{h}\Big{)}^{n},U_{3}=-i\sum\
olimits_{j=1}^{4}C_{j}\sum\
olimits_{n= 0}^{\infty}t_{nj}\Big{(}\frac{x_{3}}{h}\Big{)}^{n} \tag{22}\]  
where \(C_{j}(j=1-4)\) represents the undetermined constants. For \(n\geq 2\), the values of \(s_{nj}\) and \(t_{nj}\) can be determined using Equation (20).