# A Comparative Study of Differential Quadrature Methods for METE Nanobeam Vibrations
## 3 Method of Solution
### Discrete Singular-Convolution Differential Quadrature Method (DSCQQM)  
As defined in the references [36; 37; 38; 39; 40], a singular convolution is given by  
\[F(t)=(T\times\eta)(t)=\int_{-\infty}^{\infty}T(t-x)\eta(x)dx \tag{39}\]  
where the singular kernel is denoted by \(T(t-x)\). The DSC algorithm leverages various kernel functions to approximate the unknown function \(v\) and its derivatives at discrete nodes, \(v_{i}(i=-N:N)\), within a localized region defined by a narrow bandwidth, \((x-x_{M},x+x_{M})\)[36; 37; 38; 39; 40]. Two kernels of DSC will be employed as follows:  
1. The Delta Lagrange Kernel (DLK) serves as the shape function within the DSC algorithm. This kernel enables the approximation of the unknown function \(v\) and its derivatives through a weighted linear combination of nodal values, \(v_{i}(i=-N:N)\): \[v(x_{i})=\sum_{j=-M}^{M}\frac{\prod_{k=-M}^{M}(x_{i}-x_{k})}{ \left(x_{i}-x_{j}\right)\prod_{j=-M,j\
eq k}^{M}(x_{j}-x_{k})}v\big{(}x_{j} \big{)},\quad(i=-N:N),M\geq 1\] (40) \[\frac{\partial v}{\partial x}\Big{|}_{X=x_{i}}=\sum_{j=-M}^{M}C_ {ij}^{(1)}v\big{(}x_{j}\big{)},\quad\frac{\partial^{2}v}{\partial x^{2}}\Big{|} _{x=x_{i}}=\sum_{j=-M}^{M}C_{ij}^{(2)}v\big{(}x_{j}\big{)},\quad(i=-N:N),\] (41)  
Likewise, we can approximate \(\frac{\partial^{3}v}{\partial x^{3}}\frac{\partial^{4}v}{\partial x^{4}}\) and determine \(C_{ij}^{(3)},C_{ij}^{(4)}\), where the effective computational bandwidth is denoted by \(2M+1\). \(C_{ij}^{(1)},C_{ij}^{(2)},C_{ij}^{(3)}\), and \(C_{ij}^{(4)}\) are obtained as  
\[C_{ij}^{(1)}=\left\{ \begin{array}{l}\frac{1}{\left(x_{i}-x_{j} \right)}\prod_{k=-M,k\
eq i,j}^{M}\frac{\left(x_{i}-x_{k}\right)}{\left(x_{j}- x_{k}\right)},i\
eq j\\ \\ -\sum_{j=-M,j\
eq i}^{M}C_{ij}^{(1)},i=j\end{array} \right., \tag{42}\] \[C_{ij}^{(3)}=\left\{ \begin{array}{l}3\bigg{(}C_{ij}^{(1)}C_{ii} ^{(2)}-\frac{C_{ij}^{(2)}}{\left(x_{i}-x_{j}\right)}\bigg{)},i\
eq j\\ \\ -\sum_{j=-M,j\
eq i}^{M}C_{ij}^{(3)},i=j\end{array} \right. \tag{43}\]  
The Regularized Shannon Kernel (RSK) is also utilized as a shape function to approximate \(v\) and its derivatives in terms of the nodal values \(v_{i}(i=-N:N)\):  
\[\psi(x_{i})=\sum_{j=-M}^{M}\bigg{\langle}\frac{\sin\left[\tau\big{(}x_{i}-x_{ j}\big{)}/h_{x}\right]}{\pi\left(x_{i}-x_{j}\right)/h_{x}}e^{-\left(\frac{ \left(x_{i}-x_{j}\right)^{2}}{2\sigma^{2}}\right)}\bigg{\rangle}\psi\big{(}x_ {j}\big{)},\]  
\[(i=-N:N),\sigma=(r\times h_{x})>0 \tag{44}\]  
\[\frac{\partial v}{\partial x}\bigg{|}_{X=x_{i}}=\sum_{j=-M}^{M}C_{ij}^{(1)}v \big{(}x_{j}\big{)},\quad\frac{\partial^{2}v}{\partial x^{2}}\Big{|}_{x=x_{i} }=\sum_{j=-M}^{M}C_{ij}^{(2)}v\big{(}x_{j}\big{)},\quad(i=-N,N), \tag{45}\]  
Likewise, we can approximate \(\frac{\partial^{3}v}{\partial x^{3}}\frac{\partial^{4}v}{\partial x^{4}}\) and determine \(C_{ij}^{(3)},C_{ij}^{(4)}\). The parameters \(\sigma\) and \(\tau\) are the regularization parameter and computational parameter, respectively. The weighting coefficients \(C_{ij}^{(1)},C_{ij}^{(2)},C_{ij}^{(3)}\), and \(C_{ij}^{(4)}\) are obtained from the formulas presented in the reference [43]:\[C_{ij}^{(1)} =\left\{ \begin{array}{ll}\frac{(-1)^{i-j}}{h_{x}(i-j)}e^{-h_{x}2 \left(\frac{(i-j)^{2}}{2\sigma^{2}}\right)},&i\
eq j\\ 0&i=j\end{array} \right.,C_{ij}^{(2)}=\left\{ \begin{array}{ll}\left(\frac{2(-1) ^{i-j+1}}{h_{x}^{2}(i-j)^{2}}+\frac{1}{\sigma^{2}}\right)e^{-h_{x}^{2}\left( \frac{(i-j)^{2}}{2\sigma^{2}}\right)},&i\
eq j\\ 0&i=j\end{array} \right.\] \[C_{ij}^{(3)} =\left\{ \begin{array}{ll}c\frac{(-1)^{i-j}}{h_{x}^{2}(i-j)^{3}} \left(\frac{\pi^{2}}{h_{x}^{2}(i-j)}+\frac{6}{h_{x}^{2}(i-j)^{3}}+\frac{3h_{x}( i-j)}{\sigma^{2}}+\frac{3h_{x}(i-j)}{\sigma^{4}}\right)}e^{-h_{x}\left( \frac{(i-j)^{2}}{2\sigma^{2}}\right)},&i\
eq j\\ 0&i=j\end{array} \right.\] \[C_{ij}^{(4)} =\left\{