## 2 Basic Method
This interaction defines the operational parameters and underscores the critical role of \(\beta\) in influencing the behavior of the SGFD model under prescribed conditions, as follows:  
\[\sum_{n=1}^{N}a_{n}(2\text{n}-1)=1 \tag{12}\]  
Thus, \(a_{1}\) can be expressed as follows:  
\[a_{1}=1-\sum_{n=2}^{N}a_{n}(2\text{n}-1) \tag{13}\]  
In the subsequent section, we introduce the procedure for determining the coefficients of the SGFD based on LSM. In accordance with (10), the following integral expression is derived [28  
order partial derivatives on the right-hand side of (14) requires setting these coefficients to zero, as follows [28; 34]:  
\[\left[ \begin{array}{cccc}(\psi_{2},\psi_{2})&(\psi_{2},\psi_{3})&\cdots&(\psi_{ 2},\psi_{N})\\ (\psi_{3},\psi_{2})&(\psi_{3},\psi_{3})&\cdots&(\psi_{3},\psi_{N})\\ \vdots&\vdots&\vdots&\vdots\\ (\psi_{N},\psi_{2})&(\psi_{N},\psi_{3})&\cdots&(\psi_{N},\psi_{N})\\ \end{array} \right]\left[ \begin{array}{c}a_{2}\\ a_{3}\\ \vdots\\ a_{N}\\ \end{array} \right]=\left[ \begin{array}{c}(\psi_{2},\zeta)\\ (\psi_{3},\zeta)\\ \vdots\\ (\psi_{N},\zeta)\\ \end{array} \right] \tag{15}\]  
where \((\psi_{i},\psi_{j})=\int_{0}^{u}[\psi_{i}(\beta)\psi_{j}(\beta)]\,d\beta\), \((\psi_{j},\zeta)=\int_{0}^{u}[\psi_{j}(\beta)\zeta(\beta)]\,d\beta\), \(\psi_{i}(\beta)=\Psi_{i}(\beta)-(2i-1)sin\beta\), \(\zeta(\beta)=\beta-sin\beta\), and \(ij=2\), \(3\ldots\), \(N\). The similar representation of \"\((\psi_{N},\zeta)\)\" in (15) is due to the fact that the LSM method is mainly based on taking the first-order partial derivative of \(E\) and making it zero, so this form is inevitable [34]. Given that the numerical accuracy of the TEM surpasses that of the LSM at a smaller wavenumber range, we opt for \(\beta_{m}\in[\epsilon,\mu]\) in the application of LSM. Additionally, since \(a_{1}\) can be derived from (13), the subsequent formulation of the TLM-based matrix equations can be comprehensively deduced by integrating (11) and (15), as follows:  
\[\left[\begin{array}{cccc}3^{3}&5^{3}&\cdots&(2N-1)^{3}\\ 3^{5}&5^{5}&\cdots&(2N-1)^{5}\\ \vdots&\vdots&\vdots&\vdots\\ 3^{2D-1}&5^{2D-1}&\cdots&(2N-1)^{2D-1}\\ (\psi_{D+1},\psi_{2})&(\psi_{D+1},\psi_{3})&\cdots&(\psi_{D+1},\psi_{N})\\ (\psi_{D+2},\psi_{2})&  
Under identical conditions of operator length and wavenumber range, the accuracy of SGFD coefficients calculated via various methods can be compared by analyzing the resultant errors. [PERSON] et al. [35] established that the accuracy of these comparisons, whether based on absolute or relative errors, remains consistent. Consequently, for the sake of simplicity, we employed the absolute error derived from the following formula for our comparative analysis:  
\[\varepsilon=\sum_{n=1}^{N}a_{n}\Psi(\beta)-\beta \tag{17}\]  
Figure 1 illustrates the absolute error curves derived from TEM, LSM, and TLM, with the SGFD coefficients presented in Tables 1-3. From panels (a) and (d) of Figure 1, it is evident that within the lower wavenumber range, the accuracy of absolute errors for TLM(\(L=1\)) surpasses that of TEM; however, the absolute errors associated with LSM exceed those of TEM, while for conditions \(L=2\) to \(L=6\), TLM's absolute errors are greater than those related to LSM. Panels (b) and (e) reveal that in the intermediate wavenumber range, the accuracies of absolute errors for TLM(\(L=1\)) are superior to both LSM and TEM, with TEM exhibiting the largest relative errors. In conditions \(L=2\) to \(L=6\), the absolute errors exceed those associated with TEM. Due to minimal differences in the curves shown in Figure 1c, our comparative analysis primarily utilizes Figure 1f, which demonstrates that in the higher wavenumber range, the accuracies of absolute errors for TLM(\(L=1\)) and LSM are optimal, with nearly identical curves for these methods. Moreover, under conditions \(L=2\) to \(L=6\), TLM's absolute errors exceed those based on TEM. The results indicate that the numerical errors of TLM at \(L=1\) are significantly lower than those at other \(L\) values.