## 3 Numeric Solution of the Milling Equation
### Numerical Comparison between Methods  
In order to observe the rate of convergence of the first-, second- and third-order EMHPM, we chose the stable case with cutting conditions \(a_{p}=1.5\) mm, \(a_{d}=1\) and \(n=12\),000 rpm presented in Figure 2(a), and the unstable case with cutting conditions \(a_{p}=3\) mm, \(a_{d}=1\) and \(n=12\),000 rpm showed in Figure 2(b). The rate of convergence was analyzed by computing the absolute error between the solution with \(N\) discrete intervals and a converged solution. All methods were compared against itself using the solution provided with \(N=200\) discrete intervals, which are considered the converged solution. In Figure 2(a) it is observed that the convergence is better for the second- and third-order than the first-order, however, the difference of convergence between second- and third-order with the parameters used was negligible. On the other hand, Figure 2(b) shows that for few discrete intervals the third-order EMHPM had the fastest convergence in comparison with the second- and the first-order EMHPM. However, the second-order and third-order curves behaved very similarly after \(N=50\) discrete intervals. It is important to mention that for a typical stability solution in the ranges of spindle speed 5000-10,000 rpm, \(N=40\) discrete intervals will be enough to have accurate predictions. Figure 2: Numerical comparison of the enhanced homotopy perturbation method (EMHPM) solutions of the milling equation, Equation (23), with the dde23 MATLAB routine. (**a**) Stable milling operation with \(a_{p}=1.5\) mm, \(a_{d}=1\) and \(n=12000\) rpm and (**b**) unstable milling operation with \(a_{p}=3\) mm, \(a_{d}=1\) and \(n=12000\) rpm. Since the rate of convergence was proved for time-domain simulations, we next explored the convergence of the methods applied to the stability analysis. The stability lobes computed with the second- and third-order EMHPM for regular milling tools were compared with its predecessor for radial immersion value of \(a_{d}=\)1 and the other parameters indicated above as it was used in [44]. Figure 4 shows the stability diagrams for spindle speed in the range 2000-3000 rev/min where the precision of the method was compromised due to the higher value of the time delay. While the shaded gray area represents the stability lobes computed with \(N=200\) discrete intervals in all subfigures, in each subfigure solid black lines draw the stability frontier for a specific discrete interval and using the first-, second- or third-order EMHPM. In Figure 4 the first, second and third column represents the solution for the first-, the second- and the third-order EMHPM respectively, while the first and the second row was for \(N=60\) and \(N=100\) discrete intervals, respectively. It is observed that the error achieved in the third-order EMHPM was less than those attained for the first-order and second-order EMHPM solutions. This confirms that the third-order EMHPM had the highest rate of convergence. Figure 3: Convergence rate of absolute error between first-, second- and third-order EMHPM for down-milling operation. Cutting parameters for (**a**) \(a_{p}=1.5\) mm, \(a_{d}=1\) and \(n=12\),000 rpm and (**b**) \(a_{p}=3\) mm, \(a_{d}=1\) and \(n=12\),000 rpm. The results were also compared, in Figure 4g, with the semi-discretization method (SDM) presented by [PERSON] and [PERSON] in [43] (dot line) and with the full-discretization method (FDM) presented by [PERSON] et al. in [45] (dash line). It is observed that the EMHPM converged faster than the SDM and FDM. Table 1 list the results for computation times for a different number of discretized intervals together with the absolute error between stability frontiers. Notice that the solution obtained with the EMHPM with \(N=60\) discrete intervals was faster than the SDM and the FDM and even the error was less in the solution by the EMHPM.