## 4 Numerical Examples
### Example 2  
A steady-state AC equation was derived for these four shape domains, where \(H(x,y)=u^{3}-3\exp\cdot\sin(x)\sin(y)\) and the closed-form solution is:  
\[u=\exp\cdot\sin(x)\sin(y). \tag{32}\]  
Under the Dirichlet boundary condition, for the peanut domain, with \(n_{1}=28\), \(n_{2}=4\) (\(n_{k}=112\)), \(m=7\), and \(n=28\), a multiple-scale expansion approach was utilized by applying the CGM with a convergence criterion of \(\varepsilon_{1}=10^{-11}\). At \(\varepsilon_{2}=10^{-11}\), convergence was reached after 6 iterations and the results are shown in Figure 12a,b. The numerical error is shown in Figure 13, the maximum error being \(5.82\times 10^{-9}\). For the Cassini-shaped domain, with \(n_{1}=50\), \(n_{2}=10\) (\(n_{k}=500\)), \(m=12\), \(n=78\), and convergence criterion of \(\varepsilon_{1}=10^{-10}\). At \(\varepsilon_{2}=10^{-10}\), convergence was reached after 17 iterations and the results are shown in Figure 14a,b. The numerical error is shown in Figure 15, and the maximum error was \(2.45\times 10^{-9}\). For the irregular gear domain, with \(n_{1}=65\), \(n_{2}=3\) (\(n_{k}=195\)), \(m=5\), \(n=15\), and convergence criterion of \(\varepsilon_{1}=10^{-2}\). At \(\varepsilon_{2}=10^{-2}\), convergence was reached after 15 iterations and the results are shown in Figure 16a,b. The numerical error is shown in Figure 17, and the maximum error was \(0.61\). Comparison with the maximum absolute value of \(2.71\) of \(u(x,y)\) shows the above error to be acceptable. For the amoeba-like irregular domain, with \(n_{1}=50\), \(n_{2}=2\) (\(n_{k}=100\)), \(m=11\), \(n=66\)  
Figure 11: The numerical errors with random noise. Figure 10: The numerical solutions for the steady-state AC equation with amoeba-like irregular domain are shown in (**a**), and in (**b**) the proposed approach solution with random noise. and convergence criterion of \(\varepsilon_{1}=10^{-7}\). At \(\varepsilon_{2}=10^{-7}\), convergence was reached after more than 100 iterations and the results are shown in Figure 18a,b. The numerical error is shown in Figure 19, and the maximum error was 1.70 \(\times\) 10\({}^{-3}\). Note that the multiple-scale scheme is highly efficient, and it can offer a very accurate solution. To the authors' best knowledge, there are no existing published reports of numerical solutions to this problem that provide more accurate results than these. Concern about the stability of this scheme, where the boundary data were perturbed by random noise, was investigated by adding random noise to the boundary data. For the amoeba-like irregular domain, with \(n_{1}=35\), \(n_{2}=2\) (\(n_{k}=70\)), \(m=7\), \(n=28\), and convergence criterion of \(\varepsilon_{1}=10^{-2}\). At \(\varepsilon_{2}=10^{-2}\), convergence was reached after more than 100 iterations and the results are shown in Figure 20a,b. The numerical error is shown Figure 21, and the maximum error was 0.23. Note that the multiple-scale method can acquire accurate results with imposed noise as large as 7.5%. Figure 12: The exact solutions for the steady-state AC equation with peanut-shaped domain are shown in (**a**), and in (**b**) the proposed approach solution without random noise. Figure 13: The numerical errors of proposed method solution for the steady-state AC equation with a peanut-shaped domain. Figure 14: The exact solutions for the steady-state AC equation with Cassini-shaped domain are shown in (**a**), and in (**b**) the proposed approach solution without random noise. Figure 15: The numerical errors of proposed approach solution for the steady-state AC equation with a Cassini-shaped domain. Figure 17: The numerical errors of the proposed method solution for the steady-state AC equation with the gear shaped irregular domain. Figure 18: The exact solutions for the steady-state AC equation with amoeba-like irregular domain are shown in (**a**), and in (**b**) the proposed scheme solution without random noise.