## 4 Numerical Examples  
### Example 1  
We first consider the following AC equation in heat transfer of thin films in an arbitrary domain (peanut shape, Cassini shape, gear shape, and amoeba-like irregular shape) as shown in Figure 1.  
\[\Delta u=u-u^{3}+H(x,y), \tag{26}\]  
\[\rho(\theta)=0.3\sqrt{\cos(2\theta)+\sqrt{1.1-\sin^{2}(2\theta)}}, \tag{27}\]  
\[\rho(\theta)=\left[\cos 4\theta+\sqrt{18/5-\sin^{2}4\theta}\right]^{1/3}, \tag{28}\]  
\[x=0.5[2+0.5\sin(7\theta)]\cos[\theta+0.5\sin(7\theta)],y=0.5[2+0.5\sin(7\theta) ]\sin[\theta+0.5\sin(7\theta)], \tag{29}\]  
\[\rho(\theta)=\exp[\sin(\theta)]\sin^{2}(2\theta)+\exp[\cos(\theta)]\cos^{2}( 2\theta), \tag{30}\]  
where \(\rho\) is the radius function of \(\Gamma\), \(H(x,y)=u^{3}-u+6\exp(xy^{3}+x^{3}y)-6x^{2}y(x^{2}+2y^{2})\) and the closed-form solution is  
\[u=x^{3}y^{3}(\exp-x). \tag{31}\]  
Under the Dirichlet boundary condition, for the peanut domain, with \(n_{1}=27\), \(n_{2}=4\) (\(n_{k}=108\)), \(m=7\), and \(n=28\), we employed the multiple-scale expansion method by using the CGM under the convergence criterion \(\varepsilon_{1}=10^{-8}\). At \(\varepsilon_{2}=10^{-8}\), the iterative process reaches convergence after 7 iterations and the results are shown in Figure 2a,b. The numerical error is shown in Figure 3, the maximum error being 1.88\(\times 10^{-10}\). For the Cassini-shaped domain, with \(n_{1}=26\), \(n_{2}=2\) (\(n_{k}=52\)), \(m=8\), \(n=36\), and a convergence criterion of \(\varepsilon_{1}=10^{-12}\). At \(\varepsilon_{2}=10^{-12}\), the iterative process reaches convergence after 11 iterations and the results are shown in Figure 4a,b. The numerical error is plotted in Figure 5, the maximum error being 1.40 \(\times\) 10\({}^{-13}\). For the gear irregular domain, under \(n_{1}=25\), \(n_{2}=2\) (\(n_{k}=50\)), the same \(m\), \(n\), and convergence criterion \(\varepsilon_{1}=10^{-3}\). At \(\varepsilon_{2}=10^{-3}\), convergence is reached after 8 iterations and the results are shown in Figure 6a,b. The numerical error is plotted in Figure 7, the maximum error being 3.09. Comparison with the maximum absolute value of 124.88 of \(u(x,y)\), shows this error to be acceptable. For the amoeba-like irregular domain, with \(n_{1}=50\), \(n_{2}=2\) (\(n_{k}=100\)), the same \(m\), \(n\), and convergence criterion, convergence is reached after 11 iterations and the results are shown in Figure 8a,b. The numerical error is plotted in Figure 9 and the maximum error is 9.04 \(\times\) 10\({}^{-9}\). Note that the multiple-scale method is highly efficient, and it can provide a very accurate solution. Furthermore, to the authors' best knowledge, there are no existing published reports of numerical solutions to this problem that provide more accurate results than these. To address concern about the stability of this scheme when boundary data were perturbed by random noise, an investigation was carried out with random noise added to the boundary data. For the amoeba-like irregular domain, with \(n_{1}=60\), \(n_{2}=2\) (\(n_{k}=120\)), \(m=8\), \(n=36\), and convergence criterion \(\varepsilon_{1}=10^{-4}\). At \(\varepsilon_{2}=10^{-4}\), convergence was reached after 100 iterations and the results are shown in Figure 10a,b. The numerical error plot is shown in Figure 11 and the maximum error was 0.36. Note that the multiple-scale method still gives an accurate result with imposed noise equal to 0.5. Figure 1: _Cont._  
Figure 1: Four geometric configurations of the Allen-Cahn (AC) equation in heat transfer of thin films are shown in (**a**) the peanut shape, in (**b**) the Cassini shape, in (**c**) the gear irregular shape, and in (**d**) the amoeba-like irregular shape. Figure 2: 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. \[\Delta u=u-u^{3}+H(x,y), \tag{26}\]0 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.