## 4 Validation of the Proposed Approach
### Example 5  
We investigate a two-dimensional inverse heat conduction problem in a doubly-connected domain, as depicted in Figure 20. An annular region has an outer radius and inner radius of 3 m and 1 m, respectively. To evaluate the accuracy of the MSMM, we compare the computed results of the method with those of the GFDM [34]. The two-dimensional object boundary is described by the following equation:  
\[\Gamma_{1}=\Big{\{}(x,y)\big{|}x=\rho_{1}cos\theta_{1},\quad y= \rho_{1}sin\theta_{1}\Big{\}},\quad\rho_{1}=3\quad\text{m} \tag{26}\] \[\Gamma_{2}=\Big{\{}(x,y)\big{|}x=\rho_{2}cos\theta_{2},\quad y= \rho_{2}sin\theta_{2}\Big{\}},\quad\rho_{2}=1\quad\text{m} \tag{27}\]  
where the radial coordinate is denoted by \(\rho_{i}\) and the angular coordinate is denoted by \(\theta_{i}\). The exact solution is given by:  
\[T=-\frac{1}{2}x+2y+\frac{1}{4}\Big{(}x^{2}-y^{2}\Big{)}, \tag{28}\]  
\(N_{q}\) is the Trefftz order of the basis functions and is set as 15. For solving this example, as depicted in Figure 20, the boundary data are imposed on the inner boundaries \(\Gamma_{3}\) and \(\Gamma_{7}\). Moreover, there is no information on the outer boundaries \(\Gamma_{1}\) and \(\Gamma_{5}\). Figure 21 shows the collocation points on the physical domain. The DDM [32] is adopted to solve this problem. At the interface, continuity in flux and temperature must be achieved. We split the problem into two simply connected sub-domains, as depicted in Figure 21. For each sub-domain, 300 collocation points are collocated on the domain uniformly. The sub-domain \(\Omega_{1}\) includes \(\Gamma_{1}\), \(\Gamma_{2}\), \(\Gamma_{3}\), and \(\Gamma_{4}\) and the sub-domain \(\Omega_{2}\) includes \(\Gamma_{5}\), \(\Gamma_{6}\), \(\Gamma_{7}\), and \(\Gamma_{8}\). At the interface, \(\Gamma_{2}\), \(\Gamma_{4}\), \(\Gamma_{6}\), and \(\Gamma_{8}\) should achieve continuity in temperature and flux conditions. Therefore, the over-specified boundary conditions are given as follows:  
\[T|_{\Gamma_{2}}=\left.T|_{\Gamma_{6}},\quad\frac{\partial T}{ \partial n}\right|_{\Gamma_{2}}=\left.\frac{\partial T}{\partial n}\right|_{ \Gamma_{6}},\quad T|_{\Gamma_{4}}=\left.T|_{\Gamma_{8}},\quad\frac{\partial T }{\partial n}\right|_{\Gamma_{4}}=\left.\frac{\partial T}{\partial n}\right|_ {\Gamma_{8}}. \tag{29}\]  
To evaluate the stability of the MSMM, different noise levels of \(s=0\) and \(s=3\) are considered in the numerical model. To examine the accuracy, we collocate 200 inner points to obtain the temperature field in each sub-domain. The GFDM is also adopted to solve the same problem. The MAE by assigning different levels of noise for example 5 is depicted in Table 3. It can be seen that the absolute error is relatively small, and should be noted that the accuracy of the numerical solution decreases with the increasing noise level. The computed results demonstrate that the accuracy of the proposed method can reach to the order of \(10^{-1}\) while \(s=3\) is considered. Moreover, the numerical results demonstrate that the accuracy of the MSMM is superior to that of the GFDM, as depicted in Figure 22.  
Figure 19: Comparing of the numerical solutions of the proposed method and the exact solution along \(\Gamma_{1}\).