## 3 The Radial Basis Function  
One of the commonly used polyharmonic RBF is [21; 22; 23]  
\[\varphi(r)\ =\ r^{2k}\ln(r),\ k=1,2,3\cdots, \tag{4}\]  
where \(\varphi(r)\) is the basis function, \(r\) is the distance between points, written as \(r=|\mathbf{x}-\mathbf{x}^{s}|\), \(\mathbf{x}\) is the inner or boundary points, \(\mathbf{x}^{s}\) is the source point, and \(k\) is the order which is a certain value. In this study, we propose polyharmonic polynomials (PPs), which are a series of polyharmonic RBFs at each center or source as follows:  
\[\varphi(r)\ =\ \sum_{k=1}^{M_{N}}r^{2k}\ln(r),\ k=1,2,3\cdots \tag{5}\]  
where \(M_{N}\) is the term of the PPs. The first derivative of Equation (5) for \(\mathbf{x}\) and \(\mathbf{y}\) yields the following:  
\[\frac{\partial\varphi(\mathbf{x})}{\partial x}\ \cong\ \sum_{k=1}^{M_{N}}(x-x^{s})r^{2k-2}[2k\ln(r)+1], \tag{6}\]  
\[\frac{\partial\varphi(\mathbf{x})}{\partial y}\ \cong\ \sum_{k=1}^{M_{N}}(y-y^{s})r^{2k-2}[2k\ln(r)+1], \tag{7}\]  
where \(x^{s}\) and \(y^{s}\) are the \(x\)- and \(y\)-coordinates of the center point, respectively. The second derivative of Equation (5) yields the following:  
\[\frac{\partial^{2}\varphi(\mathbf{x})}{\partial x^{2}}\ \cong\ \sum_{k=1}^{M_{N}}\Bigl{[}(x-x^{s})^{2}r^{2k-4}[(4k^{2}-4k)\ln(r)+4k-2]+r^{2 k-2}[2k\ln(r)+1]\Bigr{]}\ \text{and} \tag{8}\]  
\[\frac{\partial^{2}\varphi(\mathbf{x})}{\partial y^{2}}\ \cong\ \sum_{k=1}^{M_{N}}\Bigl{[}(y-y^{s})^{2}r^{2k-4}[(4k^{2}-4k)\ln(r)+4k-2]+r^{2 k-2}[2k\ln(r)+1]\Bigr{]} \tag{9}\]  
Employing the PPs for the function approximation, we have:  
\[u(\mathbf{x})\ \cong\ \sum_{j=1}^{M_{Q}}\sum_{k=1}^{M_{N}}a_{j,k}r_{j}^{2k}\ln(r _{j}), \tag{10}\]  
where \(a_{j,k}\) is the unknown coefficient determined by collocation, \(M_{Q}\) is the total number of source points, \(r_{j}\) is the radial distance at the \(j\)th source point, \(r_{j}=\left|\mathbf{x}-\mathbf{x}_{j}^{s}\right|\), and \(\mathbf{x}_{j}^{s}\) is the \(j\)th  
Figure 1: Inverse problem in the domain of (**a**) simply connected (**b**) multiply connected. source point described as \(\mathbf{x}_{j}^{s}=\left(x_{j}^{s},y_{j}^{s}\right)\). Equation (10) is the function approximation that is built up by intruding the proposed PPs into the RBF. In order to satisfy the boundary conditions (as shown in Equations (2) and (3)), Equation (10) is then imposed at each boundary point. Therefore, we primarily adopt Equation (10) for collocating the boundary points in this study. Inserting the aforementioned equations into Equation (1), the governing equation can be written as:  
\[ \begin{array}{l}\sum\limits_{j=1}^{M_{D}}\sum\limits_{k=1}^{M_{N}}a_{jk}r_{ j}^{2k-2}[4k^{2}\ln(r_{j})+4k]+\alpha\sum\limits_{j=1}^{M_{D}}\sum\limits_{k=1}^{M_{N}}a_{ j,k}(x-x_{j}^{s})r_{j}^{2k-2}\big{[}2k\ln(r_{j})+1\big{]}\\ \quad+\beta\sum\limits_{j=1}^{M_{D}}\sum\limits_{k=1}^{M_{N}}a_{j,k}(y-y_{j}^ {s})r_{j}^{2k-2}\big{[}2k\ln(r_{j})+1\big{]}+\gamma\sum\limits_{j=1}^{M_{D}} \sum\limits_{k=1}^{M_{N}}a_{j,k}r_{j}^{2k}\ln(r_{j})=f(\mathbf{x}).\end{array}  \tag{11}\]  
Equation (11) is mainly utilized for the discretization of inner collocation points within the solution domain. Since the proposed PPs are not a solution to the stationary convection-diffusion equation, the inner points must be collocated inside the domain for the discretization. Thus, Equation (11) is mainly imposed to satisfy the governing equation (stationary convection-diffusion equation) at each inner point.