## 3 Proposed Theoretical Derivation
### Analytical Solution  
To simulate 2D heat transfer or irregular shapes via Gauss-Seidel iterations [19; 20; 21; 22; 23; 24; 25], a prior testing of applying Equation (14) within a script or Excel against a regular shape such as a square is required to ensure that results are accurate. To do so, an analytical solution is produced via Excel initially by implementing Equation (14) with reference to Figure 6 for a visualisation of how it is applied. This analytical solution provides the exact solution for a given square boundary which can then be compared with the numerical solution [26] obtained via the script produced. \[\theta(x,y)=\frac{2}{\pi}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}+1}{n}\sin\frac{n \pi x}{L}\frac{\sinh\left(\frac{n\pi y}{L}\right)}{\sinh\left(\frac{n\pi y}{L} \right)} \tag{14}\]  
whereby  
* \(x\) and \(y\) are coordinate points with reference to nodes;
* \(W\) is the width of the boundary condition;
* \(L\) is the length of the boundary condition. Equation (14) is valid, as the summation from one to infinity suggests a series solution rather than numerical integration. The correct approach involves solving the differential equation analytically rather than numerically integrating the series. Here is our explanation:  
Equation (14) below represents an analytical solution for the temperature distribution in a rectangular domain using a Fourier series. Furthermore,  
* \(\theta(x,y)\) is the dimensionless temperature variable;
* x and y are in meters (m);
* L and W are in meters (m). The solution is achieved through the summation of the series, but for practical research purposes, a finite number of terms is used. Numerical integration is not appropriate here; rather, the series is truncated to a reasonable number of terms to approximate the solution. The summation should be correctly handled by truncating the series after enough terms to ensure accuracy without resorting to a numerical integration of the differential equation. Figure 6: Visualization of the analytical solution. As Equation (14) will be tested on a square domain, nodes indicated with a yellow 'x' in the middle and left of the square domain as shown in Figure 6 are the ones that will have their \(\theta\) values calculated as they mirror the nodes on the right-hand side. From the \(\theta\) values obtained, Equation (15) can be used to find the temperature values at the respective nodes. \[T(x,y)=\theta(T_{2}-T_{1})+T_{1} \tag{15}\]  
Equation (15) is derived from the temperature distribution equation. \[\theta\equiv\frac{T(x,y)-T_{1}}{T_{2}-T_{1}}\]  
Equation (15) aims to express the temperature T\({}_{XY}\) at a specific node in terms of a dimensionless temperature variable \(\theta\). This is based on the temperature distribution analysis within a given boundary. Furthermore, Equation (15) also shows that \(\theta\) is a dimensionless quantity as it represents the normalized temperature difference. We have clarified that Equation (15) does not add different physical quantities. Instead, it multiplies the dimensionless temperature \(\theta\) by the temperature difference T\({}_{2}\)-T\({}_{1}\) (which has the units of temperature), and then adds the reference temperature T\({}_{1}\) (which also has the units of temperature). The entire expression remains dimensionally consistent and results in a temperature value. Therefore, Equation (15) correctly combines the dimensionless temperature variable with temperature differences to yield a temperature value at a specific node, maintaining the consistency of physical units throughout the calculation. Theta (\(\theta\)) is related to temperature distribution during the analysis of 2D heat transfer in irregular shapes. It is a dimensionless temperature variable used in the analytical solution of the temperature distribution equation.