Consider steady two-dimensional heat conduction in a square cross section \((3 \mathrm{~cm} \times 3 \mathrm{~cm}, k=20 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=\) \(6.694 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) ) with constant prescribed temperature of \(100^{\circ} \mathrm{C}\) and \(300^{\circ} \mathrm{C}\) at the top and bottom surfaces, respectively. The left surface is exposed to a constant heat flux of \(1000 \mathrm{~W} / \mathrm{m}^{2}\) while the right surface is in contact with a convective environment \(\left(h=45 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)\) at \(20^{\circ} \mathrm{C}\). Using a uniform mesh size of \(\Delta x=\Delta y\), determine \((a)\) finite difference equations and \((b)\) the nodal temperatures using Gauss-Seidel iteration method.

Based on the solution explained step by step, answer the following question: **Question**: Find the steady-state nodal temperatures of a given 2D domain with uniform grid size using Gauss-Seidel iteration method, given the following boundary and initial conditions: 1. Top surface: \(T = 100^\circ C\) 2. Bottom surface: \(T = 300^\circ C\) 3. Left surface: Constant heat flux of \(1000 W/m^2\) 4. Right surface: Convective environment with \(h=45 W/m^2K\) and \(T_\infty = 20^\circ C\) **Answer**: To find the steady-state nodal temperatures, follow the steps below. 1. Create finite difference equations using the given boundary and initial conditions: \(T_{i+1,j} + T_{i-1,j} + T_{i,j+1} + T_{i,j-1} - 4T_{i,j} = 0\). 2. Rearrange the finite difference equation to obtain an explicit equation for \(T_{i,j}\): \(T_{i,j}^{(k+1)} = \frac{1}{4}\left(T_{i+1,j}^{(k)} + T_{i-1,j}^{(k+1)} + T_{i,j+1}^{(k)} + T_{i,j-1}^{(k+1)}\right)\). 3. Apply the Gauss-Seidel iteration method by starting with an initial guess for nodal temperatures, calculating new temperatures using the explicit equation for \(T_{i,j}\), checking for convergence, and iterating until the convergence criterion is met. 4. The final converged nodal temperatures represent the steady-state temperature distribution in the 2D domain.