Consider one-dimensional transient heat conduction in a plane wall with variable heat generation and variable thermal conductivity. The nodal network of the medium consists of nodes 0,1 , and 2 with a uniform nodal spacing of \(\Delta x\). Using the energy balance approach, obtain the explicit finite difference formulation of this problem for the case of specified heat flux \(\dot{q}_{0}\) and convection at the left boundary (node 0 ) with a convection coefficient of \(h\) and ambient temperature of \(T_{\infty}\), and radiation at the right boundary (node 2 ) with an emissivity of \(\varepsilon\) and surrounding temperature of \(T_{\text {surr }}\).

**Question:** Find the explicit finite difference formulation for one-dimensional transient heat conduction in a plane wall with variable heat generation and variable thermal conductivity. The boundary conditions consist of specified heat flux and convection at the left boundary and radiation at the right boundary. **Answer:** The explicit finite difference formulation for the given problem is as follows: $$\frac{T_0^{n+1} - T_0^n}{\Delta t} = \frac{1}{\rho c_p\Delta x}\left[\dot{q}_{0} - h(T_0^n - T_{\infty})\right]$$ $$\frac{T_1^{n+1} - T_1^n}{\Delta t} = \frac{1}{\rho c_p\Delta x}\left[-k_1\frac{T_1^n - T_0^n}{\Delta x} - k_2\frac{T_2^n - T_1^n}{\Delta x}\right]$$ $$\frac{T_2^{n+1} - T_2^n}{\Delta t} = \frac{1}{\rho c_p\Delta x}\left[-k_2\frac{T_2^n - T_1^n}{\Delta x} - \varepsilon\sigma(T_2^{n^4} - T_{\text{surr}}^4)\right]$$ These equations represent the future temperatures at each node based on the current temperatures and the given boundary conditions.