Question 04 (25 points)
A rod of steel governed by $\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2} = 0$ is subjected to a temperature of 100°C on the left end and 25°C on the right end.
a) Using the following finite difference approximations $\frac{\partial T}{\partial t}|_{i,j} \approx \frac{T_{i}^{j+1} - T_{i}^{j}}{\Delta t}$ and $\frac{\partial^2 T}{\partial x^2}|_{i,j} \approx \frac{T_{i+1}^{j} - 2T_{i}^{j} + T_{i-1}^{j}}{(\Delta x)^2}$, determine the temperature at node j + 1.
b) If the rod is of length 0.05m, using the notation $T_m^p$, do a sample hand calculation of the temperatures $T_1^1$ & $T_2^1$ for $t = \Delta t$ ($p = 1$) (Use a grid graph to illustrate) if the temperature distribution in the rod can be calculated from $t = 0$ and $t = 9$ seconds. Use $\Delta x = 0.01m$, $\alpha = 1.4 \times 10^{-5} m^2/s$, the initial temperature of the rod of 20°C, and the maximum allowable value of $\Delta t$.
