Question 3. Assume the heat conduction in an aluminum rod 5-cm in length is governed by the equation: aT(x, t) a' T (x, t ) at ax2 where the thermal diffusivity is a=0.84 (cm /s). The temperature T(x,t) is in unit of 'C; the positions on the rod x/ to x6 are in unit of cm; time t is in unit of second. The temperatures at =0 have been determined and are listed in the following table, while the temperatures at the ends of the rod (at x1 and x6) are fixed at 0 "C, i.e., they are independent of time. Based on the heat conduction equation, numerical methods are used to determine the temperatures at positions x2 to x5 and at time t, which are marked by symbols in the table. Answer the following questions with At=0.05 s and Ax=1.0 cm. X1=0 X2=1 X3=2 X4=3 X5=4 X6=5 1=0 0.0 12 150 150 12 0.0 1=0.05 0.0 Ta Tb To Td 0.0 In the heat conduction equation, the first derivative is computed by backward finite difference: aT(x,t) / at = [T(x,t)-T(x,t- At) ]/ At , and the second derivative is computed by approximative a' T ( x, t ) / ax) = = [ (x, t) + Txx (x, t- At)], where Txx(x,t) and Txx(x,t-At) are computed by center finite difference: Tax (x, t) = [T(x+Ax, t)-2T(x,t) +T(x-Ax,t) ]/ (4x), Tax (x, t- At) = [T(x+ Ax,t-At)-2T(x,t-At)+T(x-Ax,t - At)]/ (4x). Write down the simultaneous linear equations for unknown temperatures Ta, Tb, Te and Ta at time =0.05 s, and then solve the equations to determine the temperatures Ta, Th, To and Ta. (20 marks)