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1. (3 points) The Fourier series solution for the heat conduction equation of has the formal solution of - when we assume that T(x,0) is a constant (i.e. T(0,0) = T(x,0)=T(L,0). If we maintain one end of our rod at a constant temperature T1 and the other at T2, we can redefine the boundary conditions as: T(0,1) T1 and T(L,0) T2 for all 1> O. Part A: Determine the expression for v(x), which is the spatial distribution of Tas a function of X at steady state (i.e. determine T(x) at 00). Part B: You can now express T(x,t) as the sum of the steady-state temperature distribution v(x) and as a transient temperature distribution w(x,t). i.e. T(x,t) = v(x) w(x,t). Starting with a dt dx² show that dw(x,2) d²w(x,1) a at dx² Part C: Since T(x,t) = v(x) + w(x,t) must be true for all x and t. determine the boundary conditions for w(x.t). In other words, determine w(0,1); w(L,t), and w(x,0). Part D: Show that - x dx 2. (2 points) Using the solutions above, graph the initial condition (t 0), intermediate conditions (==3 and 25), and the steady-state condition (t =00). Use the following information and complete the following steps: T(0,1) 20 for all >0 T(30,t) 50 for all >0 T(x,0) 60-2x for 0 dT(x,t) d²T(x,t) a for 0 0 dt dx2 a 1 Part A: Determine v(x) This is your steady-state condition. Part B: Determine w(x,0) = T(x,0) - v(x) This is your initial state condition. Part C: Determine C1 and C2 (the first two constants in the Fourier series). Hint: use integration by parts and let Part D: Create a spread sheet for 0for and 25. Part E: What do you observe?

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