Transcribed Text
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 steadystate 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 steadystate 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) 602x for 0 dT(x,t)
d²T(x,t)
a
for 0 0
dt
dx2
a 1
Part A: Determine v(x) This is your steadystate 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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