1 dimensional non homogeneous heat equation
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Consider the following problem; it can be interpreted as modeling the temperature distribution along a rod of length 1 with temperature decreasing along every point of the rod at a rate of bx (x the distance from the left endpoint, b a constant) while a heat source increases at each point the temperature by a rate proportional to the temperature at that point. In addition, the endpoints of the rod are kept at a temperature of 0. The initial temperature distribution is f(x).
(1) ut = uxx + ((pi)2 /4)*u - bx , 0 < x < 1, t>0
u(0,t) = 0 t>0
u(1,0) = 0 t>0
u(x,0) 0 < x < 1
(a) Determine the steady state (equilibrium) solution. This will require solving a relatively simple ODE (linear, second order; use undetermined coefficients). Your final answer should be
UE(x) = (4b/(pi)2)*(x-sin((Pi/2)*x)
b)Let v(x, t) = u(x, t) − (x). Show that v satisfies the following problem:
(2) vt = vxx + ((pi)2 /4)*v , 0 < x < 1, t>0
v(0,t) = 0 t>0
v(1,0) = 0 t>0
v(x,0) = g(x) 0 < x < 1
Determine how g(x) is related to f(x).
c) Let w(x, t) = e -((Pi)2*t/4)*v(t) show that w satisfies
(3) wt = wxx , 0 < x < 1, t>0
w(0,t) = 0 t>0
w(1,t) = 0 t>0
w(x,0) = h(x) 0 < x < 1
Determine how h(x) is related to f(x).
(d) Solve equation (2) by separation of variables.
(e) Write out the solution of equation (1) for f(x) = uo , uo a constant. Verify that
Lim u(x,t) = uE(x,t)
tinfinity
Solve (2) (part b) to get v(x,t). Clearly state the formula for the coefficients and explain how this formula is obtained.
wt
w(0,t) w(1, t) w(x,0)
= wxx, 0<x<1, t>0, = 0, t>0, =0, t>0, = h(x), 0<x<1.
Determine how h(x) is related to f(x).
(f )
3. Consider the following eigenvalue problem:
Determine the solution u(x, t) of problem (1) and show that limt→∞ u(x, t) = uE (x, t) for all x, 0 < x < 1.
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Solution Summary
The solution (13 pages) shows in a step by step manner how to solve a non-homogeneous heat equation on a finite rod.
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