Application to Steady-State Heat Conduction.
Use function for LU decomposition and substitution in order to solve a problem involving 1-D
steady-state heat conduction. Consider a slender bar of length L. On one end (x = 0) it is subjected
to a fixed temperature, To, while at the other end (. = L) it is insulated. There is an
external heat source, with heat per unit length, dc(I). The governing equation for the temperature
distribution, T(x), is
T(0) = To,
(L) = O,
where 6 is the thermal conductivity. You should obtain a linear set of equations, AT = b, for the
temperatures at the nodes. Your routines must set up the problem-including setting up the matrix A
for the discrete set of equations-and solve it and plot results.
In your study, set L - 1 m and To = 500 K. The thermal conductivity is N = 40W/(m K).
Discretize the problem on a grid with 200 points. The heat source has the form
Use 40 = 1000W/m². Try two different values for 20 (where both values are between 0 and L) and
two different values of OT (I suggest a = 0.5 and a = 0.05). Remember that the point of using LU
decomposition is that this only needs to be performed once for a given matrix A. Only the forward
and backward substitution need to be repeated when the right-hand side changes.
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%Application to steady state heat conduction
%Clearing the screen, workspace and closing all plots