Page 218 - Applied Numerical Methods Using MATLAB

PROBLEMS 207
for five different sets of data rates

a = [32 32 32 32], [643232 32], [128 32 32 32], [256 32 32 32], and [512 32 32 32]

and plots a 1 /x 1 (the number of subchannels assigned to user 1) versus a 1
(the data rate of user 1).

4.10 Temperature Rising from Heat Flux in a Semi-infinite Slab
Consider a semi-infinite slab whose temperature rises as a function of posi-
tion x> 0 and time t> 0as

2
Qx e −s 2 2
T(x, t) = √ − erfc(s) with s = x /4at (P4.10.1)
k πs
where the function erfc() is defined by Eq. (P4.9.3) and

2
◦
Q(heat flux) = 200 J/m s, k (conductivity) = 0.015 J/m/s/ C,
2
a(diffusivity) = 2.5 × 10 −5 m /s
In order to find the heat transfer speed, a heating system expert, Kyung-
won, wants to solve the above equation to get the positions x(t) with a
◦
temperature rise of T = 30 Cat t = 10:10:200 s. Compose the program
which does this job and plots x(t) versus t.
4.11 Damped Newton Method for a Set of Nonlinear Equations
Consider the routine “newtons()”, which is made for solving a system of
equations and introduced in Section 4.6.
(a) Run the routine with the initial point (x 10 ,x 20 ) = (0.5, 0.2) to solve
Eq. (4.6.5) and certify that it does not yield the right solution as depicted
in Fig. 4.6c.
(b) In order to keep the step size adjusted in the case where the norm of the
vector function f(x k+1 ) at iteration k + 1 is larger than that of f(x k ) at
iteration k, insert (activate) the statements numbered from 1 to 6 of the
routine “newtons()” (Section 4.6) by deleting the comment mark (%)at
the beginning of each line to make a modified routine “newtonds()”,
which implements the damped Newton method. Run it with the initial
point (x 10 ,x 20 ) = (0.5, 0.2) to solve Eq. (4.6.5) and certify that it yields
the right solution as depicted in Fig. 4.6d.
(c) Run the MATLAB built-in routine “fsolve()” with the initial point
(x 10 ,x 20 ) = (0.5, 0.2) to solve Eq. (4.6.5). Does it present you a right
solution?