MAE107 Homework #5
Prof. MCloskey
Due Date
The homework is due at 9AM, Friday, February 20, 2015, to the TA at the beginning of recitation.
Problem 1
Do problem 9.42 in the course text (page 808 in the paperback version). First derive the dierential
equat

MAE107 Homework #1
Prof. MCloskey
Due Date
The homework is due at beginning of recitation on Friday, Jan 17, 2014.
Mandatory Reading
Please read the following sections from the course text:
1. All of Chapter 1. This introductory chapter broadly discusses

20
1
2
1
IT
ikk
kii
ki
IT
ik
k
i
TTY
T
dP
P
d
P
, where
12
, ,.,
T
iiii
k k k kN
TTTT
PPP
P
. (49)
The stopping criterion. The iterative procedure does not provide the conjugate gradient
method with the stabilization necessary for the minimization of SPt

2000). Here we have adopted classification based on the type of causal characteristics to be
estimated:
1. Boundary value determination inverse problems,
2. Initial value determination inverse problems,
3. Material properties determination inverse problem

. n 0,1,. (18)
where [n/2] = floor(n/2) stands for the greatest previous integer of n/2. T-functions in 2D are
the products of proper T-functions for the 1D heat conduction equations:
Vm x,y,tvnk (x,t)vk (y,t) , n 0,1,. ; k 0,.,n ; 1
2
nn
mk
(19)
The 3D T

nonlinear heat equation: application to a coupled radiative-conductive heat transfer
problem, Inverse Problems in Science and Engineering, Vol.16, No.1, pp. 55-67, ISSN
1741-5977
Reddy, J. N. & Gartling, D. K. (2001) The finite element method in heat tran

Fig. 1. Conductivity identification by the Marching Algorithm. The dots are a multiple of the
first eigenfunction at the observation points pm. The algorithm identifies the values of the
conductivity a and its discontinuity points
(i) Find l, 0 < l < M su

x x x , (30)
where , , * t F t t x x , F is given by (29) and j
are unknown coefficients to be
determined.
For this choice of basis functions , the approximated solution T automatically satisfies the
original heat equation (20)1. Using the conditions (

Function Specification
Method GA PSO RPSO CRPSO FMLP RBFN
Linear 1.81e2 7.62e2 3.85e2 3.42e2 3.17e2 9.90e2 5.35e2
Nonlinear 2.14e2 7.71e2 4.46e2 5.12e2 4.26e2 3.57e4 2.76e4
Table 4. The L2 norm of error in the solution in an exact domain for different alg

Stopping criteria
reached?
Crossover (weighted
average) of Et members;
create offsprings (Ot)
t=t+1
(Next generation)
Apply random mutations on
some of offsprings (Ot)
Pt = Ot-1
(New population)
No
Solution = Top
ranking member
of Pt
Yes
Fig. 1. Flowchart

TP
JP
P
or i
ij
j
JT
P
(52)
where N = total number of unknown parameters, I= total number of measurements. The
elements of the sensitivity matrix are called the sensitivity coefficients, (zisik & Orlande,
2000). The results of differentiation (51) can be

MAE107 Homework #3
Prof. MCloskey
Due Date
The homework is due by 9AM, Friday, Jan 30, 2015, to the TA at the beginning of recitation.
Problem 1
Consider a damped, spring-mass system represented by the second order dierential equation
y + 2y + 17y = 17u.

MAE107 Homework #2
Prof. MCloskey
Due Date
The homework is due a the beginning of recitation on January 23, 2015.
Mandatory Reading
Please read Section 6.8, page 210, in the course text. It discusses impulsive inputs and impulse
responses. This treatment

MAE107 Homework #1
Prof. MCloskey
Due Date
The homework is due at beginning of recitation on Friday, Jan 16, 2015.
Mandatory Reading
Please read the following sections from the course text:
1. All of Chapter 1. This introductory chapter broadly discusses

MAE107 Homework #2 Solution
Prof. MCloskey
Problem 1
1. From the ODE for y we have vb = R2 C2 y + y, so vb = R2 C2 y + y. Substitute both expressions
into the ODE for vb :
1
R2 C2 y + y +
(R2 C2 y + y) = u.
R 1 C1
This is put into standard form
y+
1
1

techniques are briefly reviewed in the following section:
Genetic algorithm
This technique has been widely adopted to solve inverse problems (Raudensky et al., 1995;
Silieti et al., 2005; Karr et al., 2000). Genetic algorithms (GAs) belong to the family o

2
1
0
Jj
jjj
j
Tx
wYTx
q
and
2
1
0
Jj
jjj
j con
Tx
wYTx
T
. (12)
Heat Conduction Basic Research
8
Equations (12) involve two sensitivity coefficients which can be evaluated from (10),
T xj /q xj /k and T xj /Tcon 1 , j = 1,2,J , (Beck et al., 1985).

T
qq
qqqq
(8)
where
q j * is the initial guess of heat fluxes, Tci* is the calculated temperature vector with
the initial guess values.
Recalling equations (6) and (7), equation (8) may be rearranged and written in the following
form:
( )( * ) * X X I q q

24
K
(17)
where = c1 + c2. Here, following the recommendations in (Clerc, 2006), the initial values for
c1 and c2 are set to 2.8 and 1.3, respectively. These values will be modified in subsequent
iterations, as discussed below.
As mentioned above, the rel

or r2 ho kr
3
2
2 2
or r2(= rc) =
2k
ho
.(15.43)
Fig. 15.27
Example 15.13. A small electric heating application
uses wire of 2 mm diameter with 0.8 mm thick insulation (k
= 0.12 W/mC). The heat transfer coefficient (ho) on the insulated
surface is

1,.,
0,
xl
j
jJ
,
1,., k k K 0, f t t .
The heat flux is more difficult to calculate accurately than the surface temperature. When
knowing the heat flux it is easy to determine temperature distribution. On the contrary, if
the unknown boundary characteri

_
j(x)|2
kj
=1 2j
2k
k
2
(2k2)2k
2 = C(k).
Therefore
|k k| C(k)_a a_
L1
and the desired continuity is established.
The following theorem is established in (Gutman & Ha, 2007).
Theorem 4.2. Let a PS, PS Aad be equipped with the L1(0, 1) topology, and cfw

, and (69)
- minimizing the defect of energy of dissipation between elements:
,
2
,0
ln ln
e
ij
t
ij
ij
ijij
T T dt T T d
nn
(70)
with tf being the final moment of the considered time interval, (Ciakowski et al., 2007; Grysa
& Leniewska, 2009), and i,

pp.157-165.
Kim, S. K., and Lee, W. I. (2002). Solution of Inverse Heat Conduction Problems using
Maximum Entropy Method. International Journal of Heat and Mass Transfer, Vol. 45,
No. 2 , pp.381-391.
Krejsa, J., Woodbury, K. A., Ratliff, J. D., and Rauden

found. This is termed as a direct problem. However in many heat transfer situations, the
surface heat flux and temperature histories must be determined from transient temperature
measurements at one or more interior locations. This is an inverse problem.

0
,*1*
nn
nnnn
n
TxtfxdTgxdq
dt a dt
. (16)
with a standing for thermal diffusivity, a k /c , [m/s2]. The functions fn xand gn x
have to fulfill the conditions
2
0
df0
dx
,
2
2
21
n
n
1
dff
dx a ,
2
0
dg0
dx
,
2
2
21
n
n
1
dgg
dx a , n 1,2,.
f0 x *1 , fn

=
16964 6
0 2777 2 8881 0 7192 0 5208
.
. . . . = 3850.5 W
i.e., Rate of heat loss = 3850.5 W (Ans.)
Example 15.11. A 150 mm steam pipe has inside dimater of 120 mm and outside diameter
of 160 mm. It is insulated at the outside with asbestos. The steam

(41)
A suitable regularization parameter 2 is the one near the corner of the L-curve, (Hansen
& OLeary, 1993; Hansen, 2000).
4.9 The conjugate gradient method
The conjugate gradient method is a straightforward and powerful iterative technique for
solving

Solving by hit and trial, we get
r3 ~ 0.105 m or 105 mm
Thickness of insulation = r3 r2 = 105 80 = 25 mm. (Ans.)
15.2.8. Heat Conduction Through Hollow and Composite Spheres
15.2.8.1. Heat conduction through hollow sphere
Refer Fig. 15.22. Consider a holl