PROBLEMA 3 1 e 3 2

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Chapter 3 
PROBLEM 3.1 
Show that in the limit x 0, the difference equation for one-dimensional steady 
conduction with heat generation, Equation (3.1), is equivalent to the differential 
equation, Equation (2.27). 
GIVEN 
One dimensional steady conduction with heat generation 
SHOW 
(a) In the limit of small x, the difference equation is equivalent to the differential equation 
SOLUTION 
From Equation (3.1) 
 Ti + 1 � 2Ti + Ti � 1 = 
2
,G i
x
q
k
 
By definition 
 Ti � 1 = T (x � x) 
 Ti = T (x) 
 Ti + 1 = T (x + x) 
so we can rewrite Equation (3.1) as follows 
 
2
2T x x T x T x x
x
= G
q x
k
 
Now, in the limit x 0, from calculus, the left hand side of the above equation becomes 
2
2
d T
dx
 so we 
have 
 k 
2
2
d T
dx
 = Gq x 
which is equivalent to Equation (2.27). 
PROBLEM 3.2 
�What is the physical significance of the statement that the temperature of each node is 
just the average of its neighbors if there is no heat generation� [with reference to 
Equation (3.2)]? 
SOLUTION 
The significance is that in regions without heat generation, the temperature profile must be linear. 
Compare the subject equation with the solution of the differential equation 
 
2
2
d T
dx
 = 0 
which is T(x) = a + bx, which is also linear.