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 equi...

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).