Question

A thin rod of length, 𝐿 = 40 𝑐𝑚, was made from a material with a thermal 

diffusivity, 𝑘 = 2.5 𝑐𝑚²⁄𝑠 . The temperature distribution in terms of the time, 

𝑡 and the position, 𝑥 is denoted by 𝑇(𝑡, 𝑥). The following initial and boundary 

conditions are considered:

𝑇(0, 𝑥) = 𝑓(𝑥) 

𝑇(𝑡, 0) = 𝑓(0) 

𝑇(𝑡, 40) = 𝑓(40) where 𝑓(𝑥) has the following piecewise function form 

𝑓(𝑥) = { 𝑔(𝑥), 0 ≤ 𝑥 < 𝑎 

ℎ(𝑥), 𝑎 ≤ 𝑥 ≤ 40 

.

The functions 𝑔(𝑥) and ℎ(𝑥) are not constant and 𝑓(𝑥) satisfies the following 

condition, 

𝑓(40) > 𝑓(0) > 0 or 𝑓(0) > 𝑓(40) > 0.

By using a suitable function for 𝑓(𝑥) and, the values of ∆𝑥 = 4 𝑐𝑚 and ∆𝑡 = 

4 𝑠, consider TWO (2) finite-difference methods to compute the temperature 

distribution 𝑇(𝑡, 𝑥) over the time interval [0, 8].