Mathematics

Consider the PDE: 

2

2

x

u

t

u













where u is the temperature in degree Celsius, α is thermal diffusivity in cm2

/s, t is in seconds and 

x is in cm. It is required to compare the temperature profile of the 10 cm long rods of silver (α = 

1.71 cm2

/s) and cast iron (α = 0.12 cm2

/s), each with initial temperature of 100C and the ends 

u(x=0, t) = 30C and u(x=0, t) = 60C.

(a) Write down the Difference equation for u(x, t) using fully implicit scheme with h = 1 

cm.

(b) Solve the temperature at all unknown nodes from t = 0 to 10 s for both materials 

using 1 s time step.

(c) Plot the temperature distribution for both materials at every 2 s.

(d) Discuss the effect of material on the temperature distribution.