Question

The temperature at some instant (t = 0) of a laterally insulated solid rod of 1 unit length is given by ?(?,?) = sin ?? for 0  x  1, where x is measured from left end of the rod to the other end on the right. Also, at time t = 0, the left end of the rod is subjected to 0oC and maintained over the time, while the right end is insulated when t > 0. Note that the term x in the expression of the temperature is in radian. The temperature variation in the rod,

?(?,?), satisfies the heat equation: ?? ?? = ?( 2 ?^2? /??^ 2 )

Solve the above heat equation numerically with del x = 0.2 and del t = 0.04 using explicit formula: ?(?? ,??+1) = ??(??−1,??) + (1 − 2?)?(?? ,??) + ??(??+1,??)

Given that, c ^2 = 0.1 and ? = ? ^2 Δ? (Δ?)2 .

Estimate the numerical values for ?(?,?) at t = 0.08, giving your answers at 4 decimal points

Answer 1

Given:

The temperature variation in rod satisfies the heat equation:

Initial Condition:

Left boundary condition:

Right boundary condition:

for all t > 0 as the right end is insulated.

x	0	0.2	0.4	0.6	0.8	1.0
U	0	0.5878	0.9511	0.9511	0.5878	0

Explicit formula:

Solution:

For x = 0.2 to 0.8:

For x = 1.0:

At t = 0.04:

For x = 0.2

For x = 0.4

For x = 0.6

For x = 0.8

For x = 1.0:

At t = 0.08:

For x = 0.2

For x = 0.4

For x = 0.6

For x = 0.8

For x = 1.0:

x	0	0.2	0.4	0.6	0.8	1.0
t = 0	0	0.5878	0.9511	0.9511	0.5878	0
t = 0.04	0	0.5856	0.9475	0.9475	0.5856	0.5856
t = 0.08	0	0.5834	0.9439	0.9439	0.5892	0.5892