Exercise

EXERCISE 3.4

1. Solve the boundary value problem

4. A bar of 40 cm long has originally a temperature of 0° C along all its length. At t = 0, the temperature at the end x = 0 is raised to 50° C, while at the other end it is raised to 100° C. Determine the resulting temperature function.

5. A rod of length / has its ends A and B kept at 0° C and 100° C until steady state conditions prevail. If the temperature of A is suddenly raised to 50° C and that of B is 150° C, find the temperature distribution at any point.

6. A rod of length / has its ends A and B kept at 0° C and 120° C respectively, until steady state condition prevail. If the temperature at B is reduced to 0° C and kept so, while that of A is maintained, find the temperature distribution of the rod.

7. A rod of 30 cm long has its ends A and B kept at 20° C and 80° C respectively until steady state conditions prevail. The temperature at each end is then suddenly reduced to 0° C and kept so, find the resulting temperature distribution function u (x, t) taking x = 0 at A.

8. The ends A and B of a rod of 20 m length have temperature 30° C and 80° C until steady state prevails. The temperature of the ends are then suddenly changed to 40° C and 60° C respectively. (iii) Find the temperature distribution of the rod.

9. The ends A and B of a rod of length I have their temperature anonil kept at 10° C and 90° C until steady state conditions prevail. The 50 temperature of the end A is suddenly raised to 40° C and kept so while the end B is reduced to 60° C. Find the temperature distribution in the rod for the subsequent time.

12. A homogeneous rod of conducting material of length 100 cm has its ends kept at zero temperature and the initial temperature distribution is

13. A rod of length / has its ends A and B kept at 0°C and 100° C respectively, until steady state conditions prevail. The temperature at A is raised to 25° C while that at B is reduced to 75° C. Find the temperature u (x, t) at a distance x from A and at time t.

14. The ends A and B of a rod / c.m. long have their temperatures kept at 30° C and 80° C, until steady state conditions prevail. The temperature of the end B is suddenly reduced to 60° C and that of A is increased to 40° C. Find the temperature distribution in the rod after time t.

15. A bar 10 cm long with insulated sides, has its ends A and B kept at 50° C and 100° C respectively until steady state conditions prevail. The temperature at A is then suddenly raised to 90° C 91 and at the same instant that at B is lowered to 60° C and maintained thereafter. Find the subsequent temperature distribution in the bar.

16. The ends A A and B of a rod 7 cm long have the temperature 40°C and 90°C until steady state prevails. The temperature at A is suddenly raised to 90°C and at the same time that at B is lowered to 40°C. Find the temperature distribution in the rod at time t. Also show that the temperature at the mid point of the rod remains unaltered for all time, regardless of the material of the rod.