Exercise

EXERCISE 3.5

1. A square plate is bounded by the lines x = 0, y = 0, x = 20, y = 20. Its faces are insulated. The temperature along the upper horizontal edge is given by u(x, 20) = x(20-x)  when 0 < x < 20 while the other three edges are kept at 0° C. Find the steady state temperature in the plate.

2. A square plate has its faces and the edge y = 0 insulated. Its edges x = 0 and x = л are kept at 0° C and the edge y = л is are kept at 0° C and the edge kept at temperature f(x). Find the steady state temperature.

3. Find the steady state temperature at any at any point of a square plate whose two adjacent sides are kept at 0° C and the other two edges are kept at 100° C.

4. A rectangular metal plate is bounded by the lines x = 0, x = a, y = 0 and y = b. The three sides x = 0, x = a and y = b are insulated and the side y = 0 is kept at temperature  Show that the temperature in the steady state is

5. The three sides x = 0, x = a, y = 0 of a square plate bounded by [1-the lines x = 0, x = a, y = 0 and y = a are kept at temperature 0° C. The side y=a is kept at steady temperature given by u (x, a) bx (x - a), 0 ≤ x ≤a when b is a constant. Find the steady-state temperature u (x, y) in the plate.

6. Find the solution of the equation V2 = 0 for 0 < x < Â and 0 < y <л, given u (0,y) = u(л,у) = u(x, π) = 0 u(x,0) = sin² x

7. A rectangular plate is bounded by the lines x = = 0, x = a, y = 0 and y = b and the temperature at the e temperature at the edges are given by

Find the steady state temperature distribution inside the plate.

8. A long rectangular plate long sides as well as of the short sides are maintained at 0° C. Find an expression for the steady state temperature u (x, y) if the short side y = 0 is л cm long and is kept at u₀^o C.

9. An infinitely long rectangular plate with insulated surface is 10 cm wide. The two long edges and one short edge are kept at 0° C temperature while the other short edge x = 0 is kept at temperature is given by

Find the steady state temperature distribution in the plate.

10. A long rectangular plate has its surfaces insulated and the two long sides as well as one of the short sides are kept at 0° C. While the other sides u (x, 0) 3x and the length being 5 cm. Find u(x, y).

11. A long rectangular plate of width / cm with insulated surface has its temperature u equal to zero on both long sides one of the shorter sides so that u (0,y) = u (l,y) = u(x, ∞)=0 and u (x, 0) =kx. Find u (x, y).

12. An infinitely long vertical uniform plate plate is bounded by two parallel edges and an end at right angles to them. The breadth is л. This end is maintained at a temperature uo at all points and other edges are kept at zero temperature. Determine the temperature at any point of the plate in the steady state.

13. An infinite long plate of width л with insulated surfaces has its temperature zero on both long sides and are of the short sides. The side y = 0 is maintained at a temperature 3x. Find the steady state temperature u (x, y).z

14. A rectangular plate with insulated surface is 10 cm wide so long compared to its width that it may be considered infinite  ength. If the temperature along short edge y = 0 is given u (x, 0)=8 sin πx/10  when 0 < x < 10, while the two long edges x = 0 and x 10 as well as the other short edge are kept at 0°C, find the steady state temperature function u (x, y)..