Question

Question 3 Left end (r-0) of a copper rod of length 100mm is kept at a constant temperature of Temp-1 0 a 2 degrees and the right end and sides are insulated, so that the temperature in the ul ul where D = 111 mm2/s for copper. rod, u(x,t), obeys the heat partial DE, Ot Ox (a) Write the boundary conditions for il(x,t) of the problem above. Note that for the left end the condition is not zero, which presents a problem for the separation of variable method. Use u(x, t)-Const φ(x, t) and select Const appropriately to have the zero boundary condition for φ(x,t) at x-0. Check that if φ(x,t) is a solution to the heat equation, u(x,t)- Const +p(x,/) is also a solution. Find the boundary condition for (x,t) at the right end of the rod. (b) Find the general solution to heat equation for p(x,t) satisfying the boundary conditions found in (a) and then determine the temperature in the rod, u(x,t) (c) Using the solution in (b) find u(x,t) that satisfies the initial conditions: 11(x,0) = 10+d, + sin x +d, sin (d) Using the solution in (c) determine the temperature in the middle of the rod at t-60, record your answer to 3 significant figures (this is not three decimal places)

d1=7

d2=8

Any help would be greatly appreciated.

Answer 1

cn is, he di-P4eTe

100ノト):0 | condtfon (too, t ):0 ヒ0

o -to ave vin-ki G.J .coln β扣 Pri 200 (200ナ 20oX and Nouo,

20-1、2 2.00 300 M e (En =Bn.cn) 200 20づ tHence, oe have 200 2.00

Hence 一111 200 50 C,500 1 200o