Question

Question 3 Left end (x 0) of a copper rod of length 100mm is kept at a constant temperature of Temp - 10+d2 degrees and the right end and sides are insulated, so that the temperature in the rod, u(x,t). obeys the heat partial DE, CD11 mms copper. where D-1 mm's for copper (a) Write the boundary conditions for u(x, 1) 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 ф(x,t) at r-0. Check that if ф(x,t) is a solution to the heat equation, u(x,t) Const +o(x,t) is also a solution. Find the boundary condition for p(x,t) at the right end of the rod. (b) Find the general solution to heat equation for ф(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, 1) that satisfies the initial conditions: xd, sin (d) Using the solution in (c) determine the temperature in the middle of the rod at r 60, record your answer to 3 significant figures (this is not three decimal places).

D1 = 7
D2 = 4
Any assistance would be greatly appreciated

Answer 1

2o D lse ·习

B(t) 는 6 se,) -(to+4) (1.x ) ete bied n X(X) T (t) U(x, t): br lety

Now wmn 2 Ant Dオ (1oo) 2 9,33 na p大

lo o 0 Ai e =0

lo o 2ec