Question

d1=7

d2=8

Question 3 Left end (r-0) ofa copper rod of length 100mm is kept at a constant temperature of Temp = 10+42 degrees and the right end and sides are insulated, so that the temperature in the ou u ax2 rod, 11(X, 1) , obeys the heat partial DE, Ơ Co2 , where D-111 mm 2/s for copper. where D 111 mm*/s for copper. (a) Write the boundary conditions for u(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+p(x,/) and select Const appropriately to have the zero boundary condition for φ(x,t) at x-0. Check that if Ø(x,t) is a solution to the heat equation, 11 (x,t)-Const + φ(x,t) is also a solution. Find the boundary condition for Ø(x,t) at the right end of the rod. (b) Find the general solution to heat equation for p(x,/) satisfying the boundary conditions found in (a) and then determine the temperature in the rod, u(x,t) e) Using the solution in (b) find u(r,/) that satisfies the initial conditions u (x,0)-10+d,+ sin2+d, sin 20*d, sin 200 (d) Using the solution in (c) determine the temperature in the middle of the rod at t-60, record your 57T answer to 3 significant figures (this is not three decimal places)

Answer 1

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