# Consider a 2 d metal plate with 0 x l 0 y h on which

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method of separation of variables.Consider a 2-d metal plate with 0xL, 0yH, on which the temperatureU(t, x, y) obeysthe heat equation:Ut(t, x, y,) =α2[Uxx(t, x, y) +Uyy(t, x, y)],with boundary conditions:U(t, x,0) = 0,U(t, x, H) = 0,t >0,0< x < L.U(t,0, y) =g(y),U(t, L, y) = 0,t >0,0< y < H.Suppose that we wish to find the steady state solution of this problem, which we do by imposingthe additional restrictionUt(t, x, y) = 0. The PDE can not be solved through the method ofseparation of variables, i.e. by tryingUe(x, y) =X(x)Y(y).
(i) Solve for the steady state solution, and show that your solution can be expressed in the form:Ue(x, y) =n=Xn=1Ansinhh(L-x)HisinhnπyHi.(ii) Evaluate the constantsAnwheng(y) =ky(H-y), for some constantk.
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ECE 205 - Winter 2017
5: Marking SchemeTotal is 48 points (Questions 1,3,4) and 12 bonus points (Question 2).9
ECE 205 - Winter 2017Assignment 8 solutionsDue 02-04-2017Question 1: 24 points.1 point for the form of the solution.1 point for separating the variables and setting equal to k.1 point forUn(x, t).1 point forU(x, t).1 point for the integral ofAn.1 point forU(x, t) after evaluatingAn.1 point for the equilibrium solutionUe(x, t).1 point for showing that the general solution tends to the equilibrium solution.1 point for findingXn(x) is a function of cosine only.1 point for the general solutionU(x, t).1 point for findingd0.1 point for finding the general solutionU(x, t).1 point for finding the equilibrium solution(s)Ue(x, t).1 point for showing that the general solution tends to one of the equilibrium solutions.1 point for using Leibniz rule.1 point for substitutingUtbyα2Uxx.1 point for integration by parts.
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