method of separation of variables.Consider a 2-d metal plate with 0≤x≤L, 0≤y≤H, 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=1Ansinhhnπ(L-x)HisinhnπyHi.(ii) Evaluate the constantsAnwheng(y) =ky(H-y), for some constantk.