Question

1. Consider the problem of steady state heat flow in the half-plane, 22T a2T + ar2 =0 for ER and y>0, ay2 subject to the boundary condition T(3,0) = g(x), and T +0 as yo. You will solve the problem using the Fourier transform in 2, with T(w,y) = ZELT(, y)e-iw= ds 2 (a) Derive an ODE for T. You can assume T +0 as|a . (b) Derive conditions for I at y = 0 and as y. You can assume the Fourier transform of g(x) erists. (e) Show that f(w,y) = C(w) e-lly where C(w) is an arbitrary function of w, satisfies the ODE for Î and the condition for T as y () Apply the condition for Î at y = 0) to find C(w). (e) Use the convolution theorem for FTs to find an integral expression for T in terms of g (rather than ĝ). Hint: You can use the result F-1(elv) = 2 22 + y2 2 y

Answer 1

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