Question

Using Fourier transform, prove that a solution of the Laplace equation in the half plane: Urn+ Uyy=0,- << ,y>0, with the boundary conditions u(1,0) = f(t), - <I< u(x,y) +0,31 +0,+0, is given by r(2, y) == Love you > 0. Hint: 1. Take Fourier transform on the variable r, 2. Observe U(k, y) +0 as y → 00, 3. Use pt {e-Mliv = Vice in

Answer 1

Gli Answer soli The given equation is Taking Fouriver's trains formation on both side with respet to Y, - we have. =-pā (par) to the CPx) =0 224 -p?u=0 - © -PX :ūlo,)hely + Be _ NOW since u(0,0)= f(u), when - Cico - Taking Fouriore ris transformation on both side, with respect to X, we thave ū(PO) < f (0) 0 Now since ulx,y) is bounded, when y d then ū (P.1) Must be bounded, when yo from ③ we have A=0 PPO and B=0 :f PCO

Hence we have oply a (Pry)=ke , where K is constant Now from o we have ū (DO) = f(p) E = H (P) betely ū(P.y) = Ēiples a (P.y) = le pa fué lpix de. Taking inverse Porier's trains form on both sides we have - use.gs = F"( fase-ply + ipe de] ulk,y) met lam (5 kce je Prva toode) e podp. u in, y)-chefflere Se (ou te-3) -11%) op -@ Now let peco CP Cice-2)] – 1 Ply) dp I = 1 en 1-2)], poplice -1)-yldp Ply+ICE -2) - dp + le ya then szilen, yfiCE-x) . Y-1(E-X)

we have from 4 Camouf) - fces y legside 462,4)= ţ se concentree, when you hon