Question

Use Fourier transforms to solve the boundary van ns to solve the boundary value problem Uzr +tyy = 0, 3ERy>, u(7,0) = 2,7 32; (,0) = 0, < > 2, u is bounded

Answer 1

using infinite Fourier transform on both side IS de 02 2 (0" In juga --05 juca) e "de sco) doe sho, where ü- jucu ele de -15 , where y e da solving we ū (s, y) = get A et +Best A and Bare constants as y , for which В must Since ucx,y) is bounded is also bounded as y s be equal to zero... . ū (sy) = A e

Now ucxo) = 2 , 12152 0 ; 121 > 2: in u (5,0) = u(x,0) eist de i U (50) " Cosas tisinas - Cosas+isinas) " - 2 i sin 2s s ų Sin 25 putting this value in , we get A = 4 sinas & J (3.7) = usinas esy (by putting the value of A est in ③ By inverse Fourier transformation uc**) = 10 (8,) eise ds (**) - za je sings ēt piszds