Question

Consider the Laplace equation on a circle of radius a around the origin of the xy-plane: p?u=0, Osr<a, -Isosa. The boundary condition is u(a,0)= p cos?o, with p a positive constant. Find the solution u(r,o) by separation of variables. Require that the solution is finite at r = 0, and that the solution is continuous with a continuous derivative at 0 = Ín. To check your solution, set r = a and 0 = 0. You should get u(a,0) = p. In polar coordinates: 02-12/ ou. 1 d?u pôr l'or*2 202 20 cs Scanned with CamScanner

Answer 1

r adius "a" in the have a irdu we XY plane with centre at origin. we have the laplace equationv, inn palar co-06 dinates 수, 무)살 수 Vari ablis Seperationu ur,0) 1, TRO) =0 rak@ ((H) 82 J02

+ R dYA d02 multi ply ing whali equation by dR d'O (H) R. dy do doz R. dr H. RHS = fimetionu "o" anly ZHS fumction "R" should be egual to Hence Some consdant 11 "n2" trom -n2 (H) d02 -n H) A los (no) + B Sin(n0)

i.e, O'O = T) = HO = #) A. los no + Sinngl tromo o os no + Binno) U, ,0) = in iur,0) S An.yn cosino) + Sin (no)) at u 0,0) plos?0 n. no+Sin(no) multiplying both 8i cdus by Cos mo

trom R d8 dR ar -nR =0 d'R + rdR -n?R=D0 The salution will be RI) -h. RIU). rn. lon ditions:- finite trom D =0 RI8) A cosino) + B Ginino)- (H)(0) An Sinno + Bn los no)

p coso cosmo los no osmo odmo) + Ginlno Grm EAna" integrating bolh sides foom o to TT IT Sp losa cos mo do = Ž An an frsmn 10) n-0 TT los (nolosime) d0=H&mn TT and | Sinno Cas modo Z TT An. an &mn. m. TTAM.a los 0 Cos mo do 12, a m. in 1) jur,o) Z. Amrm los mo + Sinno) m.

whore ond 3 aives the complute salution for aur Joplace equation 12 at ond r= a %3D 13 u (a,0) = 14 from(2 %3D Tram 4) ond (Is a,0)

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