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 c

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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+ R dYA d02 multi ply ing whali equation by dR dO (H) R. dy do doz R. dr H. RHS = fimetionu o anly ZHS fumction R shouldi.e, OO = 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) pltrom R d8 dR ar -nR =0 dR + rdR -n?R=D0 The salution will be RI) -h. RIU). rn. lon ditions:- finite trom D =0 RI8) A cosino)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 10whore 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 (I

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