Question

Problem (3pt). In the disk {(r,(): r < 3, 0 < 0 < 2n} find solution (if solution does not exist, explain why) Au=0, Ur|r=3 = 0, u(0) = 0.

Solve Laplace's Equation with polar coordinates please

Answer 1

Answeri The Given information is Auzo p . 9: 30 -> Solution of Laplace equation -> Consider Laplace equation : in polzcoordinate L oo ) vu Uuturo 99 71 oo (ory 91 4 +1 4 + 40 =0 with ocrca and ococa (2) --> By Seperation of variable, let us take 449,0) =RG]. 010) --(3) - Leger = r. o Lily-Ro, o "Ro" pust in (3) - Dingo tempo tréCO >>> {"+si's']0= - RO" 9 td 00.8.2 Rthur'-*R=0 ut 9 gr A cos to +. B Sindo -----(D) (44) To solve it O Put Ragn, R' = rn R = nco-1) annie mCD1) 417-2}+ {01913-83 -0.

=){r?-o +5 - 12) =0 s mad? so as 57°40 =) nie.* - This solution (4.2): RG) =crtçral ) 263& O CO2TY ! Now we use boundary condition U(30) = 6 = R(3).OCO) = 6 2 either RC3)- / 08(0)=0 » If we assume that ucro is bounded the uno con then we have to take C2-0 20.30 otherwise it as a yo * ucro) - ef Thus required sakition is UC90) - Cod CA Cos de+B Sinto) - Since 199,0) = UC, 80-0) Casio - cos - R odez sin to = sin(25-oj A). - A must be an integer cor non- negative because for din; nex), salution co-incide with d=0) - Thuis

> 49 4910) (An COSnot By Simno) : 7:12... Ao so by principle of Stipenposition is (90) = 2 An 20soola"+ 2 Bn Sinke) sa n:o where Anat she cos no do; nenu{o} Bm = 1 none 1 hcal Sin no donen hcov - 4163,0) = 0