Question

Solve the following Laplace's equation in the semi-annular plate (on the right) Urr + - + = 0, a<r<b, O<O<T, u(a,0) = (1-0), u(b, ) = 0, ur,0) = 0, ur.) = 0. 4= (1-1)

Answer 1

- Answers tu : Bu + C + uca,) : OCT-0), ucho): 0, U(Y,0120, UCI, T)o assume that the solution will be in the form uvio) - (OJ GOY) Uy (ro) : acos G. (V) Urr (10): (O) GRY, CY) Ugo (no) - Poo (O) (V) du + + 0 $co) Gry(V) +$0) G7 (1) thg $ SGCY) = 0 9) GCr) Ø (0) Gap (v +49(0) Gr CV) = Poocomo varor (N) +r GyCY) - 40.CO). - Gero pco) → doo + 10 =0, v Gorr + YG - 1G =0 U(Y,0) = CO GLCV) -0 = 0) uch, T) :0 OCTC) GCN -0 =) CTT) = 6 consider the die boo +10=0, with boundary conditions 0 (0) 0 (IT) = 0 case 1: 1:0 Doo -0 =$(O): A0 +B Co =0 → B=0, Ø CIT)-0 =) 4:0 no solution exist for do Scanned with CamScanner

let yo so m² Poo tm?p=070 colc, cos(molte sin (md) p_2 P-12P 2PLP-1r tror nr=0 erownG=0 = r P (P-Urt CO) 0 ,20 OCT) = 0 C₂ sin (mi) = 0 Sin (mi) no mitant men eigen functions are Ancol = sin (no), NEN, eigen values are An = n² Gry trG - XG = 0 » r Gr + VG, -nG=0 this is euler differential equation. So assume G. criage rGrr tray = p (P-1) rP tereny P=0 P (P-1) + P-n²=0 ) phan =) .P=th. Therefore, two independent solutions are rn, oh Gencri e Anth + Burn un (ro) - Sin Cros (Anra Burn). by principle of superpositions, we have La crio) = Ź Sin (nos (Andy Bar-n uCb, OJ = 0 a Sin (noj (An bh+B5h) .o =) An bh + Brbn=0 uca, O) = OCH-o) Sin (ro) (Arany Bran) = OCITO A, an + Bran - Locos sin Gnojde - 2-2 Cos (hl) 2-2(-1) 79 n3 n3

Anant Bran: 2-2(-1)h n] cs Scanned with CamScanner