Question

In spherical polar coordinates (r, 0, ¢), the general solution of Laplace's equation which has cylindrical symmetry about the polar axis is bounded on the polar axis can be expressed as u(r, 0) = Rm(r)P,(cos 0), (A) where P is the Legendre polyomial of degree n, and R(r) is the general solution of the differential equation *() - n(n + 1)R = 0, (r > 0), dr dr where n is a non-negative integer. (You are not asked to show this.) (a) By solving this differential equation (A), show that (n= 0,1,2,...), R„(r) = A„r" + B„r-(n+1), %3D (3] where A, and B, are arbitrary constants. (b) Find the solution u(r,0) of Laplace's equation in the regionr >a which satisfies the boundary conditions u(a, e) = 2 + cos 0. lim u(r,0) = 0 and (4)

Answer 1

differential is, quntions The n(n+) R=0, (r >o) ye dR dr dr Solution the that assume Let us of the is - (n+) (noo,e.) R(v) = A, sat isky tue diffimuntiol equating it Then must -(n+2) d R,() dr - (n+i) B -5-1 (* dR-= n(nai) An r +(nei)n Bn dr dr -(nt) n(n) R,(^) = n(n+i) Any+n(nai) Bn r dR, (r) - (ma) -n(ntD Anr- (nti)n8Y -(nt) n(nt) Anr+ nti)nBn Y Thus the Solution give, diffemntal

equatim is, 8 ht) (m=0,2,) Kn (r)= A where An, B, Constants, 8. the solution of Rün) From Ca), Substitudting P Cese) u(r,e)= (A, Y+B " Condition, Applying Ae Boundany Lim urr,e)=0, -(nti) 5 (Anr +B,n' Lim a (^, G) = Lim E Lim P. Ccaso) An=0.

-(n+1) Pin Ccose). Applaing B.C yain, -(n+1) ula,0) = s B, Gr a ఒG6) 2+ Cose, Let Coso = x, Pm Cc) and, intguting Multiplying thrughout by to 1. -1 over (n+1) Po (x) Pm (x) dx = (2+ x) P, (x) de -(n) (2+2) (2)dx P, (x) Pm ()dx. From orthansrmalit polynimials, gandre Smn P (x) Pm C) dx = 2n+1 (n+) Ž Bn ( (24x) Px) de. 2. (2nt1) M8

-(m+) (2+x) Pm (2) dz %3D (2mt) But mtk-1 C. 2 5 x k-o Pm (x)= above equation, Sabettating in to - (m+1) mtkt 2. (2mt1) ktil Com k-o k+) dx mtk- k+1 k+2 k+2 k+i

k+1 mtk1 C. kti k+2 k+2 (2mt1) 2 B = 2. 2. k-o kt 2 (-t) + (-6) k+2 The Solution - (n+)) P, Ccase) ulr,0) = 5 BY B is ex presed abave he valuer