背景 球内区域 Laplace方程的边值问题 V ulg=f(E) 但在写出定解问题在球坐标系下的具体形式时,需要注意 ° Laplace方程在坐标原点=0也和=2π方向 上不成立,在这两点上充其量只存 在u(,0,)对的单侧导数 把 Laplace方程改写到球坐标系时,为了保持 定解问题的等价性,必须补充上周期条件 l=0=u4l=2x aplo=0 a019=2t
Associated Legendre Functions Spheroidal Harmonics Eigenproblem of Associated Legendre Eqns Orthogonality of Associated Legendre Ftns µ ¥S«Laplace§>¯K ∇2u = 0 u Σ = f(Σ) 3ѽ)¯K3¥IXeäN/ª§I5¿ Laplace§3I:φ = 0Úφ = 2π þؤ᧠3ùü:þ¿Ùþ 3u(r, θ, φ)éφüýê rLaplace§U¥IX§ ± ½)¯K d5§7LÖ¿þ±Ï^ u φ=0 = u φ=2π ∂u ∂φ φ=0 = ∂u ∂φ φ=2π C. S. Wu 1lù ¥¼ê(n)
背景 球坐标系下的定解问题 10/,0u 10 1 a2u 209 02 u0有界 ul=有界 u0有界 u==f(0,0) 令(.0.)=B()5(.0),将上面的方程和齐 边界条件分离变量
Associated Legendre Functions Spheroidal Harmonics Eigenproblem of Associated Legendre Eqns Orthogonality of Associated Legendre Ftns µ ¥IXe½)¯K 1 r 2 ∂ ∂r r 2 ∂u ∂r + 1 r 2 sin θ ∂ ∂θ sin θ ∂u ∂θ + 1 r 2 sin2 θ ∂ 2u ∂φ2 = 0 u φ=0 = u φ=2π ∂u ∂φ φ=0 = ∂u ∂φ φ=2π u θ=0k. u θ=πk. u r=0k. u r=a = f(θ, φ) -u(r, θ, φ) = R(r)S(θ, φ)§òþ¡§Úàg >.^©lCþ C. S. Wu 1lù ¥¼ê(n)
背景 球坐标系下的定解问题 10/,0u 10 1 a2u +725n2b2=0 u0有界 ul=有界 u0有界 u==f(0,0) 令(r,6,)=R(r)S(,9),将上面的方程和齐次 边界条件分离变量 尜
Associated Legendre Functions Spheroidal Harmonics Eigenproblem of Associated Legendre Eqns Orthogonality of Associated Legendre Ftns µ ¥IXe½)¯K 1 r 2 ∂ ∂r r 2 ∂u ∂r + 1 r 2 sin θ ∂ ∂θ sin θ ∂u ∂θ + 1 r 2 sin2 θ ∂ 2u ∂φ2 = 0 u φ=0 = u φ=2π ∂u ∂φ φ=0 = ∂u ∂φ φ=2π u θ=0k. u θ=πk. u r=0k. u r=a = f(θ, φ) -u(r, θ, φ) = R(r)S(θ, φ)§òþ¡§Úàg >.^©lCþ C. S. Wu 1lù ¥¼ê(n)
背景 d l 2dr(r) d AR(=0 10 sin e OS(,) a2S(0,) sin 0 a0 sin20 a2 +AS(6,)=0 2有界 有界 =0
Associated Legendre Functions Spheroidal Harmonics Eigenproblem of Associated Legendre Eqns Orthogonality of Associated Legendre Ftns µ d dr r 2 dR(r) dr − λR(r) = 0 1 sin θ ∂ ∂θ sin θ ∂S(θ, φ) ∂θ + 1 sin2 θ ∂ 2S(θ, φ) ∂φ2 + λS(θ, φ) = 0 S θ=0k. S θ=πk. S φ=0 = S φ=2π ∂S ∂φ φ=0 = ∂S ∂φ φ=2π C. S. Wu 1lù ¥¼ê(n)
背景 再将 10 OS(6,)1,1a2S(6,0) sin 0 a0 sin e in0 a02 +AS(6,)=0 S有界 S有界 aS aS 0=S =27b|0=0b|6=2x 分离变量,S(6,)=(6)()
Associated Legendre Functions Spheroidal Harmonics Eigenproblem of Associated Legendre Eqns Orthogonality of Associated Legendre Ftns µ 2ò 1 sin θ ∂ ∂θ sin θ ∂S(θ, φ) ∂θ + 1 sin2 θ ∂ 2S(θ, φ) ∂φ2 + λS(θ, φ) = 0 S θ=0k. S θ=πk. S φ=0 = S φ=2π ∂S ∂φ φ=0 = ∂S ∂φ φ=2π ©lCþ§S(θ, φ) = Θ(θ)Φ(φ) C. S. Wu 1lù ¥¼ê(n)