北京大学：《数学物理方法》精品课程电子教案（A类）第二部分 数学物理方程_第5讲 分离变量法（四）

圆形区域中的稳定问题 定解问题 a2u a2u +-< ax2 ay2 分析 以下可按照分离变量法的标准步骤求解

Dirichlet Problem of Laplace Eqn in Circular Region Helmholtz Eqn in Orthogonal Curvilinear . . . Well-posed Problem in Circular Region Solutions of Well-posed Problem in Circular Region /«¥­½¯K ½)¯K ∂ 2u ∂x2 + ∂ 2u ∂y2 = 0 x 2 + y 2 < a2 u   x 2+y 2=a 2 = f ©Û ±eŒUì©lCþ{IOÚ½¦) (?) C. S. Wu 1Êù ©lCþ{(o)

圆形区域中的稳定问题 平面极坐标中的方程与边界条件 1a(duxiu r ar(ar)T r2 a 00<r<a ul=f(o) 令u(,以)=R(r)p( C. S. Wu

Dirichlet Problem of Laplace Eqn in Circular Region Helmholtz Eqn in Orthogonal Curvilinear . . . Well-posed Problem in Circular Region Solutions of Well-posed Problem in Circular Region /«¥­½¯K ²¡4‹I¥§†>.^‡ 1 r ∂ ∂r  r ∂u ∂r  + 1 r 2 ∂ 2u ∂φ2 = 0 0 < r < a u   r=a = f(φ) ©Û -u(r, φ) = R(r)Φ(φ) C. S. Wu 1Êù ©lCþ{(o)

圆形区域中的稳定问题 平面极坐标中的方程与边界条件 1a(duxiu r ar(ar)T r2 a 00<r<a ul=f(o) 令u(,)=R(r)p(),代入方程,有 1心N∥)Rd2 dR 0 rd r2 do2 C. S. Wu

Dirichlet Problem of Laplace Eqn in Circular Region Helmholtz Eqn in Orthogonal Curvilinear . . . Well-posed Problem in Circular Region Solutions of Well-posed Problem in Circular Region /«¥­½¯K ²¡4‹I¥§†>.^‡ 1 r ∂ ∂r  r ∂u ∂r  + 1 r 2 ∂ 2u ∂φ2 = 0 0 < r < a u   r=a = f(φ) ©Û -u(r, φ) = R(r)Φ(φ)§\§§k 1 r d dr  r dR dr  Φ + R r 2 d 2Φ dφ2 = 0 C. S. Wu 1Êù ©lCþ{(o)

圆形区域中的稳定问题 平面极坐标中的方程与边界条件 1a(duxiu r ar(ar)T r2 a 00<r<a ul=f(o) 令u(,)=R(r)p(),代入方程,有 r d dR 1d2④ rdr d d2=4 C. S. Wu

Dirichlet Problem of Laplace Eqn in Circular Region Helmholtz Eqn in Orthogonal Curvilinear . . . Well-posed Problem in Circular Region Solutions of Well-posed Problem in Circular Region /«¥­½¯K ²¡4‹I¥§†>.^‡ 1 r ∂ ∂r  r ∂u ∂r  + 1 r 2 ∂ 2u ∂φ2 = 0 0 < r < a u   r=a = f(φ) ©Û -u(r, φ) = R(r)Φ(φ)§\§§k r R d dr  r dR dr  = − 1 Φ d 2Φ dφ2 = λ C. S. Wu 1Êù ©lCþ{(o)

圆形区域中的稳定问题 平面极坐标中的方程与边界条件 1a(duxiu r ar(ar)T r2 a 00<r<a ul=f(o) 分析 因此,可以分离变量 d dR 入R=0 d2④ d2+№p=0 C. S. Wu

Dirichlet Problem of Laplace Eqn in Circular Region Helmholtz Eqn in Orthogonal Curvilinear . . . Well-posed Problem in Circular Region Solutions of Well-posed Problem in Circular Region /«¥­½¯K ²¡4‹I¥§†>.^‡ 1 r ∂ ∂r  r ∂u ∂r  + 1 r 2 ∂ 2u ∂φ2 = 0 0 < r < a u   r=a = f(φ) ©Û Ïd§Œ±©lCþ r d dr  r dR dr  − λR = 0 d 2Φ dφ2 + λΦ = 0 C. S. Wu 1Êù ©lCþ{(o)