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 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 但是边界条件 R(a(o)=f(o 仍然不能分离变量,因为边界条件是非齐次的 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 ´>.^‡ R(a)Φ(φ) = f(φ) E,ØU©lCþ§Ï>.^‡´šàg 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 我们尽管能够将齐次方程分离变量,得到两个含 有待定参数的齐次常微分方程,但是并没有相应 的齐次边界条件与之配合而构成一个本征值问题 在平面极坐标杀下应用分离变量法,又遇到了新 的特殊的困难! 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 òàg§©lCþ§ü‡¹ k½ëêàg~‡©§§ ´¿vkƒA àg>.^‡†ƒÜ ¤‡Š¯K 3²¡4‹IXeA^©lCþ{§q‘ # AÏ(Jœ 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 0/au\102u r(ar)Tr2002 00<r<a =n=f() a2u a2u ax2 ay2 0x2+y2 是适定的定解问题吗?

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 ∂ 2u ∂x2 + ∂ 2u ∂y2 = 0 x 2 + y 2 < a2 u   x 2+y 2=a 2 = f ´·½½)¯Kíº 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 0/au\102u r(ar)Tr2002 00<r<a =n=f() ax2 ay2 5=0x2+g2 是适定的定解问题吗?是!

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 ∂ 2u ∂x2 + ∂ 2u ∂y2 = 0 x 2 + y 2 < a2 u   x 2+y 2=a 2 = f ´·½½)¯Kíº´œ 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 0/au\102 r(ar)Tr2002 00<r<a =n=f() 10/an)102u r(O)+2a2 00<T<a =n=f() 是适定的定解问题吗?

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 1 r ∂ ∂r  r ∂u ∂r  + 1 r 2 ∂ 2u ∂φ2 = 0 0 < r < a u   r=a = f(φ) ´·½½)¯Kíº C. S. Wu 1Êù ©lCþ{(o)