ent of the Solution Approach 矩形区域内的稳定问题 设有定解问题 a2u a-u=f(a, y) 0x2 0<x<a,0<y<b u=0=0=0=00≤y≤b 0 u=00≤x≤ 代入方程和边界条件,可得 n7\ 2 Yn(g)- a)(0)=9() (0)=0Yn(b)=0 由此即可求出Yn(y
Imhomogeneous Steady State Problems Homogenization of Imhomogeneous BVC’s Laplacian in Orthogonal Curvilinear Coordinates Illustrative Example Further Development of the Solution Approach Ý/«S½¯K k½)¯K ∂ 2u ∂x2 + ∂ 2u ∂y2 = f(x, y) 0 < x < a, 0 < y < b u x=0 = 0 u x=a = 0 0 ≤ y ≤ b u y=0 = 0 u y=b = 0 0 ≤ x ≤ a \§Ú>.^§ Y 00 n (y) − nπ a 2 Yn(y) = gn(y) Yn(0) = 0 Yn(b) = 0 dd=¦ÑYn(y) C. S. Wu 1où ©lCþ{(n)
ent of the Solution Approach 矩形区域内的稳定问题 设有定解问题 a2u a-u=f(a, y) 0x2 0<x<a,0<y<b u=0=0=0=00≤y≤b 0 u=00≤x≤ 也可设 a(x,y)=∑Xm(x) mTT m=1 f(x,0)=∑hn(x)sin"ny m=1
Imhomogeneous Steady State Problems Homogenization of Imhomogeneous BVC’s Laplacian in Orthogonal Curvilinear Coordinates Illustrative Example Further Development of the Solution Approach Ý/«S½¯K k½)¯K ∂ 2u ∂x2 + ∂ 2u ∂y2 = f(x, y) 0 < x < a, 0 < y < b u x=0 = 0 u x=a = 0 0 ≤ y ≤ b u y=0 = 0 u y=b = 0 0 ≤ x ≤ a u(x, y) = X∞ m=1 Xm(x) sin mπ b y f(x, y) = X∞ m=1 hm(x) sin mπ b y C. S. Wu 1où ©lCþ{(n)
ent of the Solution Approach 矩形区域内的稳定问题 设有定解问题 a2u a-u=f(a, y) 0x2 0<x<a,0<y<b u=0=0=0=00≤y≤b 0 u=00≤x≤ 代入方程和边界条件,可得 x%()-(m)x()=h( Xm(0)=0Xm(a)=0 由此亦可求出Xmn(y)
Imhomogeneous Steady State Problems Homogenization of Imhomogeneous BVC’s Laplacian in Orthogonal Curvilinear Coordinates Illustrative Example Further Development of the Solution Approach Ý/«S½¯K k½)¯K ∂ 2u ∂x2 + ∂ 2u ∂y2 = f(x, y) 0 < x < a, 0 < y < b u x=0 = 0 u x=a = 0 0 ≤ y ≤ b u y=0 = 0 u y=b = 0 0 ≤ x ≤ a \§Ú>.^§ X 00 m(x) − mπ b 2 Xm(x) = hm(x) Xm(0) = 0 Xm(a) = 0 dd½¦ÑXm(y) C. S. Wu 1où ©lCþ{(n)
ent of the Solution Approach 评述 这两种做法没有原则差别.主要的不同是非齐次 项gn(y)和hn()的函数形式可能不同,因而在关 于Yn(y)和Xm(x)的非齐次两个常微分方程 y02-(n)Y(0)=(0 Xm(a) m7\2 Xm( 中可能有一个更易于求解
Imhomogeneous Steady State Problems Homogenization of Imhomogeneous BVC’s Laplacian in Orthogonal Curvilinear Coordinates Illustrative Example Further Development of the Solution Approach µã ùü«{vkK O©ÌØÓ´àg gn(y)Úhm(x)¼ê/ªUØÓ§ Ï 3' uYn(y)ÚXm(x)àgü~©§ Y 00 n (y) − nπ a 2 Yn(y) = gn(y) X 00 m(x) − mπ b 2 Xm(x) = hm(x) ¥Uk´u¦) C. S. Wu 1où ©lCþ{(n)
Further Development of the Solution Approach 讲授要点 ③非齐次稳定问题 示例 方法的进一步发展 非齐次边界条件的齐次化 基本思路 特殊技巧:方程及边界条件同时齐次化 ③正交曲面坐标系下的 Laplace算符 。柱坐标系下的 aplace算符 球坐标系下的 Laplace算符
Imhomogeneous Steady State Problems Homogenization of Imhomogeneous BVC’s Laplacian in Orthogonal Curvilinear Coordinates Illustrative Example Further Development of the Solution Approach ùÇ: 1 àg½¯K «~ {?ÚuÐ 2 àg>.^àgz Ä g´ AÏE|µ§9>.^Óàgz 3 ¡IXeLaplaceÎ ÎIXeLaplaceÎ ¥IXeLaplaceÎ C. S. Wu 1où ©lCþ{(n)