Further Development of the Solution Approach 重级数展开 设有定解问题 02u ax2 Oy2=/(z, y) 0<x<a,0<y<b 0 = 0≤y<b 二a 0 0<x<a 还可以考虑更进一步的做法,即将u(x,y)和f(x,y)既按本 征函数{Xn()}、又按本征函数{Ym(y)}展开(为二重级数 T ∑∑ T nvt =∑∑ not dom sin -r sin 展开系数Cm待求
Imhomogeneous Steady State Problems Homogenization of Imhomogeneous BVC’s Laplacian in Orthogonal Curvilinear Coordinates Illustrative Example Further Development of the Solution Approach ?êÐm 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 ±Ä?Ú{§=òu(x, y)Úf(x, y)QU ¼ê{Xn(x)}! qU ¼ê{Ym(y)}Ðm(?ê) u(x, y) = X∞ n=1 X∞ m=1 cnm sin nπ a x sin mπ b y f(x, y) = X∞ n=1 X∞ m=1 dnm sin nπ a x sin mπ b y ÐmXêcnm¦ C. S. Wu 1où ©lCþ{(n)
Further Development of the Solution Approach 二重级数展开 a2u a2 ax2 ay2 u(x,y)=∑∑ mVt Cnm sin -.sin--y f(x,y)=∑∑am sIn-rsin n=1m=1 ★因为∫(x,y)已知,故cm已知
Imhomogeneous Steady State Problems Homogenization of Imhomogeneous BVC’s Laplacian in Orthogonal Curvilinear Coordinates Illustrative Example Further Development of the Solution Approach ?êÐm ∂ 2u ∂x2 + ∂ 2u ∂y2 = f(x, y) u(x, y) = X∞ n=1 X∞ m=1 cnm sin nπ a x sin mπ b y f(x, y) = X∞ n=1 X∞ m=1 dnm sin nπ a x sin mπ b y F Ïf(x, y)®§cnm® C. S. Wu 1où ©lCþ{(n)
Further Development of the Solution Approach 二重级数展开 a2u a2 ax2 ay2 u(x,y)=∑∑ mVt Cnm sin -.sin--y f(x,y)=∑∑am sIn-rsin n=1m=1 ★因为f(x,y)已知,故cm已知 ★在作二重级数展开时,已经考虑了边界条件
Imhomogeneous Steady State Problems Homogenization of Imhomogeneous BVC’s Laplacian in Orthogonal Curvilinear Coordinates Illustrative Example Further Development of the Solution Approach ?êÐm ∂ 2u ∂x2 + ∂ 2u ∂y2 = f(x, y) u(x, y) = X∞ n=1 X∞ m=1 cnm sin nπ a x sin mπ b y f(x, y) = X∞ n=1 X∞ m=1 dnm sin nπ a x sin mπ b y F Ïf(x, y)®§cnm® F 3?êÐm§®²Ä >.^ C. S. Wu 1où ©lCþ{(n)
Further Development of the Solution Approach 二重级数展开 a2u a2 ax2 ay2 f( r, y u(x,y)=∑∑ mVt Cnm sin -.sin--y f(x,y)=∑∑am sIn-rsin n=1m=1 因此只需将上面的展开式代入方程 n丌\2 nnt nat nnt a)+(b a sin n=1m=1 nnt ∑∑ am sin a- sin b y n=1m=1
Imhomogeneous Steady State Problems Homogenization of Imhomogeneous BVC’s Laplacian in Orthogonal Curvilinear Coordinates Illustrative Example Further Development of the Solution Approach ?êÐm ∂ 2u ∂x2 + ∂ 2u ∂y2 = f(x, y) u(x, y) = X∞ n=1 X∞ m=1 cnm sin nπ a x sin mπ b y f(x, y) = X∞ n=1 X∞ m=1 dnm sin nπ a x sin mπ b y ÏdIòþ¡Ðmª\§ − X∞ n=1 X∞ m=1 cnm nπ a 2 + mπ b 2 sin nπ a x sin mπ b y = X∞ n=1 X∞ m=1 dnm sin nπ a x sin mπ b y C. S. Wu 1où ©lCþ{(n)
Further Development of the Solution Approach 二重级数展开 a2u a2 ax2 ay2 f( r, y u(x,y)=∑∑ mVt Cnm sin -.sin--y f(x,y)=∑∑am sIn-rsin n=1m=1 根据本征函数的正交性,比较系数,即得 nVt m丌\2 d
Imhomogeneous Steady State Problems Homogenization of Imhomogeneous BVC’s Laplacian in Orthogonal Curvilinear Coordinates Illustrative Example Further Development of the Solution Approach ?êÐm ∂ 2u ∂x2 + ∂ 2u ∂y2 = f(x, y) u(x, y) = X∞ n=1 X∞ m=1 cnm sin nπ a x sin mπ b y f(x, y) = X∞ n=1 X∞ m=1 dnm sin nπ a x sin mπ b y â ¼ê5§'Xê§= −cnm nπ a 2 + mπ b 2 = dnm C. S. Wu 1où ©lCþ{(n)