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 (,y)=∑∑ d,sinsin n=1m=1 ★优点:无需解非齐次常微分方程
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 −cnm nπ a 2 + mπ b 2 = dnm F `:µÃI)àg~©§ 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)
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 因此 d n丌\2m丌\2 b
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 Ïd cnm = − dnm nπ a 2 + mπ b 2 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 not not y sin-r sin n=l m=l/n7 not
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 u(x, y) = − X∞ n=1 X∞ m=1 dnm nπ a 2 + mπ b 2 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 u(x,y)=∑∑ mVt Cnm sin -.sin--y f(x,y)=∑∑am sIn-rsin n=1m=1 ★这种方法,实际上扩充了“相应齐次问题本征函数” 的概念
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 ù«{§¢Sþ*¿ /Aàg¯K ¼ê0 Vg C. S. Wu 1où ©lCþ{(n)