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4讲 分离变量法（三）

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 ★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 F ù«{§¢Sþ*¿ /ƒAàg¯K¼ê0 Vg F `:µÃI)šàg~‡©§ 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 ★这种方法,实际上扩充了“相应齐次问题本征函数” 的概念 ★优点:无需解非齐次常微分方程 ★缺点:结果是二重级数.事实上,尚可求和

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*¿ /ƒAàg¯K¼ê0 Vg F `:µÃI)šàg~‡©§ F ":µ(J´­?ꩯ¢þ§ÿŒ¦Ú C. S. Wu 1où ©lCþ{(n)

讲授要点 ③非齐次稳定问题 示例 方法的进一步发展 ②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 Outlines Simultaneous Homogenization of . . . ùLJ: 1 šàg­½¯K «~ {?ÚuÐ 2 šàg>.^‡àgz Äg´ AÏE|µ§9>.^‡Óžàgz 3 ­¡‹IXeLaplaceŽÎ ΋IXeLaplaceŽÎ ¥‹IXeLaplaceŽÎ C. S. Wu 1où ©lCþ{(n)

言 到目前为止,除了在稳定问题中需要有一部分边 界条件用于定叠加系数、因而允许是非齐次的以 外,在应用分离变量法解偏微分方程定解问题 时,我们总是要求边界条件是齐次的

Imhomogeneous Steady State Problems Homogenization of Imhomogeneous BVC’s Laplacian in Orthogonal Curvilinear Coordinates Outlines Simultaneous Homogenization of . . . Úó 8cŽ§Ø 3­½¯K¥I‡kÜ©> .^‡^u½U\Xê!Ï #N´šàg± § 3A^©lCþ{) ‡©§½)¯K ž§·‚o´‡¦>.^‡´àg C. S. Wu 1où ©lCþ{(n)

言 为什么边界条件必须是齐次的?

Imhomogeneous Steady State Problems Homogenization of Imhomogeneous BVC’s Laplacian in Orthogonal Curvilinear Coordinates Outlines Simultaneous Homogenization of . . . Úó Ÿo>.^‡7L´àgº šàg>.^‡ØU©lCþ k÷vàg§Úàg>.^‡A)U \å5âEU÷vàg§Úàg>.^‡ ŠÏ9¼ê5 šàg>.^‡XÛ?nº E±ÅЧ½)¯K~ C. S. Wu 1où ©lCþ{(n)