矩形区域内的稳定问题 分离变量法乜适用于热传导方程和稳定问题(例 如, Laplace方程)的定解问题 a2u a2 0<x<a,0<y<b ax2 ay2 00≤y≤b au l=0=f(a) 00<x< 仍可用分离变量法求解 仍然按照上面总结的四个标准步骤求解
Steady State Problems Forced Vibration in a String Fixed at Both Ends Ý/«S½¯K ©lCþ{·^u9D§Ú½¯K(~ X§Laplace§)½)¯K ∂ 2u ∂x2 + ∂ 2u ∂y2 = 0 0 < x < a, 0 < y < b u x=0 = 0 ∂u ∂x x=a = 0 0 ≤ y ≤ b u y=0 = f(x) ∂u ∂y y=b = 0 0 ≤ x ≤ a E^©lCþ{¦) E,Uìþ¡o(oIOÚ½¦) C. S. Wu 1nù ©lCþ{()
分离变量 a2u a2u 0<x<a,0<y<b au =00<y<b =00<x<a
Steady State Problems Forced Vibration in a String Fixed at Both Ends ©lCþ ∂ 2u ∂x2 + ∂ 2u ∂y2 = 0 0<x< a, 0<y < b u x=0 = 0 ∂u ∂x x=a = 0 0≤y≤b u y=0 = f(x) ∂u ∂y y=b = 0 0≤x≤a C. S. Wu 1nù ©lCþ{()
分离变量 a2u a2u 0<x<a,0<y<b au =00<y<b =00<x<a 仍用分离变量法求解.令 u(a, y)=X(r(y)
Steady State Problems Forced Vibration in a String Fixed at Both Ends ©lCþ ∂ 2u ∂x2 + ∂ 2u ∂y2 = 0 0<x< a, 0<y < b u x=0 = 0 ∂u ∂x x=a = 0 0≤y≤b u y=0 = f(x) ∂u ∂y y=b = 0 0≤x≤a E^©lCþ{¦)©- u(x, y) = X(x)Y (y) C. S. Wu 1nù ©lCþ{()
分离变量 a2u a2u 0<x<a,0<y<b au =00<y<b =00<x<a 代入方程,分离变量,即得
Steady State Problems Forced Vibration in a String Fixed at Both Ends ©lCþ ∂ 2u ∂x2 + ∂ 2u ∂y2 = 0 0<x< a, 0<y < b u x=0 = 0 ∂u ∂x x=a = 0 0≤y≤b u y=0 = f(x) ∂u ∂y y=b = 0 0≤x≤a ✑ \§§©lCþ§= C. S. Wu 1nù ©lCþ{()
分离变量 a2u a2u 0<x<a,0<y<b au =00<y<b =00<x<a X"(x) X(x)+λX(x)=0 X(r) Y"(y)-AY(y)=0
Steady State Problems Forced Vibration in a String Fixed at Both Ends ©lCþ ∂ 2u ∂x2 + ∂ 2u ∂y2 = 0 0<x< a, 0<y < b u x=0 = 0 ∂u ∂x x=a = 0 0≤y≤b u y=0 = f(x) ∂u ∂y y=b = 0 0≤x≤a X00(x) X(x) =− Y 00(y) Y (y) =−λ =⇒ X00(x)+λX(x)=0 Y 00(y)−λY (y)=0 C. S. Wu 1nù ©lCþ{()