《数学物理方程》课程PPT教学课件（讲稿）8.4 泊松方程

84泊松方程 △M=∫(x1y,2)非齐次方程 无时间变量一不能用冲量定理法 特解法Mm(xy)=(xy:)+W(xy 设定 待求 △w(x,y,z)=f(x,y2z) △(xyz)=0拉普拉斯方程 例圆域 P<p△a=a+b(x2-y p-p

8.4 泊松方程 2 2  = + − u a b x y ( )  = u f x y z ( , , ) 无 时间变量－不能用冲量定理法 非齐次方程 特解法 u x y z v x y z w x y z ( , , ) ( , , ) ( , , ) = + 设定 待求  = w x y z ( , , ) 0 拉普拉斯方程 例  = v x y z f x y z ( , , ) ( , , ) 0 u c  = = 圆域    0

(x,y)=(x2+y2)+(x-y)=p2+Pcos2 △(x p=-C、q b Po cos 2o w(p,p)=Co+Do In p+2(mp"m+Dmp m)(An cos mg+mn sin mp) w(P,)=Co+Ep"(A, cosmo+B, sin mp m= 边界条件 w(Po, )=Co+2Po"(Am, cos mp +Bm sin mo b 4 po cos 20 12 C- Po

2 2 4 4 2 4 ( , ) ( ) ( ) cos 2 4 12 4 12 a b a b v x y x y x y = + + − = +     = w x y ( , ) 0 0 2 4 0 0 cos 2 4 12 a b w c  = = − −    0 0 1 ( , ) ln ( )( cos sin ) m m m m m m m w C D C D A m B m         − = = + + + +  0 0 0 1 ( , ) ( cos sin ) m m m m w C A m B m       = = + +  2 4 0 0 cos 2 4 12 a b = − − c    2 0 0 4 a C c = −  0 1 ( , ) ( cos sin ) m m m m w C A m B m       = = + +  边界条件

∑P"( Am cos m+ Bmn m) P0242cOs20=-,P 2 0 COS 20 120 w(e,=c-p Po p cos 2o (p,q)=v(p,)+w(P b +c0s20+C-,P po p cos zpp 12 4 12 c+(p2-P2)+,p(2-p62)cos20 4

0 1 ( cos sin ) m m m m    A m B m  =  + 4 0 cos 2 12 b = −   m = 2 2 4 0 2 0 cos 2 cos 2 12 b     A = − 2 2 0 12 b A = −  2 2 2 0 0 ( , ) cos 2 4 12 a b w c       = − − u v w ( , ) ( , ) ( , )       = + 2 4 2 2 2 0 0 cos 2 cos 2 4 12 4 12 a b a b = + + − −        c 2 2 2 2 2 0 0 ( ) ( )cos 2 4 12 a b = + − + − c      

例 0 0 0 V(x x(-x) +1 y x=0 X=a y=0rla-x sh=x(a-x) wxy)=∑4exp"n!+Bp any nIx 1 sin

2 xx yy u u + = − 0 0 x u = = 例 0 x a u = = 0 0 y u = = 0 y b u = = v x y x a x ( , ) ( ) = − 2 xx yy v v + = −0 w w xx yy + = 0 0 w x= = 0 w x a= = 0 ( ) w x a x y= = − ( ) w x a x y b= = − 1 ( , ) { exp[ ] exp[ ]}sin n n n n y n y n x w x y A B a a a     = = + − 

0=x(a-x) h=x(a-x) w(x,0)=2(4,+B, )sin nx=x(a-x) w(x,b)=2(A, exp[1+B, exp[-1)sin"-=x(a-x) 4的联立代数方程 xa-x)=∑cnsi nIX A,+B n兀b A, exp[-]+ B exp/-nr6

0 ( ) w x a x y= = − ( ) w x a x y b= = − 1 ( ,0) { }sin ( ) n n n n x w x A B x a x a   = = + = −  1 ( , ) { exp[ ] exp[ ]}sin ( ) n n n n b n b n x w x b A B x a x a a a     = = + − = −  , A B n n 的联立代数方程 1 ( ) sin n n n x x a x C a   = − =  A B C n n n + = exp[ ] exp[ ] n n n n b n b A B C a a   + − =

+B A, exp[]+BC、Hmb n兀b b exp[-]+(Cn-A)exp[ nzDl=C n兀b n丌b n丌b A, lexp(-)-exp(]=C,1-exp(--) 1-exp(nzb exp(nTb n兀b exp 1-exp(n6 exprn n兀b n兀b n元 b n元 b exp(-)-exp( eXp(一

A B C n n n + = exp[ ] exp[ ] n n n n b n b A B C a a   + − = B C A n n n = − exp[ ] ( )exp[ ] n n n n n b n b A C A C a a   + − − = [exp( ) exp( )] [1 exp( )] n n n b n b n b A C a a a    − − = − − 1 exp( ) exp( ) exp( ) n n n b a A C n b n b a a    − − = − − B C A n n n = − 1 exp( ) exp( ) 1 [1 ] exp( ) exp( ) exp( ) exp( ) n n n b n b a a C C n b n b n b n b a a a a       − − − = − = − − − −

nTX rla-x)= ∑ C sin nIx 4a 8a x(a-xsin -dx [(-1)”-1 (2k+1)丌 (2k+1)b 8a p (2k+1)z3 epr(2k+1)Ib )-exp( (2k+1)b 8a eb/(2k+1)丌b (2k+1)C(2k+1)zb )-exp( (2k+1)b C u(x,y)=x(a-x)+2(A, exp nny n7 nZX +B, exp[-1 sin

1 ( ) sin n n n x x a x C a   = − =  2 2 3 3 3 3 0 2 4 8 ( )sin [( 1) 1] (2 1) a n n n x a a C x a x dx a a n k    = − = − − = − +  2 3 3 (2 1) 1 exp[ ] 8 (2 1) (2 1) (2 1) exp( ) exp( ) n k b a a A k k b k b a a     + − − = − + + + − − 2 3 3 (2 1) exp[ ] 1 8 (2 1) (2 1) (2 1) exp( ) exp( ) n k b a a B k k b k b a a     + − = − + + + − − 1 ( , ) ( ) { exp[ ] exp[ ]}sin n n n n y n y n x u x y x a x A B a a a     = = − + + − 