数学物理方程-第2讲new

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2、)êÆÔn§AϼêNovember26,20083/37þgSNES£Ñ¤¡´(®ÏÆnÆ)êÆÔn§AϼêNovember26,20084/372.1k.ugdÄÄüà½ugdįKµ22∂u=a2∂u,00(2.1)∂t2∂x2u

3、x=0=0,u

4、x=l=0(2.2)u

5、=ϕ(x),∂u

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7、t)sinωxddéu§·Á¦§£2.1¤äkXe±©lCþ/ª")u(x,t)=X(x)T(t)¡´(®ÏÆnÆ)êÆÔn§AϼêNovember26,20085/372.1k.ugdÄ£Y¤·k∂2u00∂2u00=X(x)T(t),=X(x)T(t)∂x2∂t2§£2.1¤§kX(x)T00(t)=a2X00(x)T(t)½ö¤X00(x)T00(t)=X(x)a2T(t)±-X00(x)T00(t)==−λX(x)a2T(t)u´ü~©§T00(t)+λa2T(t)=0(2.4)X00(x)+λX(x)=0(2.5)¡

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9、^>.^£2.2¤§qkX(0)T(t)=0X(l)T(t)=0duT(t)Øð"§Ïd7,kX(0)=X(l)=0"u´§¦)5½)¯K©lCþ/ª)§·Ò7LÄk¦)±e~©§>¯KµX00(x)+λX(x)=0(2.5)X(0)=X(l)=0(2.6)·¡þ¡~©§>¯KA¯K§ª¥½~êλT¯KA§éAudA§")X(x)¡§A¼ê"§)£A¼ê¤òXλ<0,λ>0,λ=0ùn«ØÓ¹

10、äkØÓ/ª"¡´(®ÏÆnÆ)êÆÔn§AϼêNovember26,20087/372.1k.ugdÄ£Y¤e¡§éλ<0,λ>0,λ=0n«¹?1?ص£1¤λ<0â~©§£§d§X00(x)+λX(x)=0√√)X(x)=Ae−λx+Be−−λx"2d>.^X(0)=X(l)=0§(A+B=0√√Ae−λl+Be−−λl=0dþ¡§

11、A=B=0§?kX(x)=0§ùX´")¦ØΧÏdλØUu0"¡´(®ÏÆnÆ)êÆÔn§AϼêNovember26,20088/372.1k.ugdÄ£Y¤£2¤λ=0d

12、§CX00(x)=0§)X(x)=Ax+B"2d>.^X(0)=X(l)=0§B=0Al+B=0ÓA=B=0§?kX(x)=0§ùX´")¦ØΧÏdλØUu0"¡´(®ÏÆnÆ)êÆÔn§AϼêNovember26,20089/372.1k.ugdÄ£Y¤£2¤λ>0-λ=β2§d§Cd§CX00(x)+β2X(x)=0â~©§£§§)X(x)=Acosβx+Bsinβx2d>.^X(0)=X(l)=0§A=0Acosβl+Bsinβl=0ùp§BØU0£oº¤"u´ksinβ

13、l=0"lnπβ=(n=±1,±2,±3,...)l¡´(®ÏÆnÆ)êÆÔn§AϼêNovember26,200810/372.1k.ugdÄ£Y¤±Än>0¹"?Ú§ÒkXnπ2n2π2λ=β2==(n=1,2,3,...)(2.7)nnl2lþãλnÑ´e¡A¯KAµX00(x)+λX(x)=0X(0)=X(l)=0u´2éAA¼êµnπXn(x)=Bnsinx(n=1,2,3,...)(2.8)l¡´(®ÏÆnÆ)êÆÔn§AϼêNovember26,200811/372.1k.ug

14、dÄ£Y¤y3¦)T(t)§òλnT00(t)+λa2T(t)=0Òka2n2π2T00