数理方程2012试卷.pdf

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1、Z#8:℄f_uy{wanguftl=3<)1<2011∼2012unxuomjri9.u(x,t)eG')Y,ZB,'?o&>G-φ,ψ[x1,x2]'skdb(e3v)`+'H%T={(x,t()

2、}
1,

3、x

4、≤1(10.C^3pu(x)=u(x,t)'FourierÆ
%F(ω)=jrqh2012n11z18p)0,otherelse}
A=4B8

5、[4y/<108r118&81.u(x,y)=F(x)+g(y),F,G4Vl

6、6d#6uxx+uyy=0'=g(<308rg38)1.X
()%l'b:ta:'_.G"YZBU
"Y'-E%n2.{pn(x)}LegenderU1/*pn(1)=1,pn(−1)=−1
()-D.'-%n?:''-8%φ(x),+X(du%3.−1f1,f2'FourierÆ
F1,F2,F(F1∗F2)(x)=f1(x)·f2(x)
()
4.0F.*ZB')Y,ZB,(R5T`x;!%(−∞,+∞)2.?9W

7、w(x,t)q℄wx

8、x=0=v1(t),w

9、x=l=v2(t),d7w(x,t)%d!h3;~X
()
−x5.6uxy−2uxuy=0w.4z#86
()3.&>%φ(x)=e-%ψ(x)=0')Y,ZB,?o%6.−p1~t−1,t≥0'LaplaceÆ
dj;4T~&F(p)=ep2
()
7.6uxx+uyy=f(x,y,t)
2'
()4.B68.X′′(x)+λX(x)=0v$($4*%kn.VkF(t),.YSdu%u

10、x=0

11、=F(t)X

12、=0,X

13、=0k
()xx=0xx=l9.(4(ENeumann(
()+'>C*%
5.10.+X('X}$=(Æ+'
()?pn(x)nLegendreU1/=
6.2′′′6ρR(ρ)+2ρR(ρ)−n(n+1)R(ρ)=0'X%R"(<128r168)R(ρ)=
1.T6H%Y2y/{i2
2+yz,7.u(x,y,z)=xgrad(u)=
?'Æ

Z#8_XASuy{wanguft

14、l=3<)2<(<488)1.(128)%l'+:`#1,:'s-CDWtb'
:'m.9k2-9%l(1−2ε),F7
:`ZB,
~:',?o
(:95ÆHU*Æ)(x,0G-8%φ(x),ψ(x)~X:'ZB,?o
Z#8_XASuy{wanguftl=3<)3<4.(12

15、8)~X+X((u+u=0,x2+y2>a2xxyyu

16、x2+y2=a2=0,u

17、ρ→∞=−E0x3.(128)~X+X(u=a2u,x>0,t>0txxu

18、x=0=N0,u

19、t=0=0.4.(128)~X+X((u+u=0,x2+y2>a2xxyyu

20、x2+y2=a2=0,u

21、ρ→∞=−E0x