数理方程(中科大)讲义5

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1、Æh,e3.1Bessel,ewx%G+R$ν2′′′229Bessel$:xy+xy+(x−ν)y=0(ν≥0)X+∞k1x2k+νν9Bessel,e:Jν(x)=(−1)()k!Γ(k+ν+1)2k=0ν9Bessel$l:o:y(x)=CJν(x)+DNν(x)X+∞x(ζ−ζ−1)n2S,e:e=Jn(x)ζn=−∞ν′νn%+w`:(xJν)=xJν−1(x−νJ)′=−x−νJνν+1−1m)`:Jν−1+Jν+1=2νxJνJ−J=2J′ν−1ν+1νr2νππ7=)`:Jν≈πxcos(x−2−

2、4)r2νππNν≈sin(x−−)πx24%13.2Bessel$*si2′′′22xy+xy+(λx−ν)y=0(0

3、e2′′′$:(1−x)y−2xy+λy=0Lengedre$*si2′′′(1−x)y−2xy+λy=0(−1

4、t

5、<1)n=0m)`:P29722RU$:

6、

7、Pn

8、

9、=2n+1X+∞f(x)BFourier-Lengedre3e:f(x)=CnPn(x)n=03X+∞2
x=CnPn(x)n=0△3u=0(r

10、=cos2θ,r=a△3

11、u=0(r

12、θ=π=0u

13、r=a=u0,2
oa8

14、r=a=−=−√,ρa2−2abcosθ+b2P302−304.4tÆ0:.:1%`0:$#:→1%4→5.1δ,e(
Y:+∞(x=0)1)δ(x)=0(x6=0)Z+∞2)δ(x)dx=1−∞uH>1.-,e2.[q%~Z+∞1.℄~δ(x−ξ)ϕ(x)dx=ϕ(ξ)−∞Z+∞δ(x)ϕ(x)dx=ϕ(0)−∞2.Æ~

15、δ(x)=δ(−x)Z+∞3.′′eδ(x)f(x)dx=−f(0)−∞Z+∞δ(n)(x)f(x)dx=(−1)nf(n)(0)−∞5Z+∞4.Fourier−iλx/F[δ(x)]=δ(x)edx=1−∞F−1[1]=δ(x)Z+∞1iλxδ(x)=edλ2π−∞Z+∞1δ(x)=cosλxdλ2π−∞&pW}δ(x,y,z)=δ(x)δ(y)δ(z)5.2a$5.2.1Lu=0

16、$0:LU=δ(M)ÆoLu=0

17、$0:HU∗fbLu=f(M):1∆3u=00:bU=−4πr11∆2u=00:bU=−ln2

18、πry′+ay=0−αx0:bU=eh(x)1∆3u=−f(M):b∗f(M)4πr65.2.2',e2VuH△u=−f(x,y,z)((x,y,z)∈V)u

19、S=ϕ(x,y,z),△G=−δ(M−M0)((x,y,z)∈V)',e:G

20、=0,S',euHG(M1;M0)=G(M0;M1)ZZZZZ∂Gu(M)=−ϕ(M0)dS+Gf(M0)dM0s∂nV5.2.3?z#Y',e1111.^D5G=[−]4πr(M,M0)r(M,M1)11R12.XTG=[−]4πr(M,M0)ρ0r(M,M1)1113.^

21、UG=[ln−ln]2πr(M,M0)r(M,M1)11R14.TG=[ln−ln()]2πr(M,M0)ρ0r(M,M1)△3u=0(z>0)u

22、z=0=ϕ(x,y),Z+∞Z+∞1ϕ(x,y)ζu(ξ,η,ζ)=dxdy32π−∞−∞[(ξ−x)2+(η−y)2+(ζ−z)2]27△2u=0(y>0)u

23、y=0=ϕ(x),Z+∞1ϕ(x)ηu(ξ,η)=dxπ−∞(ξ−x)2+η25.2.4Fourier$#Y',e85.3ut=Lu

24、$Cvsi0:ut=Lu(t>0,−∞

25、(x,y,z)ut=Lu+f(t,x,y,z)Æo(1)u(0,x,y,z)=ϕ(x,y,z)0:ZtHU(t,M)∗ϕ(M)+U(t−τ,M