数理方程(中科大)讲义4new

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1、%7527u75→754.1.1Fourier52HV1.f(x)HxU#EA/}UW-?Z+∞2.H(−∞,+∞)sa.lR

2、f(x)

3、dxH−∞Z+∞−iλx
BF(λ)=f(x)edx!f(x)#FourierF[λ]−∞Fourier#38=Hf(x)#k6&AZ1+∞f(x)=F(λ)eiλxdλ2π−∞T95R1)05RF[c1f+c2g]=c1F[f]+c2F[g]12)v*′′

4、f(±∞)=0,zF[f(x)]H,
F[f(x)]=iλF[f]′(n−1)(n)f(±∞)=0,f(±∞)=0,···,f(±∞)=0,zF[f(x)]H,(n)n
F[f(x)]=(iλ)F[f]4)`5RF[f∗g]=F[f]·F[g]F−1[F(λ)G(λ)]=F−1[F(λ)]∗F−1[G(λ)]:"FourierZZZ+∞F(λ,µ,ν)=f(x,y,z)e−i(λx+µy+νz)dxdydz−∞2ZZZ+∞1i(λx+µy+νz)f(x,y,z)=F(λ,µ,ν)edλdµdν(2π)3−∞75RF[fx]=iλF[f

5、]F[fy]=iµF[f]F[fx]=iνF[f]F[f]=(iλ)2F[f]=−λ2F[f]xxF[∆f]=−(λ2+µ2+ν2)F[f]34.1.2Y^i2ut=auxx(t>0,−∞

6、32utt=auxx(t>0,−∞

7、,tZ+∞−iλxY[Fourieru(t,λ)=u(t,x)edx−∞2du22=−aλu(t>0)dt2duu

8、t=0=ϕ(λ),

9、t=0=ψ(λ),dtu(t,λ)=A(λ)eiλat+B(λ)e−iλat1iλat−iλat1iλat−iλatu(t,λ)=[ϕ(λ)e+ϕ(λ)e]+[ψ(λ)e−ψ(λ)e]22aiλZx+at11u=[ϕ(x−at)+ϕ(x+at)]+ψ(ξ)dξ22ax−at42utt+auxxxx=0(t>0,−∞

10、x)sinλxdx0Z+∞fc(λ)=f(x)cosλxdx0Z2+∞f(x)=fs(λ)sinλxdλπ0Z+∞2f(x)=fc(λ)cosλxdλπ0u=a2u(t>0,x>0)ttxxiu(0,x)=0,ux(t,0)=Qu(0,+∞)=ux(t,+∞)=062ut=a△3u(t>0,−∞

11、>B#;$B5#XYA>B>#Y∂ui∂y=f(x)∂2ui∂x∂y=0∂2u∂ui∂x∂y+∂y=0∂u∂ui∂t+a∂x=05dx−adt=0x−at=c∂2u2∂2ui∂t2=a∂x22225(dx)−a(dt)=081.2;'35x)h[1.2.1h1.#H+5h1T1)H;wqs2)H+G2

12、u

13、≪1,

14、ux

15、≪13)A;;U#j∂2u2∂2u=a+f(t,x)∂t2∂x21.2.2!5∂udQ=−k(x,y,z)dSdt∂n∂u21=a△u+f(t,x,y,z)∂tcρ91

16、.2.3(#5(Cp,!ρ(x,y,z),(!E(x,y,z)ZZ1E·dS=ρ(x,y,z)dV:*n∂Vε0VZZE·dS=∇·EdV:=∂VVρ(x,y,z)∇·E=ε0ZE·dl=02e%*nLZZE·dl=∇×EdSStocks=LS∇×E=0E=−∇ϕ,∇·∇=∆ρ(x,y,z)∆ϕ=−ε0101.3*YVD*Y'4*5rÆ;;)h#u75*YV~*;#_VVZV4*5*Y'*YV1.3.1VD'V#Bux.U#

17、ϕ(x),,ψ(x)V,(Z.#Y+#*Y':24*5utt=auxx(t>0,−∞