数学物理方程第二版(谷超豪)第二章答案

数学物理方程第二版谷超豪

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19D §9Ù½)¯K Ñ

~1.1 þ![\ » l,b §3Ó ¡þ §Ý´ Ó ,\ L¡Ú± 0 u)9 ,¿Ñl5Æ

dQ=k1(u u1)dSdt.

b \ —Ý ρ§'9 c,9D Xê k,Á Ñd §Ýu÷v §.):

\¶ x¶, \ u[x,x+ x] ã 9þ²ï.ü mlý

dQ1= k1(u u1)πl x;

ü mlx?,x+ x?6\ 9þ

πl2 u

,dQ2= k(x)(x,t)·

4

uπl2

dQ3=k(x+ x)(x+ x,t)·,

41

¡6\ 9þ

数学物理方程第二版谷超豪

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ü m6\(x,x+ x) 9þ

uπl2

dQ=dQ1+dQ2+dQ3=k(x)· x k1(u u1)πl x.

x 4nþ,l t1 t26\ u[x1,x2]\ã 9þ

t2 x2

uπl2

k(x) k1(u u1)πldxdt.4t1x1

3ùã mS[x1,x2]\ãS :§Ýlu(x,t1)C u(x,t2),ÙáÂ9þ

x2 t2 x22

πl2 uπl

cρ(u(x,t2) u(x,t1))dx=cρdxdt.

44x1t1x1 â9þÅð,¿5¿ x1,x2,t1,t2 ?¿5, ¤¦ § 1 u4k1 u=k(x) (u u1).cρcρl~1.2Á í *ÑL§¤÷v © §.):

N(x,y,z,t)L«3 t,(x,y,z):?*ÑÔ ßÝ,D(x,y,z) *

N

dSdt.ÑXê,3á mãdtS,ÏLá -¡¬dS þ

dm= D(x,y,z)

Ïdl t1 t26\« (Γ L¡) þ

t2 t2

N

D(x,y,z)dSdt=div(DgradN)dxdydzdt.

t1Γt1 , ,l t1 t2, ¥TÔ O\

[N(x,y,z,t2) N(x,y,z,t1)]dxdydz=

t1

t2

N

dtdxdydz. â þÅð,¿5¿ ,t1,t2 ?¿5, ¤¦ §

N N N N=D+D+D.~1.3¬(·Yè)SÜ;õX9þ,¡ Yz9,3§ Ó Åì Ñ, 9 ÝÚ§¤;õ Ys9¤ '.±Q(t)L«§3ü NÈ¥¤; 9þ,Q0 Щ ¤; 9þ,KdQ= βQ,Ù¥β ~ê.qb ¬ '9 c,—Ý ρ,9D Xê k,¦§3 Ó §Ýu÷v §.à°7

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数学物理方程第二版谷超豪

5êÆÔn §6SK‘ùÂìÀ Æ%°©

):

¬S:(x,y,z)3 t §Ý u(x,y,z,t),w,

dQ = βQ, dt Q(t)=Q0e βt. Q(0)=Q0,

´ t1 t2 ,¬S? « ¥ 9þ O\ ul Ü6\ 9þ9¬¥ Yz9 Ú,=

t2

u

cρdtdxdydz=(Q(t1) Q(t2))dxdydz+t1

t2

u u uk+k+kdxdydzdtt1

t2

dQ

= dtdxdydz+

dtt1

t2

u u uk+k+kdxdydzdt.t1

5¿ t1,t29 ?¿5,k

u1 u u uβ=k+k+k+Q0e βt.cρcρ

~1.4 þ! ?3± ~ê§Ýu0 0 ¥,Áy:3~>6 ^e §Ý÷v © §

k 2uk1P0.24i2r u

= (u u0)+,cρ2cρωcρω

Ù¥i9r©OL« N >69>{,PL«î ¡ ± ,ωL«î ¡ ¡È, k1L« éu0 9 Xê.):

11Kaq, ¶ x¶,3 t1 t20u[x1,x2] ã 9

þO\ :l ٧ܩ6\ 9þ,lý¡6\ 9þ±9>6ÏL[x1,x2]ùã ) 9þ Ú,= t2 x2 t2 x2 x2 t2 ui2rkωdxdt k1P(u u0)dxdt+0.24dxdt.t1x1t1x1x1t1Ïd â9þ²ïÒ §Ý÷v §

uk 2uk1P0.24i2r

= (u u0)+.cρ2cρωcρω

à°7

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数学物理方程第二版谷超豪

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~1.5 ÔNL¡ ýé§Ý u,d § .Ë Ñ 9þ dA}-À [ù(Stefan-Boltzmann)½Æ 'uu4,=

dQ=σu4dSdt.

b ÔNÚ± 0 m k9Ë vk9D ,qb ÔN± 0 ýé§Ý ® ¼êf(x,y,z,t),¦d TÔN9D ¯K >.^ .):

>.þ ¡È dS.3dt mS,²>. 6Ñ 9þ (k

k

dT Ë Ü0 9þ

σu4dSdt.

Ü0 ÏLT Ë ÔNL¡ 9þ

σf4dSdt.

â9þ²ïk

k

¤¦>.^

k

u

dSdt=σu4dSdt σf4dSdt. u

=σ(u4 f4). u

dSdt.9D Xê)

2Ð> ¯K ©lCþ{

~2.1^©lCþ{¦e ½)¯K ):

ut=a2uxx(t>0,0<x<π),

u(0,t)=ux(π,t)=0(t>0), u(x,0)=f(x)(0<x<π).):

u(x,t)=X(x)T(t),K

T +λa2T=0,

X +λX=0,

X(0)=X (π)=0.

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数学物理方程第二版谷超豪

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2

1

λk=k+,k=0,1,2,...

2

∞ 1122

u(x,t)=Cke (k+)atsink+x.

2k=0

dЩ^

1

f(x)=Cksink+x

2k=0

π

12

f(ξ)sink+ξdξ. Ck=

02

∞

ìÀ Æ%°©

~2.2^©lCþ{¦)9D § Ð> ¯K:

ut=uxx(t>0,0<x<1), 0<x≤1, x, u(x,0)= 1 x,1<x<1, u(0,t)=u(1,t)=0(t>0).):

u(x,t)=