数学物理方程-陈才生主编-课后习题答案1-3章.pdf

1、1c 1.1 = 411.1.1 Z 5(1)5, ZHq s;(2) pX5;(3) p) ai OT a d.1.1.2 p ZZEnZEoE(DAlembertE)as M EasMEaGreenfEa ? sZEaMsZE. 1 P - ZE.1.1.3 Z1.y ()Z(1)V “dBs,s #sBlsMT (2) ? p,1 Newton= pa ? o p, T0VrT (3)e 5 sZ.2.y ()Z H p(1) Newton= p(F = ma).(2) Fourier L p( . p).8 =i H, 3 . w Y , w 1 “1,ZE dS = 1“Zd;“ ,

2、 .(7) X(Hooke) p.K =,8 8M 1,f = kx, k8( “ :ut = a2uxx; 0 0;u(x;0) = 12x(Lx); 0 6 x 6 L;u(0;t) = 0; ux(L;t) = qk; t 0:1.2 !BL ( ?Sl, x uW0;L. uVUM,?L ,f vlT.V, sBE ,E vlM 1,1 “ k. ! SM(x), S 0.x = 0 %,x = L B, :utt = Tuxx kut; 0 0;u(x;0) = (x); ut(x;0) = 0; 0 6 x 6 L;u(0;t) = 0; (Tux +ku)flflflx=L= 0

3、; t 0:1.3 I nZ u = f(x;y)j0 :uxx uyy = 0; (x;y) 2 ;u(x;0) = f1(x); u(x;1) = f2(x); 0 6 x 6 1;u(0;y) = g1(y); u(1;y) = g2(y); 0 6 y 6 1;f1;f2;g1;g2 Xf . k5 a? I ? .B ,5 B,yN a.1 , u = u(x;y) 5, * v = u + sinn xsinn y(n 1)9 5.yN 5B.1.4 5(ut = uxx; (x;t) 2R1 (0;1);u(x;0) = 1; x 2R1 a. f un(x;t) = 1+ 1n

4、en2t sinnx (n 1), 80flflun(x;t)1flfl = supx2R1;t0flflflfl1nen2t sinnxflflflfl = supt0flflflfl1nen2tflflflfl 1nen2 !1;5 .1.3 5s1.5 8 , X 8ZFEulerZ1.3 5s 5 vt +(vr)v +rp = F; (1.3.1) Zt +r(v) = 0; (1.3.2)# Zp = f(); (1.3.3)F, Z(1.3.1) AT s vx;vy;vzZ v;p;sY a ; F s . k F = 0 H, 2o b .l 2oZ. !p00 b / ,i

5、:S = 00, =0(1+S).ZL n,L ! ,. 2o b .l H, Sv l, 0. T(1.3.1) T(1.3.2)=Q ( V , T(1.3.1)Mvt = 1rp; (1.3.4)7 T(1.3.2)Mt +0rv = 0;St +rv = 0: (1.3.5)2o.lV , V Z(1.3.3) p = p0(1+S) , V Lf p = p0(1+ S); (1.3.6) 1 1 1. T(1.3.4) T(1.3.6),vt = p00rS: (1.3.7)7 T(1.3.5) T(1.3.7)Stt a2S = 0; (1.3.8) 2oZ, a2 = p00.

6、1.6B ( %, ( , , f E,:Z_Mu(x;t). k S:_Z2ut2 =Ex2xx2ux: (1.3.9) ( %Sl, VHooke p,i OL ! .uV Um1.1 Ux;x + xl M,l Sx, S % 6 1c .Newton= p,Sx2ut2 = P(x+x;t)S(x+x)P(x;t)S(x) =x(PS)x;P(x;t) x S(x) t H YxZ_ s . 7x ! 0,S2ut2 =x(PS); (1.3.10)7Hooke p,P = Eux.yS = r2 = (xtanfi)2,PS Z(1.3.10), x2 tan2 fi2ut2 =xx

7、2 tan2 fiEux;e, T(1.3.9).8. TUSm1.2 U,5:_Z1 xh2 2ut2 = Ex1 xh2 ux; (1.3.11)h. Y L ,N H S = r2;r = (hx)tanfi.| T(1.3.10),L T(1.3.11).m1.1m1.21.7 bHMaxwellZFs T8:rE = 0;rE = 1cHt;rH = 0;rH = 1cEt:(1.3.12)kBFZHoZ1.3 5s 7 (Ett = c2E;Htt = c2H;EHsY b :sinfi tanfi = uy(x;y;t);sinfl tanfl = uy(x;y +y;t);si

8、n tan = ux(x;y;t);sin tan = ux(x+x;y;t):1.3 5s 11 | T(1.3.13), VTxuy(x;y +y;t)uy(x;y;t)+Tux(x+x;y;t)ux(x;y;t) = xyutt;Tuyy +uxx= utt:7x ! 0;y ! 0,T(uxx +uyy) = utt:a2 = T=,5utt a2(uxx +uyy) = 0: (1.3.14) =oZ.T suZ_ T, ( s)F(x;y:t), 7f(x;y;t) = F(x;y;t)=,5Z(1.3.14)Mutt a2(uxx +uyy) = f(x;y;t): (1.3.1

9、5) d Q=oZ.m1.71.13 ! = f(x;y)j x 2R1;y 0g. I n O5(u+u = 0; (x;y) 2 ;u(x;0) = (x); uy(x;0) = (x); x 2R1;(x);(x)R1 f .5 a$ 12 1c 5B a. , |un(x;y) = 1n2eny sin(pn2 +1x).A,n !1 H,supx2R1j(x)j+ supx2R1j(x)j = (n2 +n1) supx2R0jsin(pn2 +1x)j! 0: ,n !1 H,sup(x;y)2jun(x;y)j!1:1.14 Z./ I nB 0. !B ( %5, A8(1 )

10、, B(1 ), * 1 .u(x;t)VUx) H Yt HA8 i. i |Bl x0;x,nm1.8.x0;xm1.8B, c M(t) =Z xx0u(y;t)dy; , M0(t)=Z xx0ut(y;t)dy.6,B, M 7 3. Fick pM0(t) = = Kux(x;t)Kux(x0;t);Z xx0ut(y;t)dy = Kux(x;t)Kux(x0;t):H1x p,ut = (Kux)x:B Z, K “ . bW u 5, | 7 uD . * Fick p,ZZDZutdxdydz =ZZDKundS =ZZDZdiv(Kru)dxdydz: uD i,#ut

11、= div(Kru) = xKux+ yKuy+ zKuz: Z. TK = a2 ,5 Zut = div(Kru) = a22ux2 +2uy2 +2uz2= a2u:1.3 5s 13 1.15L5,B , 6B7b.5 b c 8, iu0,_5 = . 55. !x = 0 ,5 8 ,# pux(0;t) = 0. 7bB 5MY,5 b 8 iB“. u(L;t) = u0.# 558:ut = Duxx; 0 0;ux(0;t) = 0; u(L;t) = u0; t 0;u(x;0) = 0; 0 6 x 6 L:1.16 !BCc, Su0,i ! =V 9F HWtL1“

12、,V Newton p . k s 5. m1.9 V U, !c =r1,r2,5 u,1 I nu r HWtM f.A 5WZ SHqsYut = Du = D(urr +r1ur);r1 0;u(r;0) = u0;7 =V m1.9u(r1;t) = at+b;a;b .u(r;0) = u0, V pb = u0.#u(r1;t) = at+u0: !u1,5Newton pkur(r2;t) = H(u(r2;t)u1);(u+hur)flflr=r2= u1;h = k=H, kHsY .“ “ . HHq.1.17BL %,S:_ H, x = 0 %./ Hq/x = L

13、HHq.(1) x = Ls:_ T; 14 1c (2) x = Ls F(t) = ku(L;t)T, k“ , u(L;t)x = L:_M.(1)Hooke p,uxflflx=L = F(t)ES ;TE f , S .(2)F(t) = ku(L;t),uxflflx=L = kESu(L;t);(ux +hu)flflx=L = 0;Th = k=ES:2c=L sZs S2.1 = 412.1.1 1M =L sZs S1.Zs I n 1M x;y=L sZa11uxx +2a12uxy +a22uyy +a1ux +b1uy +cu = f(x;y); (x;y) 2 (2

14、.1.1) L“ f a11(x;y);a12(x;y);a22(x;y) , Y T = a212a11a22.(1) (x0;y0), Y T 0,5Z(2.1.1)(x0;y0)w;(2) (x0;y0), Y T = 0,5Z(2.1.1)(x0;y0) ;(3) (x0;y0), Y T 0.N HZF(2.1.3) B Ls wL,1(x;y) = c1; 2(x;y) = c2;M = 1(x;y); = 2(x;y);Z(2.1.1) VBSu = A2u +B2u +C2u+F2: (2.1.4)TMfi = 12( +); fl = 12( ); (2.1.5)Z(2.1.

15、1) V=Sufifi uflfl = A3ufi +B3ufl +C3u+F3: (2.1.6) = 0. HZF(2.1.3)BsZdydx =a12a11;BB L+ wL, !(x;y) = c. | = (x;y) Bf = (x;y),P(;)(x;y) 6= 0, * T(2.1.1) V ZSu = A4u +B4u +C4u+F4: (2.1.7) Tx 6= 0;Y | = y Ty 6= 0;Y | = x. 0: l, /s (4) Tp 0;q 0;p+q = n,5“ sZ(2.1.10) w.+Y,(p;q) = (n1;1),(p;q) = (1;n1) H,Z

16、(2.1.10) w. 18 2c=L sZs S(5) Tp 0;q 0;p+q 0,Z w.(2)“ a11 = a2;a12 = a;a22 = 1, Y T = a212 a11a22 = 0,Z .(3)“ a11 = x2;a12 = 0;a22 = y2, Y T = a212 a11a22 = x2y2.xy 6=0 H,Z w.xy = 0 H,Z .(4)“ a11 = x2;a12 = 0;a22 = (x+y)2, Y T = a212a11a22 = x2(x+y)2.x(x+y) 6= 0 H,Z ;x(x+y) = 0 H,Z ./ ZS2.2 uxx +4uxy

17、 +5uyy +ux +2uy = 0. Y T = 1 :uxx 2sinxuxy (3+cos2 x)uyy +ux +(2sinxcosx)uy = 0;(x;y) 2R2;u(x;cosx) = 0; uy(x;cosx) = ex=2 cosx; x 2R1: Y T = 4,Z w.+ZF dydx = 2sinx;dydx = 2sinx;+L y cosx2x = c1; y cosx+2x = c2:7 = y cosx2x; = y cosx+2x,5ZS4u = u;Yu = F()+e4G(),#ZY u(x;y) = F(y cosx2x)+eycosx2x4 G(y

18、 cosx+2x):B SHq,u(x;cosx) = F(2x)+ex2 G(2x) = 0;G(2x) = ex2 F(2x);p 2G0(2x) = 12ex2 F(2x)+2ex2 F0(2x):= SHqex2 cosx = uy(x;cosx) = F0(2x)+ 14ex2 G(2x)+ex2 G0(2x);NF(2x) = ex2 (sinx+c1); G(2x) = sinx+c1; S5 u(x;y) = F(y cosx2x)+eycosx2x4 G(y cosx+2x)= 2cosxsiny cosx2eycosx2x4 : 22 2c=L sZs S2.9 1M =“

19、 wZa11uxx +2a12uxy +a22uyy +a1ux +b1uy +cu = f(x;y) (2.2.1)B VV1M # d sMu = Ve+| V V +CV = F(;) T, C , FXf . = a212a11a22 0 H,Z(2.2.1) wZ,#i = (x;y); =(x;y), (;)(x;y) 6= 0, PZ(2.2.1)u u +A1u +A2u +A3u = f1(;);B 7u = Ve+,5V V +(A1 2)V +(A2 2)+(2 2 +A1 +A2 +A3)V = f1e;7 = A12 ; = A22 ,5ZMV V +CV = F(;

20、);C = 14(A22 A21)+A3;F = f1e:8.2.10 a | , Mv(x;y) = ex+yu(x;y),e/ Z(1) uxx +uyy +fiux +fluy + u = 0;(2) uxx = a2uy +flux +fiu;(3) uxx a2uyy = fiux +fluy + u;(4) uxy = fiux +fluy.(1) !u(x;y) = exyv(x;y), Z, Vvxx +vyy +(fi2)vx +(fl 2)vy +(2 +2 fifl + )v = 0;| = 12fi; = 12fl; 0 = fi2 +fl24 ,5 Zvxx +vyy

21、 + 0v = 0:(2) , !u(x;y) = exyv(x;y), = 12fl; = a2fi+ fl24.5Zvxx = a2vy:2.25s 23 (3) , !u(x;y) = exyv(x;y), = 12fi; = 12a2fl.5Zvxx a2vyy + 0v = 0; 0 = 14(fl2a2 fi2) .(4) , !u(x;y) = exyv(x;y), = fl; = fi, 0 = fifl,5Zvxy + 0v = 0:2.11 pZuxy = 1xy (ux uy):Y.Z V(xy)uxy = ux uyxuxy +uy = yuxy +ux; V(xux

22、+u)y = (yuy +u)x(xu)xy = (yu)yx(xuyu)xy = 0;s Q, Y(xy)u = F(x)+G(y)u = F(x)+G(y)xy ;FG i= Vf .2.12 !ZAuxx +Buxy +Cuyy = 0 (2.2.2)“ A;B;C B2 4AC = 0;A 6= 0. ZY / Tu(x;y) = f(mx+y)+xg(mx+y);f;g i = Vf m = B2A.y Y T = B2 4AC = 0; Z Z+Zdydx =B2A = m;+ wL y +mx = c1.TM = y +mx; = x, * Zu = 0 24 2c=L sZs

23、SZ1s Q, Yu = f()+g();ZYu = xf(y +mx)+g(y +mx):p/ ZS2.13 uxy uxz +ux +uy uz = 0.Z“ A =0BBBBBB0 12 1212 0 012 0 01CCCCCCA;/ p B, PBABT = diagfi1;i2;i3g = D3, ik 2f1;0;1g. M p B.XE 5 (A;E3), E3 . (A;E3)B“ MM M,M (D3;B). * B p . L M M:|.y(A;E3) =0BBBBBB0 12 12 1 0 012 0 0 0 1 012 0 0 0 0 11CCCCCCAr3+r2!

24、0BBBBB0 12 12 1 0 012 0 0 0 1 00 0 0 0 1 11CCCCCAc3+c2!0BBBB0 12 0 1 0 012 0 0 0 1 00 0 0 0 1 11CCCCAr1+r2!0BBBB1212 0 1 1 012 0 0 0 1 00 0 0 0 1 11CCCCAc1+c2!0BBBB1 12 0 1 1 012 0 0 0 1 00 0 0 0 1 11CCCCAr212r1!0BBBB1 12 0 1 1 00 14 0 12 12 00 0 0 0 1 11CCCCAc212c1!0BB1 0 0 1 1 00 14 0 12 12 00 0 0

25、 0 1 11CCA2r2!0BB1 0 0 1 0 00 12 0 1 1 00 0 0 0 1 11CCA2.25s 25 2c2!0B 1 0 0 1 1 00 1 0 1 1 00 0 0 0 1 11CA;5B =0B 1 1 01 1 00 1 11CA:70B 1CA = B0B xyz1CA =0B x+yy xy +z1CA; = x+y; = y x; = y +z,5ZMu u +2u = 0; pS.2.14 uxx +2uxy +2uyy +2uyz +2uyt +2uxt +2utt = 0:Z“ A =0BBB1 1 0 11 2 1 10 1 0 01 1 0

26、21CCCA;* (A;E4) =0BBBB1 1 0 1 1 0 0 01 2 1 1 0 1 0 00 1 0 0 0 0 1 01 1 0 2 0 0 0 11CCCCAr2r1r4r1!0BBBB1 1 0 1 1 0 0 00 1 1 0 1 1 0 00 1 0 0 0 0 1 00 0 0 1 1 0 0 11CCCCAc2c1c4c1!0BBBB1 0 0 0 1 0 0 00 1 1 0 1 1 0 00 1 0 0 0 0 1 00 0 0 1 1 0 0 11CCCCAr3r2!0BBBB1 0 0 0 1 0 0 00 1 1 0 1 1 0 00 0 1 0 1 1

27、1 00 0 0 1 1 0 0 11CCCCAc3c2!0BBBB1 0 0 0 1 0 0 00 1 0 0 1 1 0 00 0 1 0 1 1 1 00 0 0 1 1 0 0 11CCCCA; 26 2c=L sZs SB =0BBBB1 0 0 01 1 0 01 1 1 01 0 0 11CCCCA;70BBBB1CCCCA= B0BBBBxyzt1CCCCA=0BBBBxy xxy +ztx1CCCCA; = x; = y x; = xy +z; = tx,5ZMu +u u +u = 0:2.15 2ux21 +2nXk=22ux2k 2n1Xk=12uxkxk+1 = 0:

28、Z“ A =0BBBBBBB1 1 0 01 2 1 00 1 2 0. . . .0 0 1 21CCCCCCCA;/ p B, PBABT = diagfi1;i2; ;ing, ik 2 f1;0;1g./ M pB.y(A;En) =0BBBBBBB1 1 0 0 1 0 0 01 2 1 0 0 1 0 00 1 2 0 0 0 1 0. . . . . . . .0 0 1 2 0 0 0 11CCCCCCCAr2+r1!0BBBBBBB1 1 0 0 1 0 0 00 1 1 0 1 1 0 00 1 2 0 0 0 1 0. . . . . . . .0 0 1 2 0 0 0

29、 11CCCCCCCA2.25s 27 c2+c1!0BBBBBBB1 0 0 0 1 0 0 00 1 1 0 1 1 0 00 1 2 0 0 0 1 0. . . . . . . .0 0 1 2 0 0 0 11CCCCCCCAr3+r2.rn+rn1!0BBBBBBB1 0 0 0 1 0 0 00 1 1 0 1 1 0 00 0 1 0 1 1 1 0. . . . . . . .0 0 0 1 1 1 1 11CCCCCCCAc3++cn1!0BBBBBBB1 0 0 0 1 0 0 00 1 0 0 1 1 0 00 0 1 0 1 1 1 0. . . . . . . .0

30、 0 0 1 1 1 1 11CCCCCCCAB =0BBBBBB1 0 0 01 1 0 01 1 1 0. . . .1 1 1 11CCCCCCA70BBBB12.n1CCCCA= B0BBBBx1x2.xn1CCCCA1 = x1; 2 = x1 +x2; ; k = x1 +x2 +xk (k = 1;2; ;n),5ZMSu = u11 +u22 +unn = 0: nLaplaceZ.2.16 2ux21 2nXk=2(1)k 2uxk1xk = 0: 28 2c=L sZs S ZE, V pB =0BBBBBB1 0 0 01 1 0 01 1 1 0. . . .1 1 1

31、 11CCCCCCA70BBBBB12.n1CCCCCA = B0BBBBBx1x2.xn1CCCCCA;1 = x1; 2 = x1 +x2; ; k = x1 +x2 +xk(k = 1;2; ;n).5ZMSnXk=1ukk(1)k+1 = 0:2.17 “ ZnXk=1uxkxk +nXk=1bkuxk +cu = 0;V ak(k = 1;2; ;n), PMu = vexp(1x1 +2x2 +nxn)/,ZenXk=1vxkxk +bv = 0:i O pk(k = 1;2; ;n): M Z,nXk=1vxkxk +nXk=1(bk +2k)vxk +nXk=1(2k +bkk)v +cv = 0;|k = 12bk,5 ZnXk=1vxkxk +bv = 0;b = c 14(b21 +b22 +b2n).2.3 5s 29 2.3 5s2.18Zuxx 2fiuxy 3fi2uyy +fiuy +ux = 0S,i O p Y,fi .y Y T = 4fi2,#fi 6= 0 H,Z w;fi = 0 H,Z .fi 6= 0 H,+ZF dydx = 3fi;dydx = fi;+L y +