数学物理方程讲义(第3章答案) 谷超豪.pdf

5n6SK%? ? v|= u|+ M m 4R2 (x x0)2+ (y y0)2|. ?, 73 S,:(x1,y1) ?, 3:A k vxx 0, vyy 0,vxx+ vyy 0. ?, vxx+ vyy= uxx+ uyy+ M m 4R2 0 ?g. dAkM = m. 4.3 yK ut a2uxx= f(x,t), u|x=0= 1(t),(ux+ hu)|x=l= 2(t) (h 0), u|t=0= (x) ?)u(x,t) 3Rt1: 0 t t1,0 x l v u(x,t) et1max 0,max 0xl (x), max 0tt1 et1(t), et2(t) h ! , 1 max Rt1 (etf) ! , ?. ):Cv(x,t) = etu, ?. du ?Kv v vt a2vxx+ v = etf(x,t), v|x=0= et1(t), v n + hv !? ? ? ?x=l= et2(t), v|t=0= (x). v 3Rt1?, XJv(x,t) 3Rt1k?, K3: kvt 0, vxx 0 v 0, ? v = 1 e t f(x,t) (vt a2vxx) 1 e t f(x,t). |u(x,t)| et1 1 max Rt1 (etf(x,t). 2?P62y?O, =?(. 7htqi200814 5n6SK%? 5)?C5? 5.1 ye?9D?K ut a2uxx= 0, u|x=0= u|x=l= 0, u|t=0= (x) ?)?t + /Pu, Y, (0) = (l) = 0. ):d)K?) u(x,t) = X k=1 Ake k22a2 l2 t sin k l x, Ak= 2 l Z l 0 (x)sin k l xdx. )?C5? d(x) C0,l , k, |Ak| C1, C1= ?k?. |u(x,t)| C1 1 + X k=2 e a22(k21) l2 t e a 22 l2 t Ce a22 l2 t. 5.2 y: ?(x,y) R2?k.Y, L1(R2) , ?9D? K?), ?t + , t1Pu. ): |u(x,y,t)| 1 4a2t R2 |(,)|e (x)2+(y)2 4a2t dd 1 4a2t R2 |(,)|dd = Ct1. 5.3 y: ?(x,y,z) R3?k.Y, L1(R3) , n9D ?K?), ?t + , t3/2Pu. ): |u(x,y,z,t)| 1 (2a t)3 $ R3 |(,)|e (x)2+(y)2+(z)2 4a2t ddd 1 (2a t)3 $ R3 |(,)|ddd = Ct3/2. 7htqi200815 1nN 7 2008 c 12 ? 9 F 8 1!)1 2?9A8 3?11 4r4?n!1?K)?519 1!) 1.1 ?u(x1,., xn) = f(r) (r = q x2 1 + + x2 n) n N, y f(r) = c1+ c2 rn2 (n , 2), f(r) = c1+ c2ln 1 r (n = 2), c1, c2?. y:dur = q x2 1 + + x2 n, u xi = f 0(r)r xi = f 0(r)xi r , 2u x2 i = x2 i r2 f 00(r) + 1 r x2 i r3 ! f 0(r), (i = 1,2,.,n) 1 5n6SK%? “N? f 00(r) + n 1 r f 0(r) = 0, = f00(r) f0(r) = n 1 r . =?(. 1.2 y: .df3I(r,) e? 4u = 1 r2 r r2 u r ! + 1 r2sin sin u ! + 1 r2sin2 2u 2 . y: IX?IXm?CX x = rsincos, y = rsinsin, z = rcos, O, dIC x = Rcos, y = Rsin, z = z, 9 R = rsin, = , z = rcos. dIXeLaplace f?L 4u = 2u R2 + 1 R2 2u 2 + 1 R u R + 2u z2 .(1) 2d u r = u R sin + u z cos, u = u Rrcos u z rsin, )? u z = cos u r sin r u , u R = sin u r + cos r u .(2) 5? R2+ z2= r2,tan = R z , ?k r z = cos, z = sin r , r R = sin, R = cos r .(3) 7htqi20082 5n6SK%? d(2) 9(3) 2u z2 = cos2 2u r2 + sin2 r2 2u 2 + sin2 r u r + sin2 r2 u sin2 r 2u r , 2u R2 = sin2 2u r2 + cos2 r2 2u 2 + cos2 r u r sin2 r2 u + sin2 r 2u r . ?9(2) “(1) ?n, =?I(J. ?: ?-IX(q1,q2,q3) q1= q1(x,y,z),q2= q2(x,y,z),q3= q3(x,y,z), ,(x,y,z) L(q1,q2,q3) ? x = x(q1,q2,q3),y = y(q1,q2,q3),z = z(q1,q2,q3). P.rXH1, H2, H3 H1= s x q1 !2 + y q1 !2 + z q1 !2 , H2= s x q2 !2 + y q2 !2 + z q2 !2 , H3= s x q3 !2 + y q3 !2 + z q3 !2 , Kk ds2= H2 1dq 2 1+ H 2 2dq 2 2+ H 2 3dq 2 3. dLaplace f3-IX?L 4u = 1 H1H2H3 q1 H2H3 H1 u q1 ! + q2 H3H1 H2 u q2 ! + q3 H1H2 H3 u q3 !# .(4) 3IXeq1= r, q2= , q3= , ds2= dr2+ r2d2+ r2sin2d2, H1= 1,H2= r,H3= rsin H1, H2, H3“(4) =?IeLaplace f?L. 7htqi20083 5n6SK%? 1.3 y: .df3I(r,z) e? 4u = 1 r r r u r ! + 1 r2 2u 2 + 2u z2 . y: IX?IXm?CX x = rcos, y = rsin, z = z, r = (x2+ y2)1/2, = arctan y x, z = z. l? r x = x r = cos, r y = y r = sin, x = y x2+ y2 = sin r , y = x x2+ y2 = cos r . dd? u x = u r cos u sin r , u y = u r sin + u cos r , 2u x2 = 2u r2 cos2 2 2u r sincos r + 2u 2 sin2 r2 + u r sin2 r + u sin2 r2 , 2u y2 = 2u r2 sin2 + 2 2u r sincos r + 2u 2 cos2 r2 + u r cos2 r u sin2 r2 , ?, ?n, =?I(J. ?: K, 3IXeq1= r, q2= , q3= z, K ds2= dr2+ r2d2+ dz2, H1= 1,H2= r,H3= 1, “(4) =?IeLaplace f?L. 1.4 ye?N: 1. ax + by + c(a, b, c ); 2. x2 y22xy; 3. x3 3xy23x2y y3; 4. shnysinnx, shnycosnx, chnysinnx chnycosnx (n ); 7htqi20084 5n6SK%? 5. sh x(chx + cosy)1siny(ch x + cosy)1. y: ?y; ?: |EC?y, = $)?JN?5y. 1. 4(ax + by + c) = 0; 2. ?ECf(z) = z2= (x + iy)2; 3. ?ECf(z) = z3= (x + iy)3; 4. ?ECf1(z) = sh(nz) = sh(n(y + ix), f2(z) = ch(nz) = ch(n(y + ix); 5. ?ECf(z) = th z 2 = sh z 2/ch z 2, ?x = 0 y = (2k + 1) (k = 0,1,2,.) ?N. 1.5 y4IL?e?N: 1. lnr ; 2. rncosn rnsinn (n ); 3. rlnrcos rsin rlnrsin + rcos. y: ?y; ?: Pz = rei, |)? JN?5y. 1. ?ECf(z) = lnz = lnr + i; 2. ?ECf(z) = zn= rncosn + irnsinn; 3. ?ECf(z) = zlnz = r(cos + isin)(lnr + i). 1.6 lC)deN?1K?/ (0 x a, 0 y b) ?-: uxx+ uyy= 0, u(0,y) = u(a,y) = 0, u(x,0) = sin x a , u(x,b) = 0. 7htqi20085 5n6SK%? ):-u(x,y) = X(x)Y(y) “uxx+ uyy= 0 ?X Y Ov X00+ X = 0,X(0) = X(a) = 0;(5) Y00 Y = 0.(6) (5) k 0 k), = k= k22 a2 , Xk(x) = Cksin k a x,Yk(y) = Ake ky+ Bke ky. ?)K?) u(x,y) = X k=1 ? Ake k ay+ Bke k ay?sin k a x. d.? X k=1 (Ak+ Bk)sin k a x = sin x a , X k=1 ? Ake k ab+ Bke k ab?sin k a x = 0. )? A1= e b a 2sh b a ,B1= e b a 2sh b a ,Ak= Bk= 0 (k , 1). n? u(x,y) = sh (by) a sh b a sin x a . 1.7 3?.A?O, ?3Ye? , 3/0 x a, 0 y b )Xe?gN? K: 4u = py + q(p 0), ux|x=0= 0, u|x=a= 0, u|y=0,y=b= 0. 7htqi20086 5n6SK%? ):-v(x,y) = u(x,y) + (x2 a2)(fy + g), ?f = p/2, g = q/2, Kv e K) 4v = 0, vx|x=0= 0, v|x=a= 0, v|y=0= q 2(x 2 a2) = (x), v|y=b= 1 2(x 2 a2)(pb + q) = (x). dlC)? v(x,y) = X k=0 ? Ake (2k+1) 2a y + Bke (2k+1) 2a y?cos2k + 1 2a x = 2 X k=0 (1)k ?(2k+1) 2a ?3 sh 2k+1 2a b cos (2k + 1)x 2a (pb + q)sh (2k + 1) 2a (y b) qsh (2k + 1)y 2a # . 1.8 3?N?)|X?K, X)u(x,y) 3 ?k.?, o)K?)?. ):XeDirichlet ?K 4u = 0, r = p x2+ y2 1, u|r=1= 1. w,u 1, u = cln 1 r + 1 (c R) d?K?), =d)K?) . 1.9 ? J(v) = $ 1 2 v x !2 + v y !2 + v z !2 dxdydz + (1 2v 2 gv ) ds, ?CK: u V, J(u) = min vV J(v), V = C2() C1(). ?d?K, y?d5. y:dCKXe)K?d 4u = 0, u n + u !? ? ? ? ? = g. 7htqi20087 5n6SK%? 2?9A 2.1 y(2.7) uM03 ? ?/. y:?M03 , ?B(M0) ?M0(M0%, ). PS , = B(M0), Ku 9 1 rM0M 3SN. |Green 1? 0 = $ u4 1 rM0M 1 rM0M 4u ! dVM = S(M0) u n 1 rM0M 1 rM0M u n ! dSM, 1 rM0M u n u n 1 rM0M ! dSM= S 1 rM0M u n u n 1 rM0M ! dSM. S 1 rM0M u n u n 1 rM0M ! dSM = 1 S u(M) n dSM+ 1 2 S u(M)dSM = 1 u n(M )22 + 1 2 u(M)22 = 2u n(M ) + 2u(M) 2u(M0), 0. ?k 1 rM0M u n u n 1 rM0M ! dSM= 2u(M0). ?M03 ?, u 9 1 rM0M 3 SN. |Green 1?. 2.2 eu(x,y) ?N, q3 ? u = sin, L4?, u 3?:?u? ):d?N? u(M0) = 1 2a Z a uds, u(0,0) = 1 2 Z uds = 1 2 Z 2 0 sind = 0. 7htqi20088 5n6SK%? 2.3 XJ.dL|v?, ?SKk)?!fds = 0 ?n. ): ! fds = 0 =LL.-96?“, =?u- G?N, lL66?9?. 2.4 y: ?u(M) 34- ?N, 3? u(M) = O 1 rOM ! , u r = O 1 r2 OM ! (rOM ), ?M0 ?:, K(2.6) E. y:?M0 ?:, R ?KR, 9M03 S, PKR?R, K +R u n 1 rM0M ! 1 rM0M u n ! dS + 4u 4 u n ! = 0. u ? u n ? Ou u n 3M0?. -R , 5?u(M) = O ? 1 rOM ? , u r = O ? 1 r2 OM ? k lim R R u n 1 rM0M ! 1 rM0M u n ! dS = 0. ?u(M) 3 ?N u(M0) = 1 4 u(M) n 1 rM0M ! 1 rM0M u(M) n # dSM. 2.5 yN)|X?K)?-5. y:?u1, u2Dirichlet ?K?), = 4ui= 0, ui|= fi, lim r ui= 0, i = 1,2. -v = u1 u2, Kv v 4v = 0, v|= f1 f2, lim r v = 0. dulim r v = 0, ? 0, ?R ?KR, 3KR S, |v|R| . 3KR v A4?n, ?3KR |v| max ? ,max |f1 f2| ? |v| K)?5- 5. y:5: eu1, u2v Lu = n X i,j=1 aij 2u xixj + n X i=1 bi u xi + cu = 0, u|= f. -v = u1 u2, K Lv = 0,v|= 0. 7htqi200810 5n6SK%? dK, v 3 S?K?, ?K? 3.?, dd?v 0, ?u1 u2. -5: eu1, u2Ov Lui= 0,ui|= fi, i = 1,2. Kv = u1 u2v Lv = 0,v|= f1 f2. dK, ?min (f1 f2) 0 , 0 v 0 min (f1 f2) 0 , min (f1 f2) v max (f1 f2). n, ?|f1 f2| . ?0, ?x = 1 2 q 2 c, y = 1 2 q 2 c u = 1, 3 Su 1, u ? 3S?, l?4?n. 3? 3.1 y?53955. y:(53) du G(M, M0) = 1 4rM0M g(M, M0),M0 g(M, M0) v 4g(M, M0) = 0,M , g(M, M0)|= 1 4rM0M . w, g| 0, d4?ng(M, M0) 0, ? G(M, M0) 0. 4G(M, M0) = 0, K, G|= 0,G|= 1 4rM0M g(M, M0) 0. d4?n?, 3 KS, G 0. ? ?, ?G 0 3 S, dd =? 0 K?) uxx+ uyy+ uzz= 0,x2+ y2+ z2?”?N3?)|XK?) 4u = uxx+ uyy= 0,y 0, u|y=0= f(x). ):m/aq? G(M, M0) = 1 2 ln 1 rM0M 1 2 ln 1 rM1M . 7htqi200816 5n6SK%? dd? u(x0,y0) = y0 Z 1 (x0 x)2+ y2 0 f(x)dx. 5?u vu(M) = O ? ln 1 rOM ? 9u n = O ? 1 rOM ? u(M0) = 1 2 Z u(M) n ln 1 rM0M ! ln 1 rM0M u(M) n # dsM. 3.9 ? ?/3?:O %!R ?K , u(r,) d?N, (r,) L :M ?I. ?r1= R2 r , K :M1= (r1,) :M uK ?:, lM(r,) ?M1(r1,) ?C _CCCCCC. 1L ?, y v(r1,) = R r1 u R2 r1 , ! 1?N(?:?). XJ K ?., Kv(r1,) 31? :O ?N?. v(r1,) u(r,) ?ppp?(Kelvin) CCC. y:|IXeLaplace f?L, dE?K?: ? 4r,.u(r,) = 0 , 7k 4r1,.v(r1,) = 0. 3.10 |p?C9:?5nr)|X?Kz)|X SK. ):)Dirichlet ?K 4u = 0,0S u|= f, lim r = 0. R ?K ?3 S, 0uK ? Sk. 1, .1, dK v(r1,) = R r1 u R2 r1 , ! 7htqi200817 5n6SK%? v 4(r1,)v = 0,1S v|1= f1(r1,), f1(r1,) = R r1 f ? R2 r1, ? . 5? lim r10 r1 v = lim r R u(r,) = 0 r1= 0 v ?:, ?-#v 3r1= 0 ?, v 31SN. 3.11 ym.?NX3?u, ou ?O ? 1 r ? . y:dK, ?u(r,) 3.SN, 3?u, v(r1,) = R r1 u R2 r1 , ! 3k.1SN()?:), ?v 731Sk., =3A, |v| A. dv ?, ?r k ? ? ? ? ? r Ru(r1,) ? ? ? ? ? A|u(r1,)| c r . y?u 3?u?O ? 1 r ? . 3.12 y?v(2.11) ?YN. y:u Y, 3 S?v, ?:M0%, K, ?3 S, 3KS)Dirichlet K 4v = 0,KS v|= u|. dPoisson 3)v, v KS?N, , ? ?u v 3KS, du v 4?n, ?dv ? (u v)|= 0, dd?3KSu v 0, =u 3KSN. dM0?5= ?u 3 SN. 7htqi200818 5n6SK%? 4r4?n!1?K)?5 4.1 r4?n5y4?n. y:eu ?3S:M