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- 2006 1. .... ....- Cauchy .... - 2. 3. i 4. LAPLACE Laplace Laplace ( ) ii1 -1.1 , (...). . Laplace ( )2F(x) = 0 (1){ (, - , ).{ ( ).{ ( - )... Poisson2F(x) = q (2){ Poisson Laplace ( ..) 2F(x, t) = 1a2F(x, t))t (3){ .1{ (){ ( ).{ .{ Maxwell ( ) .k 2F(x, t) +k F(x, t) = cF(x, t)t (4) k , c . 2F(x, t) = 1c22F(x, t)t2 (5){ c ,{ ,( , ) c2= T . Helmholtz(2+ k2)F(x) = 0 (6) Helmholtz . ( ,, ..)2F(x, t) = 1c22F(x, t)t2 + G(x, t) (7)2 - , , . :LF(x, t) + kF(x, t) = G(x, t) (8) L (,).G(x, t) F(x, t) . (8) ( F(x, t) ) . G(x, t) = 0 .. 1.2 .... .... F(u(xi), xi, u(xi)xi, 2u(xi)xixj, ..., Nu(xi)xl11 xl22 ....xlnn) = q(xi) (9) xi ,u(xi) , , l1 + l2 + l3 + ...ln = N .. F u(xi) - .. . q(xi) = 0 . , . , 3 , ( -). ..(...) ... . .... (n) (m) - (n) . (m 1) . - , . -, , ( ...). .... Cauchy-Kovalevski , . , , , u(xi) - Taylor. (-Cauchy) . .... :A (x, y) 2F(x, y)x2 + 2 B(x, y) 2F(x, y)xy + C(x, y) 2F(x, y)y2 =G (x, y, F(x, y), F(x, y)x , F(x, y)y ) (10) A(x, y) , B(x, y) C(x, y) 4 x,y G(x, y, F(x, y), F(x,y)x , F(x,y)y ) F(x, y) - .1.3 .... .... .... = B(x, y)2 A(x, y)C(x, y) (x, y) :1. = B(x, y)2A(x, y) C(x, y) > 02. = B(x, y)2A(x, y) C(x, y) = 03. = B(x, y)2A(x, y) C(x, y) < 0 ( 10 ) (x, y), B(x, y)2 A(x, y) C(x, y) (x, y) , . Ticomi, yuxx + uyy = 0, y < 0 y > 0 . .... .5 (. 10 ) , Cauchy-Kovalevski, - Cauchy (x, y) . Cauchy Cauchy (. 10 ) -, s. u(s) -0.5 1 1.5 2 2.5 31.61.822.22.42.62.83tn sy 1: Chauchy N(s) . x = x(s), y = y(s), u(s) N(s)6N(s) = u. (11) = dy(s)ds ex + dx(s)ds ey (12)N(s) = (dy(s)ds ex + dx(s)ds ey).(ux ex + uy ey)= uxdyds |s +uydxds |s (13) t t = dx(s)ds ex + dy(s)ds ey (14)du(s)ds = uxdxds |s +uydyds |s (15) (13) (15) ( Cauchy) - u ,p(s) = ux |s q(s) = uy |s . 2ux2, 2uy2, 2uxy - ( 10 ) dp(s) = (pxdx + pydy) = (2ux2dx + 2uxydy) |s (16)dq(s) = (qxdx + qydy) = (2uy2dx + 2uxydy) |s (17)7 , (10 ,16 ,17 ),

A(x, y) 2B(x, y) C(x, y)dx dy 00 dx dy

s .A(x, y)(dydx)22B(x, y)dydx + C(x, y) |s= 0 (18) , - , - - , s. (. 18). Taylor su(x, y) = u(s) + (x x(s))ux |s +(y y(s))uy |s+(x x(s))2u2x2 |s +(y y(s))2u2y2 |s +(x x(s))(y y(s)) u2xy |s +...(19) - .. (x, y) .. - :A(x, y)(dydx)22B(x, y)dydx + C(x, y) = 0 (20)8 .... ( > 0) - , (x, y) =. (x, y) =. , , (x, t) = x + ct (x, t) = x ct. (=0) (x, y) = (x, y) =, - ( < 0) . . - ( ), , - - . Caushy . ( ) - , . : Cauchy : - , , - . Cauchy . Dirichlet : -. Neumann : .9 . . .. . Cauchy . . Dirichlet Neu-mann . .. Dirichlet Neu-mann . , , Laplace , , 1 ....1. 2F(x,t)x2 = 1c22F(x,t)t2:B(x, t) = 0, A(x, t) = 1 C(x, t) = 1c2 > 0. .. Cauchy Cauchy Dirichlet Neu-mann 22. Laplace1 2U- 102F(x,y)x2 + 2F(x,y)y2 = 0:B(x, y) = 0, A(x, y) = 1 C(x, y) = 1 < 0. Laplace .. - Dirichlet .3. 2F(x,t)x2 = 1c2F(x,t)t:B(x, t) = 0, A(x, t) = 1 C(x, t) = 0 = 0. .. - Dirichlet Neu-mann . ( 1 , 3 , 5) (x, t).1.4 .... .... , . , , 11 = (x, y) = (x, y). 2u = F(u, , , u, u) (21) . , , = (x, y) = x. 2u2 = F(u, , , u, u) (22) - = (x,y)+(x,y)2i = (x,y)(x,y)2i . 2u2 + 2u2 = F(u, , , u, u) (23) (Laplace) . , 12A(x, t) = 1 , B(x, t) = 0 C(x, t) = 1c2 (. 20) (dxdt)2 1c2 = 0 (24) (x, t) = x + ct =. (x, t) = x ct =.-3 -2 -1 1 2 3-2246xctxctxt 2: , (x, t) = x + ct (x, t) = x ct, 2u = 0 (25) (x, t) 1c 1c . .131.5 - - - . , . :1. , , 2. 3. , 4. ... , . : . Laplace,Fourier Green Fourier , . 14L . -, , - (.. ) . L - . , . (x1, x2, x3) x1 = c1 x2 = c2 x3 = c3 , c1, c2, c3, F(x1 = c1, x2, x3) = a a = 0 x1 = c1 nodal nodes F. x1 = c1 nodal F(x1, x2, x3) :F(x1, x2, x3) = F1(x1) F2(x1, x2) (26) nodal .. x2 = c2 :F(x1, x2, x3) = F1(x1)F2(x1)F3(x3) (27) F1(x1)F2(x1)F3(x3) . . . 15a = 0 F(x1, x2, x3) = Fp(x1, x2, x3) + F(x1, x2, x3) (28) Fp(x1, x2, x3) Fp(x1 = c1, x2, x3) = a F(x1, x2, x3) nodes . (. 28) .. . , , Poisson .. .. - ( , ).162 2.1 ().0.5 1 1.5 20.60.811.21.41.61.8TAABTBTAyTByw1w2x xdxdsy 3: A(x, y) B(x +dx, y

) - ds. TA( ) TB, y TAy = TAcosw1 = TAdyds T dydx, dx ds (29)TBy = TBsinw2 = TBdy

ds T dy

dx|x

17dy

= y + dy, dx|x = x + dx TAy + TBy = T(dy

dx|x dydx) = Tdxd2ydx2 = Tdx2y(x, t)x2 (30) dmd2ydt2 = dx2y(x, t)t2 = Tdx2yx2 (31)2y(x,t)x2 = 1c22y(x,t)t2 c2=

T , ( = dmdx ). , f(x, t), , 2y(x, t)x2 1c22y(x, t)t2 = 1c2f(x, t) (32) (2 1c22t2)u(x, t)1c2f(x, t) (33) . P, v . t0 , (

P,, v) . (2 1c22t2)(x, t) = 1c2

.

f(x, t) (34)18 f(x, t) . c2 c2= P |t0 .2.2 - u(x, t = 0) = a(x) u(x,t)t |t=0= b(x) . . - ( y < x < ). (. 5) u(x, t) . - Cauchy t = 0 (Cauchy) .., x ct =. - . 2u(x, t)x2 = 1c22u(x, t)t2 (35) . (, ) : = x + ct = x ct3. (. 35) :2u(, ) = 0 (36)3 (x, t) = = x +ct (x, t) = = x ct . , . .19 . (36) :u(, ) = f1() + f2() = f1(x + ct) + f2(x ct) (37) f1(), f2() . :u(x, t = 0) = a(x) u(x,t)t |t=0= b(x) x1 x x2. t = 0 :a(x) = f1(x) + f2(x) b(x) = ut|t=0 + ut|t=0 = cdf1d cdf2d df1(x)dx = 12[da(x)dx + b(x)c ] df2(x)dx = 12[da(x)dx b(x)c ] x1 x

x :f1(x) = 12[a(x) + 1c

xx1b(x

)dx

+ A1] (38)f2(x) = 12[a(x) 1c

xx1b(x

)dx

+ A2] (39)20A1, A2 . f1(x) + f2(x) = a(x) A1 + A2 = 0. (38) (39) x x1 x x2 ( ) (x, t) xct : x1 xct x2 :f1(x + ct) = 12[a(x + ct) + 1c

x+ctx1b(x

)dx

+ A1] (40)f2(x ct) = 12[a(x ct) 1c

xctx1b(x

)dx

+ A2] (41)u (x, t) = f1(x + ct) + f2(x ct) == 12[a(x + ct) + a(x ct) + 1c

x+ctx1b(x

)dx

1c

xctx1b(x

)dx

+ A1 + A2]= 12[a(x + ct) + a(x ct) + 1c

x+ctxctb(x

)dx

] (42) :u(x, t) = 12[a(x + ct) + a(x ct) + 1c

x+ctxct b(x

)dx

] (42) DAlembert - ( x, t) (x ct) ( ) (x1, x2). (x1, x2) . . x1 , x2 DAlembert (x, t). : f1(x+ct) 21 c, f2(x ct) . 1 - u(x, t = 0) = Exp[4x2]. t. DAlembert u(x, t) = 12(Exp[4(x ct)2] + Exp[4(x + ct)2]). - .. c = 4 . ( 4) -20 -10 10 200.51t1xct xct-20 -10 10 200.51t1xc xct-20 -10 10 200.51t0-20 -10 10 200.51t1xct xct 4: Exp[4x2] (t1 = 0 < t2 < t3 < t4) 22 . - . , . 2 u(x, t = 0) = f(x) =0 : < x < La(1 x2L2) : L < x < L0 : L < x < () . t > 0 u(x, t) u(x, t) = 12f(x + ct) + f(x ct)) (43) f(xct) f(x) x xct.f(xct) =0 : < xct < La(1 (xct)2L2 ) : L < xct < L0 : L < xct < (.43) f(x) xct (x, t). 0 < t < Lc ( 5)u(x, t) =0 : < x < L cta2(1 (x+ct)2L2 ) : L ct < x < L + cta(1 (x2+c2t2)L2 ) : L + ct < x < L cta2(1 (xct)2L2 ) : L ct < x < L + ct0 : L + ct < x < V I ( < x < xo), III (xo < x < ) 23-4 -2 2 4123456I II IIIIV VIVxctxctx L Lttoxo xo 5: t < to =. t > Lc 3L.2.3 0 x < , . u(x, t = 0) = f(x) , ut |t=0 = g(x) 0 x < Cauchy x 0. DAlembert (x, t) x ct > 0 ( 24 ). , - Dirichlet Neumann t .. u(x, t = 0) = 0, (x, t) ( ) 4. 5 -. DAlembert . f(x) g(x) u(x, t) x.f(x) = f(x) , g(x) = g(x)u(x, t) = 12[f(x + ct) + f(x ct) + 1cx+ct

xctg(x

)dx

] (44) x

yu(x, t) = 12[f(x ct) + f(x + ct) 1cxct

x+ctg(y)dy]= 12[f(x ct) + f(x + ct) = 1cx+ct

xctg(y)dy]= u(x, t) (45) f(x) g(x) u(x, t) u(x, t) =u(x, t) . u(x = 0, t) = 0 u(x, t) x < 0 f(x) g(x) x < 04 U Cauchy Dirichlet5 x = 025-4 -2 2 4-4-224 xctxctxI IIIIIIV 6: x, t x ct = 0 (x, t) x ct ( ) . x ct > 0u(x, t) = 12[f(x + ct) + f(x ct) + 1cx+ct

xctg(x

)dx

] (46) x ct < 0u(x, t) = 12[f(x ct) f(x + ct) 1cx+ct

xctg(x

)dx

] (47) x ct < 0 x + ct > 0u(x, t) = 12[f(x+ct) f(x+ct) 1c0

xctg(x

)dx

+x+ct

0g(x

)dx

](48)26 IV x ct > 0 x + ct < 0u(x, t) = 12(f(xct) +f(xct) + 1c0

xctg(x

)dx

x+ct

0g(x

)dx

)(49) (0 < x < ) - u(x = 0, t) = (t) t > 0 . f(x) g(x) , u(x, t) . u(x, t) = U(x, t) + (x, t) U(x, t),(x, t) U(x, t = 0) = f(x), Ut(x, t) |t=0= g(x) ,U(x = 0, t) = 0(x, t = 0) = 0, t(x, t) |t=0= 0 ,(x = 0, t) = (t) U(x, t) 1 (x, t) ( DAlembert (x, t) = f1(x + ct) + f2(x ct) (50) f1(x + ct) f2(x ct) x > 0 t > 0) f1(x) + f2(x) = 0) cf

1(x) cf

2(x) = 0) f1(ct) + f2(ct) = (t) (t) = 0 t < 027 () () f1(x) = f2(x) = C = x > 0. () f1(z) + f2(z) = (z/c) z zf1(z) + f2(z) = (z/c) = 0 f1(z) = f2(z) = f1(x) f2(x) f2(z) = (z/c), z > 0 z = x + ct > 0 (x, t) =

f2(x ct) = (t xc) : x < ct0 : ct < x (x ) .2.4 . , . Fourier. - 28, , .) L . - u(x, t) :2u(x, t)x2 = 1c22u(x, t)t2 f(x, t) (51) f(x, t) = 0 , f(x, t) = 0 f(x, t). , Cauchy , -u(x, t = 0) = f(x) u(x,t)t |t=0= g(x) Cauchy 0xL. - (xct) (0, L). Dirichlet x = 0 x = L , u(x = 0, t) = u(x =L, t) = 0. (. 51) f(x, t) = 0, x=0 x=L nodes u(x, t) -. 6u(x, t) = X(x).T(t). 6 29 (. 51) :T(t)d2X(x)dx2 = X(x)c2d2T(t)dt2 (52) X(x)T(t)= 01X(x)d2X(x)dx2 = 1c2T(t)d2T(t)dt2 (53) (. 53) x t , 1X(x)d2X(x)dx2 = (54)1c2T(t)d2T(t)dt2 = (55) . d2X(x)dx2 =X(x):1. = k2> 0X(x) = c1ekx+ c2ekx(56) X(x = 0) = 0 X(x = L) = 0 c1 + c2 = 0 c1ekL+ c2ekL= 0. c1 = c2 = 0, .302. = k2= 0X(x) = c1x + c2 (57)X(0) = 0 X(L) = 0 c1 = c2 = 03. = k2< 0X(x) = c1 sin kx + c2 cos kx (58)X(0) = 0=c2 = 0 (59)X(L) = 0 :c1 sin(kL) = 0 (60) c1 = 0 , kL =n k = nL7. (. 2.4) = k2:Tn(t) = ansin ncL t + bncos ncL t (61) Sturm-Liouvilleu(x, t) =n=0sin(nL x) (an sin(ncL t) + bn cos(ncL t)) (62)( ). an, bn - , -. t = 0,7. L = d2dx2 X(0) = X(L) = 0 , k2= n22 1L2 sin kx . Sturm-Liouville31u(x, t = 0) = f(x) u(x,t)t |t=0= g(x)u(x, t = 0) =n=0sin(nL x) bn = f(x) (63)bn Fourier f(x) sin(nL x) :bn = 2LL

0f(x)sin(nL x)dx (64) :u(x, t)t |t=0= g(x) =n=0ncL sin(nL x) an (65) an Fourier g(x) :ncL an = 2LL

0g(x) sin(nL x)dx (66)u(x, t) =n=0an sin(nL x) sin(ncL t) +n=0bn sin(nL x) cos(ncL t) ==n=01n(x, t) +n=02n(x, t) (67) 1n(x, t) 2n(x, t) - . - Tn = 2Lnc sin(nL x). n = 2Ln sin(nL ct) cos(nL ct). 1 = c2L 32 , , , . 1n(x, t) 2n(x, t) . (. 4) - , t = 0 , t = 3 . - Fourier, . bn = 0 ,0.5 1 1.5 2 2.5 3-1-0.50.51Sin3x0.5 1 1.5 2 2.5 3-1-0.50.51Sin4x0.5 1 1.5 2 2.5 30.20.40.60.81Sinx0.5 1 1.5 2 2.5 3-1-0.50.51Sin2x 7: ( - ) an = 0. - . (. 67) . 33 (.67) u(x, t) =n=0an2 [ cos(nL (x ct)) cos(nL (x + ct))] +n=0bn2 [ sin(nL (x ct)) + sin(nL (x + ct))]= G(x ct) + F(x + ct) (68)G(x ct) =n=0(an2 cos(nL (x ct)) + bn2 sin(nL (x ct))) (69)F(x + ct) =n=0(an2 cos(nL (x + ct)) + bn2 sin(nL (x + ct))) (70) DAlembert. G(x ct) F(x + ct) t = 0 f(x) = G(x) + F(x) (71)g(x) = c(F

(x) G

(x)) (72) f(x), g(x) . DAlembert.u(x, t) = G(x ct) + F(x + ct) == 12[f(x + ct) + f(x ct) + 1c

x+ctxctg(x

)dx

] (73) u(x, t) , (. 68 ) , 34f(x) g(x) Fou-rier 0xL. L . - . ()u(x, t = 0) = f(x) =

axh : 0 x ha(Lx)(Lh) : h x L . .xux,t0L L2 ( 62) . , Fourier an = 0 , bn bn = 2LL

0f(x)sin(nL x)dx= 2aL2h(L h)2n2 sin(nL h) (74)35n |bn|2|bn|21 1 12 0.125 03 0.0234 0.014 0.0 05 0.0016 0.001176 0.0015 07 0.00041 0.00038 0 09 0.00015 0.000110 0.0002 0 1: h = L2 ( 2) h = L2 ( 3). ( 62) u(x, t) = 2aL2h(L h)2n=11n2 sin(nL h)sin(nL x) cos(ncL t) (75) :|bn|2= | 2aL2h(Lh)21n2 sin(nL h)|2 , 1n2, ( ) h = L2, . , .. , , , = cL . - , -. 10 .36 , .) L . - . 2u(x, t)x2 = 1c22u(x, t)t2 (76)u(x, t = 0) = f(x) ,u(x,t)t |t=0= g(x) .u(x,t)x |x=0= 0 .u(x,t)x |x=L= 0 . u(x, t) = (Asin(kx) + Bcos(kz))(C sin(kct) + Dcos(kct)) (77) x = 0 A = 0, x = L k = nL n = 0, 1, 2, 3, ... u(x, t) =n=0an cos(nL x) sin(ncL t) +n=0bn cos(nL x) cos(ncL t) (78) an bn .37u(x, t = 0) =n=0cos(nL x) bn = f(x) (79)bn Fourier f(x) cos(nL x) :bn = 2LL

0f(x)cos(nL x)dx (80) :u(x, t)t |t=0= g(x) =n=0ncL cos(nL x) an (81) an Fourier g(x) :ncL an = 2LL

0g(x) cos(nL x)dx (82) :g(x) = Asin(5nL x)sin(3nL x) (83) . - . bn = 0. an Fourier 8.ncL an = 2ALL

0sin(5nL x)sin(3nL x)cos(nL x)dx8 - , cos(nL x) - (sin(a) sin(b0 = 12(cos(a b) cos(a + b))38= 2ALL

0(12(cos(2L x) cos(8L x)) cos(nL x)dx= A2 (2nn8) (84) a22 = (AL4c)2 2 = 2cL a28 = (AL16c)2 8 = 8cL .2.5 , T , :E = 12L

0(|u(x, t)t |2+ T|u(x, t)x |2)dx (85)u(x, t) . (.85) . :u(x, t) =n=0an sin(nL x) sin(nt) +n=0bn sin(nL x) cos(nt) (86) n = nL

T . (. 86) (. 85) sin(nL x) [0, L] :E = 2T4Ln=0n2(|an|2+|bn|2) (87) . dEdt = 0. , (. ), .392.6 - f(x, t). u(x, t = 0) = a(x) u(x,t)t |t=0 = b(x) . :2u(x, t)x2 1c22u(x, t))t2 = f(x, t) (88)f(x, t) (x, t). (. 88) .. . :u(x, t) =n=0Tn(t) sin(nL x) (89) - Sturm-Liouvill - . Fourier f(x, t) .f(x, t) =n=0fn(t) sin(nL x) (90) fn(t) Fourier f(x, t). Fourier u(x, t) f(x, t) - . (.88) Tn(t).T

n(t) + c2n22L2 T(t) = c2fn(t) (91)fn(t) = 2LL

0f(x, t) sin(nL x)dx (92)40 (91) - Tn(t) = A1T1n(t) + A2T2n(t) + Tp(t) (93) T1n(t) , T2n(t) Tp(t) .T1n(t) = sin(ncL t) (94)T2n(t) = cos(ncL t) (95)Tp(t) = T2n(t)t

0c2fn(t

)T1n(t

)W[T1n(t

), T2n(t

)]dt

T1n(t)t

0c2fn(t

)T2n(t

)W[T1n(t

), T2n(t

)]dt

(96) W[T1n(t

), T2n(t

)] Wronsky T1n(t) , T2n(t). T1n, T2n Tp(t) (. 93) (. 88).u(x, t) =0[An sin(ncL t) + Bn cos(ncL t)+ Lcnt

0fn(t

) sin(ncL (t t

))dt

] sin(nL x) (97) An, Bn -, .An = 2cnL

0b(x) sin(nL x)dx (98)Bn = 2LL

0a(x) cos(nL x)dx (99)41An, Bn Fourier.u(x, t) =n=0(An sin(ncL t) + Bn cos(ncL t)) sin(nL x) ++n=1Lcnt

0fn(t

) sin(ncL (t t

))dt

sin(nL x) (100) . , .u(x, t) =n=1Lcnt

0fn(t

) sin(ncL (t t

))dt

sin(nL x) (101) (. 101) u(x, t = 0) = 0 Leibnitz - u(x,t)t |t=0= 0 - ( ). . ( ) (. 85) (. 100). (. 100) - .42 : L . ( f(x)). . u(x, t) 2u(x,t)x2 1c22u(x,t)t2 = gc2 g . Fourier g.gc2 =n=0fn sin(nL x) (102)fn = 2LL

0gc2 sin(nL x)dx = 2g(1 cos n)nc2 (103) u(x, t) = U(x, t) + u(x, t) U(x, t) . (. 101) u(x, t) =1Lcnt

0fn sin(ncL (t t

)) dt

sin(nL x)=n=12gL2c2n33 (1 cos(n)) (1 cos(nctL )) sin(nL x)=n=12gL2c2n33 (1 + (1)n)) cos(nctL )) sin(nL x)+n=12gL2c2n33 (1 + (1)n)) sin(nL x)43=n=12gL2c2n33 (1 + (1)n)) cos(nctL )) sin(nL x)+ gx2c2(L x) (104) (. 104) Fouriergx2c2(L x) =n=12gL2c2n33 (1 + (1)n)) sin(nL x) (105) (. 104) , u(x, t) = u1(x) + u2(x, t) ( . 100) u(x, t) =0(An sin(ncL t)) + Bn cos(ncL t)) sin(nL x) ++n=12gL2c2n33(1 + (1)n)) cos(nctL )) sin(nL x)+ gx2 (L x) (106) Fourier An Bn - . u(x, t) = u1(x) +u2(x, t) , - u(x, t) = u1(x) + U(x, t) , U(x, t) u1(x) . f(x) u1(x) , . , . (.5) ( ) 440.20.40.60.8 1 x-1.2-1-0.8-0.6-0.4-0.2ux00.20.40.60.81x 0246810t-2-1.5-1-0.50ux,t 0.20.40.60 8 x 8: .92.7 L . u(x, y, t) - . .. ( -) .2u(x, y, t)x2 + 2u(x, y, t)y2 1c22u(x, y, t)t2 = 0 (107) u(x = 0, y, t) = u(x = L, y, t) = u(x, y = 0, t) = u(x, y = L, t) = 0 u(x, y, t = 0) = f(x, y) ut(x, y, t) |t=0= g(x, y) Cauchy Cauchy 0 < x < L 0 < y < L (x, y).9. u(x, t) = us(x) + U(x, t)45 u(x, y, t) = X(x)Y (y)T(t) (. 107) 1X(x)d2X(x)dx2 + 1Y (y)d2Y (y)dy2 = 1T(t)1c2d2T(t)dt2 = 0 (108) (. 108) t, (x, y). ..1X(x)d2X(x)dx2 = 1 (109)1Y (y)d2Y (y)dy2 = 2 (110)1T(t)1c2d2T(t)dt2 = (111) 1 + 2 = . X(0) = X(L) = 0 Y (0) = Y (L) = 0 Sturm-Liouville (x, y) . . Sturm-Liouville ,Xn(x) = sin(nxL ) Ym(y) = sin(myL ) n, m = 0, 1, 2, 3... Xn(x) Ym(y)u(x, y, t)) =n,m=0sin(nL x) sin(mL y) (anm sin(nmt) + bnm cos(nmt))=n,m=0unm(x, y, t) (112)46 nm = cLn2+ m2 . 11 - ,12 - 21. , - ( ) , u12(x, y, t)= u21(x, y, t), (), . (9) - . u11(x, y, t = 0) ,u12(x, y, t = 0) , u13(x, y, t = 0) , u32(x, y, t = 0) , - u11(x, y, t = 1), u12(x, y, t = 1) ,u13(x, y, t = 1) ,u32(x, y, t = 1). nodal lines - . - . u12(x, y, t = 0) - x , u21(x, y, t = 0) y. - . - ,.. u12(x, y, t) u21(x, y, t) = 0 x = y. - (). (. 112) - anm ,bnm Fourier .u(x, y, t = 0) = f(x, y) =n,m=0bnm sin(nL x) sin(mL y) (113)bnm = 4L2L

0sin(nL x) dxL

0f(x, y) sin(mL y) dy (114)47ut(x, y, t) |t=0= g(x, y) =n,m=0anm nm sin(nL x) sin(mL y) (115)anm = 4L21nmL

0sin(nL x) dxL

0g(x, y) sin(mL y) dy (116) 9: (. 112) . Fou-rier .482.8 R . f(, ) - g(, ) (, ) -, u(, , t), . u(, , t) ( ) :2u(, , t) = 1c22u(, , t))t2 (117) 10 112= 1 + 1222 (118) (1 + 1222 1c22t2)u(, , t) = 0 (119) u(, , t) = X(, )T(t). u(, , t) = X(, )T(t) (. 117 ) X(, ) (2+ k2)X(, ) = 0 (120) Helmholtz :(1 + 1222 + k2)X(, ) = 0 (121)10 :2= 1 + 1222 + 2z2. z = 0 .11 . - .49 T(t) ,T

(t) + k2c2T(t) = 0 (122) k2 12. (. 122) :T(t) = C1 sin(kct) + C2 cos(kct) (123) (. 121) (, ). = R =. - ( - ) nodal . 2 X(, ) = R()(). (. 121) 2R()d2R()d2 + R()dR()d + 1()d2()d2 + k22= 0 (124) (. 124) , . 2 ..d2R()d2 + 1dR()d + (k2 22)R() = 0 (125)d2()d2 + 2() = 0 (126) - 2, (. 126) 2.12 k2> 0 k2< 0 . k2> 050() = ( + 2),(0) = (2)

() =

( + 2) . ( ) 2= m2 m = 0, 1, 2, 3, 4... (. 126) :m() = A1 sin(m) + B1 cos(m) (127) m() Fourier sin(m) cos(m) [0,2]. (. 125)

=k, (k = 0), :d2R(

)d2 + 1

dR(

)d

+ (1 m22 )R(

) = 0 (128) (. 128) Bessel m Bessel Jm(

) Neumann Nm(

) . Neumann = 0.5 10 15 20x-0.20.20.40.6J5 10 15 20x-1-0.75-0.5-0.250.25Y 10: BesselJm(x) Neumann Ym(x) ,

( - ). (128) AJm(

) + BNm(

). 51, B = 0, R1 <

< R2 A, B = 0. Bessel - . (128) -, R(

)|

=R = 0 R(

)

|

=R = 0 , R , Sturm-Liouville 13 - Jm(kmnR ). kmn Jm(kR) = 0, kR = kmn( kmn n Bessel, Jm(kR)). - Bessel Jm(kmnR ), , [0, R] . X(, ) umnumn(, , t) = J m(kmnR )[Amn cos(m) + Bmn sin(m)][ A

cos(kmnctR ) + B

sin(kmnctR )] (129) mn = kmncR u (, , t) =m,nJm(kmnR )[Amn cos(m) + Bmn sin(m)] cos(kmnctR )+mnJm(kmnR )[A

mn cos(m) + B

mn sin(m)] sin(kmnctR ) (130) Amn , Bmn , A

mn , B

mn .u(, , t = 0) = f(, ) ==m,nJm(kmnR )[Amn cos(m) + Bmn sin(m)] (131)ut|t=0 = g(, ) =13 Sturm-Liouville - = 0 . Bessel Legendre Sturm-Liouville -52=m,nJm(kmnR )[A

mn cos(m) + B

mn sin(m)]kmnR (132) (. 131) (. 132) f(, ) g(, ) (, ) Fourier-Bessel (sin(m), Jm(kmnR )), (cos(m), Jm(kmnR )). Fourier-Bessel Bessel.Amn = 2R2(Jm+1(kmn))22

0dR

0f(, )Jm(kmnR ) sin(m)d (133)Bmn = 2R2(Jm+1(kmn))22

0dR

0f(, )Jm(kmnR ) cos(m)d (134)kmncR A

mn = 2R2(Jm+1(kmn))22

0dR

0g(, )Jm(kmnR ) sin(m)d(135)kmncR B

mn = 2R2(Jm+1(kmn))22

0dR

0g(, )Jm(kmnR ) cos(m)d(136) Fourier-Bessel - Bessel(). R = 1cm . - m = 0 . (nodal lines) Bessel. m = 0 53-1-0.500.51-1-0.500.5100.511-0.500.51-0.500.5-1-0.500.51-1-0.500.51-0.5-0.2500.250.510.500.5-1-0.500.51-1-0.500.5100.250.50.7511-0.500.5 -1-0.500.51-1-0.500.5100.511-0.500.51-0.500.5 11: n = 1 . . - u(, , t = 0) = f() . u(, , t = 0) = f() = 2(R ) t = 0 54 (. 130). A

mn B

mn . , , - , 14, m = 0. - u(, t) =nJ0(k0nR )A0n cos(k0nctR ) (137)J0(k0nR ) Bessel k0n J0(k0nR ). Fourier-Bessel A0n A0n = 1R2(J1(k0n))2R

0f()J0(k0nR )d (138) A0n . - Bessel k01 = 2.4 , k02 = 5.22 , k03 = 8.65 .... R = c = 1 A01 = 0.68 , A0,1 =0.06 , A03 = 0.05 .....2.9 ( ) (. 5) :[ 1r2rr2 r + 1r2sin (sin ) + 1r2sin22214 2

0sin(m)d = 0m55 1c22t2]u(r, , , t) = 0 (139) -. - (r, t). (139) :[ 1r2rr2 r 1c22t2]u(r, t) = 0 (140) u(r, t) = R(r,t)r (.140) R(r, t) :2R(r, t)r2 1c22R(r, t)t2 = 0 (141) (. 141) 15R(r, t) = F1(r +ct) +F2(r ct) F1(r + ct) F2(r ct) , :u(r, t) = F1(r+ct)r + F2(r+ct)r (. 140) . r =. a - . :u(r, , , t) = R(r)()()T(t) (142) (. 139). - () () 15 56 (k, , ) . - k2> 0 . .. :d2T(t)dt2 = k2c2T(t) (143)d2()d2 = () (144)( 1sin()ddsin() dd sin2())() = () (145)d2R(r)dr2 + 2rdR(r)dr + (k2 r2)R(r) = 0 (146) -. (. 143) k = kc:T(t) = Asinkct + Bcoskct (. 144) = m2 (m = 0, 1, 2, 3, ...), () 2.() = A

sin(m) + B

cos(m) (. 145) , x = cos d()d = sin d(x)dx , 1 x 1, :(1 x2)d2(x)dx2 2xd(x)dx + ( m21 x2)(x) = 0 (147)57 (. 147) m = 0 Legendre. Legendre -. = l(l + 1), l = 0, 1, 2, 3, ..., Legen-dre16Pl(x) Ql(x), Legendre. x = 1 Ql(x) - . Ql(x) . . m = 0 (. 147) - Legendre Pml (x) . 17Legendre m - |m| l. (. 147) :(cos ) = A

Pml (cos ) (. 146) R(r) = u(r)r - Besseld2u(r)dr2 + 1rdu(r)dr + (k2 (l+12)2r2 )u(r) = 0 Bessel (l + 12) , Jl+12(kr) Nl+12(kr).R(r) = AJl+12(kr)r + BNl+12(kr)r 16Pl(x) = 12nn!dndxn(x21)l17Pml (x) = (x21)m2 dmdxmPl(x)58R(r) = Ajl(kr) + Bnl(kr) jl(kr) nl(kr) Bes-sel Neumann . Neumann - r0 B . l, m ,|m| l:u (r, , , t) = R(r)()()T(t) ==m,l(Asinkct + Bcoskct)(A

sin(m) + B

cos(m))Pml (cos )(A

jl(kr) + B

nl(kr)) ==m,l(A

jl(kr) + B

nl(kr))(Asinkct + Bcoskct)Yml (, ) (148) Yml (, ) Y ml (, ) =

2l+14(lm)!(l+m)!Pml ()eim, |m| l . :2

0d

0sin Yml (, )Ym

l (, )d = ll mm

Fourier Y ml (, ).59f(, ) =l=0lm=lflmYml (, )flm Fourierflm =2

0d

0sin Y ml (, )f(, )d (. 148) - .: a , , - u0 . t = 0 . . (z = 0) t < 0. , (u = ) . . - . - . . - .1. :60(r, , t = 0) = u0rcos 2. (r,,t)t |t=0 = 0 3. - .(r,,,t)r |r=a = 0 . (. 148). B

= 0 = 0 m = 0.(r, , t) =l(Asinkct + Bcoskct)Pl(cos )jl(kr) (149)(r, , , t)r |r=a = 0 ==l(Asinkct + Bcoskct)Pl(cos )dJl+12(kr)rdr |r=a (150) (2) A = 0.(r, , t) =l(Bcoskct)Pl(cos )Jl+12(kr)r (151) (3) J

l+12(ka)a 12a32Jl+12(ka) = 0 (152)(r, , t) =l=0n=1BcoskctPlcos Jl+12(klnra )r (153)61kln = ka (152). (r, , t = 0) =l=0n=1BPl(cos )Jl+12(klnra )r = u0rcos u0r32cos =l=0n=1BlnPl(cos )Jl+12(klnra ) (154) (. 154) Bln Fourier u0r32cos Legendre-Bessel. Legendre,(P0(cos ) = 1, P1(cos ) = cos()) l = 1, .u0r32=n=1BnJ32(k1nra ) (155)Bn =2u0r=a

r=0r52J32(k1nra )dra2J32(k1n)(1 2k21n) (156)(r, , t) =n=1Bncos J32(k1nra )r cos(k1ncta ) (157) u(r, , ) = (r,,)r2.10 1. .622. . t = 0 x = 0 ut(x, t = 0) = a(x) a (x) Dirac. u(x, t) t > 0.3. (xct) . t = 0 .4. (x = 0) . u(x, t = 0) = f(x) =0 : 0 < x < LhL(x L)(4 x+LL ) : L < x < 3L0 : 3L < x < , (x, t) t = Lc, t = 3Lc , t = 5Lc , t = 7Lc .5. (x = 0), - . - x0 t = 0. t > 0.(. ut (x, t = 0) = A (x x0))6. x > 0, , t > 0 u(x, t = 0) = f(x) , u(x,t)t |t=0 = g(x)63u(x,t)x |x=0= 0 t = 0 f(x) g(x) x = 0.7. f(x, t) = 4(x ct)2(x + ct) .8. L > 2 , -. u(x, t = 0) = f(x) =

1 : L2 1 < x < 1 + L20 : 0 < x < L2 1, L2 + 1 < x < L .9. 3a - u(x, t = 0) = f(x) =exa : 0 < x < ae3a2xa : a < x < 2aex3aa : 2a < x < 3a .10. ( 2x2 1c22t2)u(x, t) = xsin(x) (158)64 u(x, t = 0) = 0 ut(x, t) |t=0=0. ;11. L g(x) = I(x L2).(. .. ( 2x2 1c22t2 hc2t)u(x, t) = 0)12. L1 L2 - ut(x, t) |t=0= 1. .13. L u(x, y, t = 0) = sin2( x/L) sin( y/L). .14. L - - u(x, y, t = 0) = f(x, y) . - .15. - - .16. - L . . (. -65 - ).17. ) R = 1 u(, , t = 0) = g J0 () ut(, , t = 0) |t=0) = 0 , J0 () Bessel .18. a - u(, , t = 0) = J3 (k31a ) cos(3) ut(, , t = 0) |t=0) = 0 , k31 Bessel . ) .19. R g J1 (k01R ) . (. - Bessel 0 < < R)20. a < b . - f()21. a - . u(, , t = 0) = AJ0(k01a) + g4c2(2 a2) .663 3.1 V S. dV dQ = cdV , c (x, t) .VndS 12: Q S t Q1 = K(x)(S)(t)(x, t). (159) . . K(x) - S. - q = Q(t)(S) (160)67 (t1, t2) Q1 = t2

t1dt

SK(x)

.d

S= t2

t1dt

V.(K(x))dV (161) dV 1(x, t1) 2(x, t2), Q2 = c(dV)(21) Q2 =

Vc(2(x, t2) 1(x, t1))dV= t2

t1dt

Vt cdV= t2

t1dt

Vt cdV (162) - Q3 =t2

t1dt

VF(x, t)dV (163)F(x, t) . Q3 = Q1 + Q2 (164) (164) (162) (163) 68(x, t)t = 1c[K(x).(x, t) + K(x)2(x, t) + F(x, t) (165) - , (x, t)t = 1cK2(x, t) (166) (166) - (.. ), .3.2 - :2T(x, t) = 1a2T(x, t)t (167) (167) Dirichlet ( - - ) Neumann( . - ) Robin( ) -18 T0(x, t = 0) = f(x). t > 019.18 19 - . , 69 (167) T(x, t) = X(x)T(t) . 1X(x)d2X(x)dx2 = 1a2 T(t)dT(t)dt (168) T(t) dT(t)dt = k2a2 T(t) (169) k2 . (169) T(t) = ek2a2t. k2> 0 T(t ) 0 X(x) = Aeikx+ Beikx. k Fourier .T(x, t) =

A(k)eikxk2a2tdk (170)A(k) Fourier 20T(x, t = 0) = f(x)=

A(k)eikxdk (171)A(k) = 12

f(x

)eikx

dx

(172) t > 0 , t > 0. . .20

ei(kk

)xdx = 2(k k

) (k k

) Dirac.70T(x, t) = 12

dk

f(x

)eik(xx

)k2a2tdx

= 12

a2tf(x

)e(xx

)24a2t dx

==

f(x

)T0(x, t)dx

(173) T0(x x

, t) . (.173) . - t 0 limt0T(x, t) =

limt0

14a2tf(x

)e(xx

)24a2t dx

= f(x) (174) Dirac210.5 1 1.5 20.250.50.7511.251.51.75t1t2t3t4 13: t1 < t2 < t3 < t4 . Q 21limt0

14a2te(xx

)24a2t = (x x

)71 x0 , . 2 Q = 2cT0, T0 . Q t :T(x, t) =

T0e(xx

)24a2t2atdx

= Q2c2atx0+

x0e(xx

)24a2t dx

(175) 0 - x0 t = 0 .lim012x0+

x0e(xx

)24a2t dx

= e(x0x)24a2t (176)lim0T(x, t) T0(x, t) (177) . t = 0 x0 Q. -

T(x, t)dx = Qpc

T0(x, t)dx = Qpc (178) . 2222

e(x0x)24a2t dx = 2at723.3 Dirichlet L T0(x) . t - . T(x, t) (167), - T(x = 0, t) = T(x = L, t) = 0 T(x, t = 0) = T0(x). (167) T(x, t) = X(x) T(t) . T(t) T(t) = ek2a2t,k2 , - t > . X(x) d2X(x)dx2 + k2X(x) = 0 (179) X(0) = X(L) = 0 ( ) - Sturm-Liouville sin(nL x) kn = nL . (. 167) - ( )T(x, t) =nAn sin(nL x)ek2na2t(180) An Fourier .T(x, t = 0) = T0(x) =nAn sin(nL x)73An = 2LL

0T0(x) sin(nL x)dx (181) 100 Fourier (. 181) :An =

400n : n = 2k + 10 : n = 2kT(x, t) =k=0400(2k + 1) sin((2k + 1)L x)e((2k+1)aL )2t(182) t e((2k+1)aL )2t -.T(x, t) 400 sin(Lx)e(aL )2t(183) t (182 ) 23, t T(x, t) 0 ( ).. 1e - . 23 .74 - :1 = limt1tln|T(x, t)| (184) .: - . (. 180) :T(x, t) = A1 sin(Lx)e(aL )2t+n=2An sin(nL x)e(naL )2t= A1sin(Lx)e(aL )2t[1 + O(e(3aL )2t)]ln|T(x, t)| = lnA1 + lnsin(xL ) + lne(aL )2t+ ln[1 + O(e(3aL )2t)]1 = limt1tln|T(x, t)| (a)2L2 (185) (.185) - , ( t ) , ( ) . - . Neumann75 L T0(x) - . , - t . :T(x, t) = X(x)T(t) T(t) = ek2a2t( k2> 0 ) X(x) = Asin(kx) + Bcos(kx). . :T(x,t)x |x=0 = T(x,t)x |x=L = 0 A = 0 k = nL . Sturm-Liouville cos(nL x) kn = nL . (167) - T(x, t) =n=0Bn cos(nL x)ek2na2t(186) Bn Fourier .T(x, t = 0) = T0(x) =n=0Bn cos(nL x)76Bn = 2LL

0T0(x) cos(nL x)dx (187) n = 0 B0 = 1LL

0T0(x)dx (188) :T(x, t) = B0 +n=1Bn cos(nL x)ek2na2t(189) - :limtT(x, t) = B0 = 1LL

0T0(x)dx (190) ( . 190) , - , , . Robin ( ) L T(x, t = 0) = T0(x). ( ) - . .77 T(x,t)x hT(x, t)|x=0 = 0T(x,t)x + hT(x, t)|x=L = 0 - - . T(x, t) = X(x)ek2a2t X(x) d2X(x)dx2 + k2dX(x)dt = 0 (191)dX(x)dx hX(x)|x=0 = 0dX(x)dx +hX(x)|x=L = 0 (192) (. 191) , X(x) = Asin(kx) + Bcos(kx). A = hBk cot = (2h2L2)2hL (193) = Lk . (. 193) n Sturm-Liouville Xn(x) = Bn[hLnsin(nxL ) + cos(nxL )] (194) Xn(x) (0, L). 7824Xn(x)X(x) =n=1Bn(hLnsin(nxL ) + cos(nxL ))e2a2tL2(195)2 4 6 8 10 12-30-20-10102030mCotm 14: (. 193)3.4 L . : T(x = 0, t) = T1(t) , T(x =L, t) = T2(t) T(x, t = 0) =T0(x) . t > 0. ( ) . 24 Xn(x) (0, L) .79( ). ( , - .) T(x, t) = U(x, t) + U(x, t) U(x, t) U(x, t) . .T(x = 0, t) = T1(t) T(x = L, t) = T2(t) T(x, t) = T0(x) :T(x, t) = U(x, t) + U(x, t) (196) U(x, t) - :U(x = 0, t) = T1(t) U(x = L, t) = T2(t) U(x, t) :U(x = 0, t) = 0 U(x = L, t) = 0 U(x, t = 0) = T0(x) U(x, t = 0) = (x).80 U(x, t) :U(x, t) = L xL T1(t) + xLT2(t) (197) U(x, t)2U(x, t)x2 1a2U(x, t)t = 1a2xLT2(t) + L xa2LT1(t) F(x, t) (198) (.198) .. - U(x, t) . (. 198) - F(x, t) . (. 198) (.88). sin(nL x) - . Fourier F(x, t)U(x, t) =nAn(t) sin(nL x) (199)F(x, t) =nbn(t) sin(nL x) (200)bn(t) = 2LL

0F(x, t) sin(nL x)dx (201) (198). Fourier, An(t) :n=1[n22L2 An(t) 1a2An(t) bn(t)] sin(nL x) = 0 (202)81 (202) .. An(t) = Cne(n2a22L2 t)+ e(n2a22L2 t)t

0bn(t

)e(n2a22L2 t

)dt

(203)U(x, t) =n[Cne(n2a22L2 t)+ e(n2a22L2 t)t

0bn(t

)e(n2a22L2 t)dt] sin(nL x) ==nCne(n2a22L2 t)sin(nL x)+ne(n2a22L2 t)t

0bn(t)e(n2a22L2 t

)dt

sin(nL x)= U(x, t) + U(x, t) (204) Cn - Cn = 2LL

0(x) sin(nL x)dxT(x, t) = L xL T1(t) + xLT2(t)+nCne(n2a22L2 t)sin(nL x)+ne(n2a22L2 t)t

0bn(t)e(n2a22L2 t

)dt

sin(nL x)(205) 1 -, T(x = 0, t) = C1 T(x = L, t) = C2 T(x, t = 0) = g(x).82 (. 205) T1(t) = C1 , T2(t) = C2 . T(x, t) = L xL C1 + xLC2+nCne(n2a22L2 t)sin(nL x) (206) Fourier , Cn Cn = 2LL

0(g(x) L xL C1 xLC2) sin(nL x)dx , . ( ) . , , T(x) :d2T(x)dx2 = 0 (207) . (207) T(x) = a1x + a2 (208) a1, a2 - T(x = 0) = C1 T(x = L) = C2.a2 = C1a1 = C2C1L83T(x) = C2C1L x + C1 (209) :T(x, t) =nCnen2a2L2 tsin(nL x) + C2C1L x + C1 (210)T(x, t = 0) =nCn sin(nL x) + C2C1L x + C1 = g(x)g(x) C2C1L x C1 =nCn sin(nL x) Fourier Cn :Cn = 2LL

0(g(x) C2C1L x C1) sin(nL x)dx (211) ( 210) :limtT(x, t) = limtT(x) + limtTo(x, t) = T(x) (212) t limtTo(x, t)0 :limt[T(x, t) T(x)] = 0 (213) T(x) . ( ) , . T(x, t) - , T(x, t) .. t > 0 :limt[T(x, t) T(x, t)] = 0 (214)84 ( 2) , . C1(t), C2(t) T(x, t) = C2(t) C1(t)L x + C1(t) (215) Fourier Cn Cn = 2LL

0(g(x) T(x, t = 0)) sin(nL x)dx (216)3.5 .. Dirichlet Neumann . LT(x = 0, t) = a(t), T(x = L, t) = b(t) T(x, t = 0) = f(x) T1(x, t) (167) . T2(x, t) W(x, t) = T1(x, t) T2(x, t), W(x, t) (167) :W(x = 0, t) = 0, W(x = L, t) = 0 W(x, t = 0) = 085 :L

0W(x, t)2W(x, t)x2 dx 1a2L

0W(x, t)W(x, t)t dx = 0 ==L

0[ xW(x, t)W(x, t)x (W(x, t)x )2]dx 12a2L

0W(x, t)2t dx == W(x, t)W(x, t)x |L0 L

0(W(x, t)x )2dx 12a2L

0W(x, t)2t dx = 0(217)L

0(W(x, t)x )2dx = 12a2ddtL

0W(x, t)2dx (218) tt

0dtL

0(W(x, t)x )2dx = 12a2L

0W(x, t)2dx (219) (219) (). W(x, t) - T1(x, t) =T2(x, t). - .3.6 L R - . T0 ,. . :(1 + 1222 + 2z2 1a2t)T(, , z, t) = 0 (220)86 :T( = R, , z, t) = T(, , z = 0, t) = T(, , z = L, t) = 0 (220) - . T(, , z, t) = R()()Z(z)T(t) (221) (221) (220) - . , k2> 0 ( k2> 0 ) :dT(t)dt = a2k2 T(t) (222) z, , ():d2Z(z)dz2 + 2Z(z) = 0 (223)d2()d2 + m2() = 0 (224)d2R()d2 + 1dR()d + (k22 m22 )R() = 0 (225) -. m 2 ( 2).87 2 z, (Z(z = 0) = Z(z = L) = 0). k2 - , - . (.223) T(, , z = 0, t) = T(, , z = L, t) = 0 Sturm-Liouville sin(nzL ) - 2= (nL )2. (225) Bessel, (128), Bessel Neumann, m (m) , k22, :R() = AJm(k22) + BNm(k22) = 0 Neumann (Nm(x 0)). ( - ) B = 0R() = AJm(k22) , R( = R) = 0,R( = R) = AJm(k22R) = 0 (226) (.225) (.226) - Sturm-Liouville ( ) Jm(kmiR ) (kmiR ) , k2miR2 = k2 n22L2 . kmi i - Bessel m , Jm(kmi) = 0. 88 (220) Sturm-Liouville :T ( = R, , z, t) =m,n,iJm(kmiR )sin(nzL )[ Amnisin() + Bmnicos()]e(kmi2R2 +n22L2 )a2t(227) t = 0 :T( = R, , z, t = 0) = T0 ==m,n,iJm(kmiR )sin(nzL )[Amnisin(m) + Bmnicos(m)] (228) Amni Bmni Fourier - T0 Jm(kmiR ), sin(nzL ) , sin(m) cos(m).Amni = (2L)(1)( 2[RJm+1(kmi)]2)L

0sin(nzL )dz2

0sin(m)dR

0T0Jm(kmiR )d (229)Bmni = (2L)(1)( 2[RJm+1(kmi)]2)L

0sin(nzL )dz2

0cos(m)dR

0T0Jm(kmiR )d (230) Amni = 0 m Bmni = 0 m m = 0,B0ni = 16[1 (1)n]Ln2[RJ1(k0i)]R

0T0J0(k0iR )d (231)B0ni= 0 n = 2l + 1.T ( = R, z, t = 0) = T0i,l=0ea2[k20iR2 +(2l+1)22L2 ]tFil(, z) == T0ea2[k201R2 +2L2 ]t(F10(, z) + O(t)) (232)89 (184)1 = limt1tln|T(x, t)| a2(k201R2 + 2L2) (233) :|t = 0.1729R2a2 . , , . L L < , L 0 x = 0 t .2. T(x, t) L, , tT(x, t) a2 2x2T(x, t) = b2T(x, t) (234) x = 0 , x = L T(x, t =0) = T0 cos2(xL )3. L T(x, t = 0) = 3 sin(xL sin(3xL - , .4. L T(x, t = 0) = T0 +T1 cos xL . - t > 0 -.5. k, L . Q091 . - , ( T(x,t)x |x=0= kQ0 ).6. - L , H T(x =0, y, t) = T(x = L, y, t) = 0 , Ty |y=0= 0 T(x, y = H, t) = g(x) T(x, y, t = 0) = f(x, y) . . ( -).7. R1 =1cm R2 =2cm L1 = 1cm L2 = 0.5cm -. 500 . ( ).8. 2(r, , , t) + b(r, , , t) = 1kt(r, , , t) k, b . 0 . . , ( ).9. R . t = 0 - T0 T . 92 , ( - tT(x, t) a2 2x2T(x, t) = )(: T(, , t) = U(, , t) + U())934 Laplace4.1 Laplace Poisson Helmholtz - .. . Laplace, , - , , ( ). , , - . - . - . Laplace - .: G Rn. G - Laplace G. - . ( ) .94 - , . ( ). - ( ) . Laplace - Dirichlet Neumann , Cauchy - . : T1(x) 2T1(x) =0 V T1|S(V ) = T0 - Laplace, 2T2(x) = 0 - T2|S(V ) = T0. (x) = T1(x) T2(x) (x)|S(V ) = 0.

V . ((x)(x)) dV =

V| (x) |2dV +

V2(x) dV=

V| (x) |2dV (235) Gauss

V . ((x)(x)) dV =

S(V )(x) (x) . dS (236)95 (235) (236)

V| (x) |2dV = 0 (237) (x) V , .T1(x) = T2(x) . , , - , , .4.2 Laplace , , . z. - z (0 x L (0 y L ) . () . T(x, y) (x, y) Laplace .T(x = 0, y) = g1(y)96T(x = L, y) = g2(y)T(x, y = 0) = g3(x)T(x, y = L) = g4(x) g1,2(y) g3,4(x) . ( , Di-richlet, ) . - , - , . , : :T(x, y) = T1(x, y) + T2(x, y) + T3(x, y) + T4(x, Y ) (238)=2T = 0g4g3g1g2+ 2T1 = 000g10+ 2T2 = 0000g2+ 2T3 = 00g3002T4 = 0g4000( ) Ti(x, y) Laplace - :T1(x, y = 0) = g1(x) , T3(x = 0, y) = T3(x, y = L) = T3(x = L, y) = 0(239)97T2(x = L, y) = g2(y) , T2(x = 0, y) = T2(x, y = 0) = T2(x, y = L) = 0(240)T3(x, y = L) = g3(x) , T4(x = 0, y) = T4(x = L, y) = T4(x, y = 0) = 0(241)T4(x = 0, y) = g4(y) , T1(x = L, y) = T1(x, y = 0) = T1(x, y = L) = 0(242) Ti(x, y). - . T4(x, y) 2T4(x, y)x2 + 2T4(x, y)y2 = 0 (243) :T4(x = 0, y) = g4(y) , T4(x = L, y) = T4(x, y = 0) = T4(x, y = L) = 0(244) :T4(x, y) = X4(x)Y4(y) (245) (.243)1X4(x)d2X4(x)dx2 = 1Y4(y)d2Y4(y)dy2 = 2(246) y = 0 y = L , Y4(y = 0) = Y4(y = L) =0, 25 Y4(y) :Y4n(y) = sin(nyL ) (247)25 98 2= (nL )2. X4(x) (.243) :X4n(x) = A sinh(nxL ) + B cosh(nxL ) (248) (247)T4(x, y) =n=1[An sinh(nxL ) + Bn cosh(nxL )] sin(nyL ) (249) x = L T4(x = L, y) =n=1(An sinh(n) + Bn cosh(n)) sin(nyL ) = 0 (250) y An Bn,An sinh(n) + Bn cosh(n) = 0 (251)An = Bncosh(n)sinh(n) (252) x = 0T4(x = 0, y) = g4(y) =n=1Bn sin(nyL ) (253) ( 253) Fourier sin(nyL ) g4(y) . Bn :Bn = 2LL

0g4(y) sin(nyL )dy (254) T1(x, y) , T2(x, y) , T3(x, y) . : - 75 100 . 99 . . , () :=2T = 0757575100+ 2T1 = 0757575752T2 = 000025 T1(x, y) = 75 . T2(x, y) .T2(x = 0, y) = T2(x, y = 0) = T2(x, y = L) = 0 T2(x = L, 0) = 25 T2(x, y) =n=1(An sinh(nxL ) ) sin(nyL ) (255)T2(x = L, y) =n=1(An sinh(n) ) sin(nyL ) = 25 (256)An sinh(n) = 2LL

025 sin(nyL )dy = 50n (1 (1)n)T2(x, y) =k=1( 100(2k + 1) sinh((2k + 1)xL ) ) sin((2k + 1)yL ) (257) T(x, y) = 75 +k=1( 100(2k + 1) sinh((2k + 1)xL ) ) sin((2k + 1)yL )(258)1004.3 Laplace ( ): R - 75 30026.: - Laplace (144) (145), - (146) k = 0 = l(l +1), l = 0, 1, 2, 3.. .d2R(r)dr2 + 2rdR(r)dr l(l + 1)r2 R(r) = 0 (259) (259) rl r(l+1) 27. Laplace :T(r, , ) =l,m(Arl+ Br(l+1))Yml (, ) (260) Y ml (, ) . ( ) -.T(r, , ) = T1(r, , ) + T2(r, , ) (261) :T1(r = R, , ) = 75 (262)26: - ( ) ( -).27 (259) ra. - a,a1 = l a2 = (l + 1)101T2(r = R, , ) =

0 : 1 < cos < 0225 : 0 < cos < 1 T1(r, , ) T2(r, , ) Laplace T1(r, , ) = 75 . T2(r, , ) m = 0T2(r, , ) =l,m(Arl+ Br(l+1))Yml (, )=l=0(Arl+ Br(l+1))Pl(cos ) (263) , , =0,(Br(l+1) r 0) (Arl r ) A = 0 .T2(r, , ) =l,mArlPl(cos ) (264) :ARl= 2l + 121

1Pl(cos )T2(r = R, , )d cos == 2l + 12 [0

1Pl(cos )T2(r = R, , )d cos +1

0Pl(cos )T2(r = R, , )d cos ] == 2l + 12 2251

0Pl(cos )d cos (265)102 28 , .T2(r, , ) = 225[12 + 34rRP1(cos ) 716( rR)3P3(cos ) + ..] (266) :T(r, , ) = 75 + 225[12 + 34rRP1(cos ) 716( rR)3P3(cos ) + ..] (267): - S R, , V (R) = Q4Rl=0Pl(cos ) Pl(cos ) Legendre Q . S .: V (r, , ) Laplace . (260). . . r V (r , , ) 0. r A = 0 :V (r, , ) =l,mBlr(l+1)Y ml (, ) (268)28 Legendre Pl(x) = P

l+1(x)P

l1(x)2l+1103 R m = 0 :V (r, , ) =l=0Blr(l+1)Pl(cos ) (269) Bl .V (r = R, , ) = Q4RmPm(cos ) ==l=0BlR(l+1)Pl(cos ) (270)BlR(l+1)= (2l + 1)Q8R1

1mPm(cos )Pl(cos )d cos == (2l + 1)Q8Rm1

1Pm(cos )Pl(cos )d cos == (2l + 1)Q8Rmml22l + 1 = Q4R (271) :V (r, ) =lQ4R(Rr )l+1Pl(cos ) (272) . , ... R.

S

E.d

S = Qtotal .En = V (r, )r = l=0Q4RRl+1rl+2 Pl(cos )(l 1)104En .Qtotal = 2

0

0l=0Q4RRl+1Rl+2Pl(cos )(l 1)R2sin dd ==l=0Q(l + 1)2

0Pl(cos ) sin d ==l=0Q(l + 1)222l + 10l = Q (273) - Legendre 2929

0Pl(cos )sin d =1

1Pl(cos )P0(cos )d cos = 22l+1l01054.4 1. Laplace z = x + iy z = x iy 2x2 + 2y2 = 4 2z z2. L . T1 = 1000 T2 = 250 . T(x, y, z) .3. Laplace -. .(2= 1rr(r r) + 1r222 + 2z2)4. R . T(R, , ) = T0 cos() sin() ( ).5. L R . - 1000 .6. R . T[R, ] = 100(1 cos()) - T(, ) . , 1067. , R1 , R2 , ( R1 < R2 ) V (R1, ) = V0 V2 = V0 (1 + cos ) . .107


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