1、 cBsZiBc1 BsZPicardiB V Qb3.1 PicardiB 3.1.1BA TsZI nBA TsZ58 uR : jxx0j a; jy y0j b f bl3.1 Ti L 0 P/ Tjf(x;y1)f(x;y2)j Ljy1 y2j8(x;y1);(x;y2) 2 R 5f f(x;y) uR =1y LipschitzHqLLipschitz b 3.1 (PicardiB ) f f(x;y) uR = x0 a;x0 + a y0 b;y0 + b 7 O1y LipschitzHq * sZ5(3.1) uWI = x0 h;x0 +h iB h = mina
2、; bM; M = max(x;y)2Rjf(x;y)j | Vs3b12 cBsZiB(1) n5 sZ5(3.1)NsZy(x) = y0 +Z xx0f(t;y(t)dt (3.2) T Vf y(x) sZ(3.1)5 B sZ(3.2) T f y(x) sZ(3.2)5 B V7 O sZ(3.1)b BBf TsZ$1 p V7TsZ1p b 8 /Ty = (x)(x 2I) Z(3.1)d(x)dx = f(x;(x) T HVx0x |s V(x)(x0) =Z xx0f(t;(t)dt5(3.1) SHq(x0) = y0 T V(x) = y0 +Z xx0f(t;(t
3、)dtyNy = (x) sZ(3.2)lI bQ Ty = (x)(x 2I) (3.2) 5(x) = y0 +Z xx0f(t;(t)dt; x 2I (3.3) T 1x p Vd(x)dx = f(x;(x) Hx = x0 (3.3)(x0) = y0yNy = (x);x 2I 5(3.1) O Vb(2)PicardQ/E ib 8ZE/sZ(3.2)BF K sZ(3.2)V7 ib |BR f y = 0(x) sZ(3.2) 1(x) = y0 +Z xx0f(t;0(t)dt3.1 PICARDiB 3A 1(x)9 f b T1(x) = 0(x) * 0(x) sZ
4、(3.2)b5?|y = 1(x) sZ(3.2) 2(x) = y0 +Z xx0f(t;1(t)dtf(x;y) R f yNAjxx0j h Hj1(x)y0jb5f(t;1(t)Bl * /BS/ b 8 /jxx0j h Hj1(x)y0j =flflflflZ xx0f(t;0(t)dtflflflflflflflflZ xx0jf(t;0(t)jdtflflflfl Mjxx0j Mh byN2(x)9 f b T2(x) = 1(x) * 1(x) sZ(3.2)b5?BB Vn(x) = y0 +Z xx0f(t;n1(t)dt (3.4)“ B ,E Vx 2I Hjn(x
5、)y0j b(n 1)bNL!n = k H jk(x)y0j b; x 2I* n = k +1 Hk+1(x) = y0 +Z xx0f(t;k(t)dt; x 2I x 2I Hjk+1(x)y0j =flflflflZ xx0f(t;k(t)dtflflflflflflflflZ xx0jf(t;k(t)jdtflflflfl Mjxx0j Mh b“ B f 0(x);1(x);n(x);Picard b 3f H Tn+1(x) = n(x) * n(x)sZ(3.2) 5 V n(x)(n = 1;2;)x 2 I HB l B f (x)b 8 /n(x) = 0(x)+1(x
6、)0(x)+2(x)1(x)+n(x)n1(x)# k) 0(x) +1Pn=0n+1(x)n(x)I B l Vb B ,E +Y |0(x) = y05j1(x)0(x)j =flflflflZ xx0f(t;0(t)dtflflflflflflflflZ xx0jf(t;0(t)jdtflflflfl Mjxx0j4 cBsZiBj2(x)1(x)j flflflflZ xx0jf(x;1(t)f(x;0(t)jdtflflflfl LflflflflZ xx0j1(t)0(t)jdtflflflfl LMflflflflZ xx0jtx0jdtflflflfl = LM2! jxx0j
7、2BLXjn(x)n1(x)j Ln1Mn! jxx0jnL Vwjn+1(x)n(x)j flflflflZ xx0jf(t;n(t)f(t;n1(t)jdtflflflfl LflflflflZ xx0jn(t)n1(t)jdtflflflfl LnMn!flflflflZ xx0jtx0jndtflflflfl = LnM(n+1)!jxx0jn+1+Yjxx0j h Hjn+1(x)n(x)j LnM(n+1)!hn+1) 1Pn=0LnM(n+1)!hn+1 l V7f ) 0(x) + 1Pn=0n+1(x) n(x)I B l bLlimn!1n(x) = (x) k) 0(x)
8、 +1Pn=0n+1(x) n(x) BI (x)9I b“ /Picard/f fn(x)gB l f (x)bB V (x) sZ(3.2)lI b8 / LipschitzHqjf(x;n(x)f(x;(x)j Ljn(x)(x)j#fn(x)gI B l (x) Vff(x;n(x)gI9B l f(x;(x)byN V(3.4) H1n |K(x) = limn!1n(x) = y0 + limn!1Z xx0f(t;n1(t)dt = y0 +Z xx0limn!1f(t;n1(t)dt= y0 +Z xx0f(t;(t)dt (x) sZ(3.2)lI b3.1 PICARDiB
9、 5(3)Bb!(x)(x) sZ(3.1)I b:fM = maxx2Ij(x) (x)jLipschitzHqx 2I Hj(x)(x)j flflflflZ xx0jf(t;(t)f(t;(t)jdtflflflfl LflflflflZ xx0j(t)(t)jdtflflflfl (3.5) fMLjxx0j (3.6) T(3.6) T(3.5) Vj(x)(x)jfM(Ljxx0j)22!NQw Vj(x)(x)jfM(Ljxx0j)nn! ; x 2I T 7n !1 V(x) = (x)b Bb1iB h+ ilb2 L= PLipschitzHq 4_#f(x;y)R y 9
10、b3.1.2B TZs:F (x;y;y0) = 0 (3.7) f i 1F(x0;y0;y00) # = OF(x0;y0;y00) = 07Fy0 6=05A Vy0BV Ux;yf y0 = f(x;y) (3.8)7 Of(x;y)9(x0;y0) B # = O y00 = f(x0;y0)TF1 M * f(x;y)x;y9i i Ofy = FyFy0(3.9) T fy(x0;y0) B # =b 3.1Z(3.8) SHqy(x0) =y0i OBb6 cBsZiB 3.2 T(x0;y0;y00) B #”1 F(x;y;y0) M(x;y;y0) Oi ”2 F(x0;
11、y0;y00) = 0 ”3 F(x0;y0;y00)y0 6= 05Z(3.7)iBy = y(x); jxx0j h hl SHqy(x0) = y0; y0(x0) = y00 (3.10)3.2 i|sZiB T| bl3.2 !XBd b“ T8x;y 2X9(x;y) 2R O / Hq(1)ddd(x;y) 0 O Ox = y H(x;y) 0 (2)(x;y) = (y;x) (3) T T T(x;y) (x;z)+(z;y)z 2Xb5X X bWbl3.3 bWX fxng T8“ 09N 0 Pm;n N H(xm;xn) 09 0 P(x;x0) 0b8 2 0;1
12、):h = mina; bM; L8 cBsZiBXV U uWx0 h;x0 + h f F bWbsZ5(3.1)N/sZ(3.2)y(x) = y0 +Z xx0f(t;y(t)dtyN X =l (Ty)(x) = y0 +Z xx0f(t;y(t)dtX (y1;y2) ky1 y2k = maxx2x0h;x0+hjy1(x)y2(x)j; 8y1;y2 2X* (Ty1;Ty2) = maxx2x0h;x0+hflflflflZ xx0f(t;y1(t)f(t;y2(t)dtflflflfl maxx2x0h;x0+hflflflflZ xx0Ljy1(t)y2(t)jdtflf
13、lflfl Lh maxx2x0h;x0+hjy1(t)y2(t)j = Lh(y1;y2) (y1;y2)yNT X b 3.3iB f y0(x)(x 2 x0h;x0+h) Py0(x) = y0 +Z xx0f(t;y0(t)dtZ(3.1)x 2 x0h;x0 +h Bbx0h;x0 +h I = x0h;x0 +hyN T 3.1 b V sZ5(3.1) SHq /ZE| I b3.35: 3.1Cauchy5iB i uWl jxx0j hb L= P Pb ?|i uW v ;$B1 f(x;y)i uRv5i uW9vb L=b 18uSb7 OSZ f(x;y)1y Li
14、pschitzHqb f(x;y) LipschitzHqbe T0 V U8(x1;y1) 2G;9a1 0;b1 0; s.t. S =n(x;y)flflfljxx1j a1;jy y1j b1oGOi L x1;y1;a1;b118(x;y0);(x;y00) 2Sjf(x;y0)f(x;y00)j Ljy0 y00jZE: 3.1 TZ(3.1) f(x;y) i uG =1y LipschitzHq59h1 0 PZ(3.1)x0 h1;x0 + h1 iB1(x)b x0 + h1;1(x0 + h1)“ 3.1i 6Bh2 0 PZ(3.1)x0 +h1 h2;x0 +h1 +
15、h2 iB2(x)b“i uW$_ b“x0 h1;1(x0 h1) V P_PbQ“ Vvi uWb 3.4 ( ) TZ(3.1) f(x;y) uG = i O1My LipschitzHqb !P0(x0;y0)G = iBy = (x)VP0Z(3.1) SHqBHs wLb * y = (x)Kvi uW B 7 uW(fi;fl) Os wL| uG =_P Z_ %H u(P0 2 G)s wL| %b10 cBsZiB1K TG u5f(x;y)G LipschitzHqN G 8LipschitzHqbG 7 u HG LipschitzHq5 G 8LipschitzHq
16、b i uG Tf(x;y)G y 5fy LipschitzHqb2Kvi uW(fi;fl) fl A ? 3/ fB(1) fl = +1 (2) fl 09 0j00j “bl B(x;y)Ai3.4 V137C D P =f f(x;y)1y LipschitzHqb K VsK 7:Ci(i = 1;2;N)i O 8 Vs wL lb !“iCii:MLipschitz Li(i = 1;2;N)b:“0 = min1iNf“igL = max1iNfLigb 7 =nSi=1Ci * l D OHl 0 09(x) 0(x) 2 Cx0;X Zdydx = g(x;y) (3.
17、25)1g(x;y) uD LipschitzHq#/ Tjf(x;y)g(x;y)j (x); x0 x X; 1 09 0j1 0j :dvdx =f(x;)y +fi3vv(x0) = y0flflflflx=x0= y(x0;x0;y0 +y0;)y(x0;x0;y0;)y0= (y0 +y0)y0y0= 1(3.31)bv = eRxx0hf(t;)y +fi3idt16 cBsZiBfi3 y0 ! 0 Hfi3 ! 0 Oy0 = 0 Hfi3 = 0bV7y0 = limy0!0 y0 = eRxx0f(t;)y dtA 9 x;x0;y0; f byi# ZE =T b3.5
18、sZ+5il: L=5KBsZ5 H p u u9 sZH5bB H5Bi Pi9BBb“) “1Sturm-LiouvilleH5b3.5.1 Sturm-Liouville5 p 5 H p / TsZ+5 Sturm-Liouville58:ddxk(x)dydxq(x)y(x)+(x)y(x) = 0; (a 0;(x) 0;q(x) 0b x = ak(x)B ,51 p+f y(x)x = a x = bk(x)B ,51 py(x)x = b b (x) f b(3.33)? fij 0;flj 0;fij +flj 0 (j = 1;2)b(3.33)c vB HHq+5b f
19、i1 = 05x = a)B HHq fl1 = 05x = a)= HHq fi1 6= 0afi2 6= 05x = a) HHqbx = b)HHq9 ) b(3.32)c vB sZ+5b k(x) = x;q(x) =”2x ;(x) = x (0 0 H= H5 H5(3.32)a(3.33) + 0b | !+f y(x)b(3.32) H y(x)a;b sZ ba(x)y(x)2dx = Z bay(x) ddxk(x)dydxdx+Z baq(x)y(x)2dx= k(a)y(a)y0(a)k(b)y(b)y0(b)+Z bak(x)y0(x)2 +q(x)y(x)2“dx
20、(3.34)(3.33)P HHqfi1y0(a) = fl1y(a) Hy0(a)fi1y0(a)2 = fl1y0(a)y(a) (3.35)fi1y0(a) = fl1y(a) Hy(a)fl1y(a)2 = fi1y0(a)y(a) (3.36)(3.35)(3.36)MF Vy0(a)y(a) = fi1y0(a)2 +fl1y(a)2fi1 +fl1 (3.37)“(3.33) HHqfi2y0(b) = fl2y(b) Hy(b)fi2y0(b)y(b) = fl2y(b)2 (3.38)fi2y0(b) = fl2y(b) Hy0(b)fl2y0(b)y(b) = fi2y0(
21、b)2 (3.39)(3.38)(3.39)MF Vy0(b)y(b) = fi2y0(b)2 +fl2y(b)2fi2 +fl2 (3.40)18 cBsZiB(3.37)(3.40) (3.34) V =k(a)ffi1y0(a)2+fl1y(a)2gfi1+fl1 +k(b)ffi2y0(b)2+fl2y(b)2gfi2+fl2 +Rbank(x)y0(x)2 +q(x)y(x)2odxRba (x)y(x)2dx 0(3.41) B/ I Hq/ Ts0 ,b n5BA1Hq y0(x) 0y(x) A yy(x)d ,#AA 6= 0b5 y0(x) 6 05yx 2(a;b) Hk
22、(x) 0AZbak(x)y0(x)2dx 0QAq(x) 0b5yq(x) 0 q(x) 6 0A(a;b) l uWa1;b1q(x) 0yNAZ baq(x)y(x)2dx = A2Z baq(x)dx A2Z b1a1q(x)dx 0Kyy(x) A(3.41)s0y0(a) = 0;y0(b) = 0;y(a) = A;y(b) = Ab1 Ps0 ,Afl1 = fl2 = 0(3.33) = HHqbyN(3.32)q(x) 0i O(3.33) = HHq H ,+f d , b3.2 V kd+1;2; 0 1 2 ; limn!1n = +1 A a usZ =v2004b
23、3.3+f a;b b | ! 6= M+f sY y(x)z(x)5(x)y(x) = ddxk(x)y0(x)+q(x)y(x) (3.42)(x)z(x) = ddxk(x)z0(x)+q(x)z(x) (3.43)(3.42)(3.43)sYz(x)y(x) TMh V()Z ba(x)y(x)z(x)dx =Z bay(x) ddxk(x)z0(x)z(x) ddxk(x)y0(x)dx= k(b)y(b)z0(b)y0(b)z(b)k(a)y(a)z0(a)y0(a)z(a)(3.44)3.5sZ+5196(3.33)x = a)HHq V(fi1y0(a)fl1y(a) = 0f
24、i1z0(a)fl1z(a) = 0 (3.45)(3.45) V A1(fi1;fl1)BL ZFbyfi1fl1 H , “ TA ,y(a)z0(a)y0(a)z(a) = 0 (3.46)“(3.33)x = b)HHq V(fi2y0(b)+fl2y(b) = 0fi2z0(b)+fl2z(b) = 0 (3.47)yfi2fl2 H ,#Ay(b)z0(b)y0(b)z(b) = 0 (3.48)(3.46)(3.48) (3.44) V+f (x)Z ba(x)y(x)z(x)dx = 0 (3.49)3.4B+f KKb +L1+f B ZE V Pt+f M (x)b+f (
25、x)“L a;b ! (x)+f “fyn(x);n = 1;2;g Pa;b B Z Vf f(x) VN+f “FourierZ 7f(x) =1Xn=1cnyn(x) (3.50)cn = 1 nZ baf(x)yn(x)(x)dx; n = 1;2; (3.51)(3.51) nn =Z ba(x)yn(x)2dx; n = 1;2; (3.52)(3.50) l L2a;bS il/limn!10Z baflflflflflf(x)nXk=1ckyk(x)flflflflfl2dx1A12= 0 (3.53)f(x)a;b s; O (3.33)HHq H(3.50) V B l b
