超越摄动同伦分析方法导论廖世俊著陈晨徐航译

超越摄动同伦分析方法导论廖世俊著陈晨徐航译北京

内容简介图书在版编目 (CIP) 数据超越摄动同伦分析方法导论 / 廖世俊著, 陈晨徐航译. 北京 : 科学出版社, 2006 ISBN 7-03-000000-0 Ⅰ. 超 Ⅱ.1 廖 2 陈 3 徐 Ⅲ. Ⅳ. 中国版本图书馆 CIP 数据核字 (2005) 第 000000 号责任编辑 : 田士勇鄢德平于宏丽 / 责任校对 : 责任印制 : 钱玉芬 / 封面设计 : 王浩出版北京东黄城根北街 16 号邮政编码 : 100717 http://www.sciencep.com 印刷科学出版社发行各地新华书店经销 * 2006 年 2 月第一版开本 : B5(720 1000) 2006 年 2 月第一次印刷印张 : 00 0/0 印数 : 0 0 000 字数 : 000 000 定价 : 00.00 元 ( 如有印装质量问题, 我社负责调换科印 )

ii 前言谨以此书献给我的妻子

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目录 i 目第一部分录基本思想第 1 章引论... 3 第 2 章范例性描述... 8 2.1 范例... 8 2.2 由传统解析方法得到的解... 9 2.2.1 摄动方法... 9 2.2.2 Lyapunov 人工小参数法... 10 2.2.3 Adomian 分解法... 11 2.2.4 δ 展开法... 12 2.3 同伦分析解... 13 2.3.1 零阶形变方程... 13 2.3.2 高阶形变方程... 15 2.3.3 收敛定理... 18 2.3.4 一些基本原则... 19 2.3.5 不同形式的解表达... 20 2.3.6 辅助参数 h 的作用... 33 2.3.7 同伦 - 帕德近似... 41 第 3 章系统性描述... 46 3.1 零阶形变方程... 46 3.2 高阶形变方程... 48 3.3 收敛定理... 50 3.4 基本原则... 52 3.5 收敛区域和收敛速度之控制... 54 3.5.1 h 曲线和 h 之有效区域... 55 3.5.2 同伦 - 帕德近似... 55 3.6 进一步一般化... 57 第 4 章与传统解析方法之关系... 59 4.1 与 Adomian 分解法之关系... 59

iv 目录 4.2 与人工小参数法之关系... 62 4.3 与 δ 展开法之关系... 64 4.4 非摄动方法之统一... 68 第 5 章优点局限性及有待解决之问题... 69 5.1 优点... 69 5.2 局限性... 70 5.3 有待解决的问题... 70 第二部分应用第 6 章具有简单分岔的非线性问题... 73 6.1 同伦分析解... 74 6.1.1 零阶形变方程... 74 6.1.2 高阶形变方程... 76 6.1.3 收敛定理... 77 6.2 结果分析... 78 第 7 章具有多解的非线性问题... 84 7.1 同伦分析解... 85 7.1.1 零阶形变方程... 85 7.1.2 高阶形变方程... 86 7.1.3 收敛定理... 88 7.2 结果分析... 89 第 8 章非线性特征值问题... 95 8.1 同伦分析解... 96 8.1.1 零阶形变方程... 96 8.1.2 高阶形变方程... 97 8.1.3 收敛定理... 100 8.2 结果分析... 101 第 9 章托马斯 - 费米原子模型... 108 9.1 同伦分析解... 108 9.1.1 渐近性质... 108 9.1.2 零阶形变方程... 109

目录 v 9.1.3 高阶形变方程... 111 9.1.4 递推表达式... 112 9.1.5 收敛定理... 113 9.2 结果分析... 114 第 10 章 Volterra 生态学模型... 120 10.1 同伦分析解... 120 10.1.1 零阶形变方程... 120 10.1.2 高阶形变方程... 122 10.1.3 递推表达式... 124 10.1.4 收敛定理... 126 10.2 结果分析... 127 10.2.1 选取一般的初始猜测解... 127 10.2.2 选取最佳的初始猜测解... 129 第 11 章具有奇非线性的自由振动系统... 132 11.1 同伦分析解... 132 11.1.1 零阶形变方程... 132 11.1.2 高阶形变方程... 134 11.2 范例... 137 11.2.1 例 1... 137 11.2.2 例 2... 139 11.2.3 例 3... 140 11.3 收敛区域之控制... 142 第 12 章具有二次型非线性的自由振动系统... 144 12.1 同伦分析解... 144 12.1.1 零阶形变方程... 144 12.1.2 高阶形变方程... 147 12.2 范例... 149 12.2.1 例 1... 149 12.2.2 例 2... 153 第 13 章多维动力系统之极限环... 157 13.1 同伦分析解... 158

vi 目录 13.1.1 零阶形变方程... 158 13.1.2 高阶形变方程... 161 13.1.3 收敛定理... 164 13.2 结果分析... 165 第 14 章布拉休斯黏性流... 171 14.1 用幂函数表达的解... 171 14.1.1 零阶形变方程... 171 14.1.2 高阶形变方程... 173 14.1.3 收敛定理... 174 14.1.4 结果分析... 175 14.2 用指数和多项式表达的解... 178 14.2.1 渐近性质... 178 14.2.2 零阶形变方程... 179 14.2.3 高阶形变方程... 180 14.2.4 递推表达式... 180 14.2.5 收敛定理... 182 14.2.6 结果分析... 183 第 15 章呈指数衰减的边界层流动... 188 15.1 同伦分析解... 189 15.1.1 零阶形变方程... 189 15.1.2 高阶形变方程... 191 15.1.3 递推公式... 192 15.1.4 收敛定理... 194 15.2 结果分析... 195 第 16 章呈代数衰减的边界层流动... 202 16.1 同伦分析解... 202 16.1.1 渐近性质... 202 16.1.2 零阶形变方程... 203 16.1.3 高阶形变方程... 205 16.1.4 递推公式... 206 16.1.5 收敛定理... 207

目录 vii 16.2 结果分析... 209 第 17 章冯卡门黏性涡流... 214 17.1 同伦分析解... 215 17.1.1 零阶形变方程... 216 17.1.2 高阶形变方程... 219 17.1.3 收敛定理... 222 17.2 结果分析... 223 第 18 章深水中的非线性前进波... 229 18.1 同伦分析解... 230 18.1.1 零阶形变方程... 230 18.1.2 高阶形变方程... 233 18.2 结果分析... 237 参考文献... 241 附录一第二章 Mathematica 程序... 248 附录二第六七章 Mathematica 程序... 254 附录三第八章 Mathematica 程序... 260 附录四第九章 Mathematica 程序... 265 索引... 268

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b { = = 6 ƒ % ) *U7U8UEUÏ = +,. i Û SU ï H[Φ(t; q); q] = (1 q) L [Φ(t; q) u 0 (t)] + q A[Φ(t; q)] (1.6) - ¼/. ÌUMUc ¼ b ), q [0, 1], Φ(t; q) t q. b q = 0 q = 1, \Ua H[Φ(t; q); q] q=0 = L[Φ(t; 0) u 0 (t)] 0 G 5U;U< UL, 5U;U< (1.5),, H[Φ(t; q); q] q=1 = A[Φ(t; 1)] Φ(t; 0) = u 0 (t) H[Φ(t; q); q] q=0 = 0 Φ(t; 1) = u(t) UL. UH, H[Φ(t; q); q] q=1 = 0. ÌUMUc q 0 U'UŽ 1, ;U< H[Φ(t; q); q] = 0 AL 0 Φ(t; q) {.0ÌAMAc q, & ' (AßAL M0AŽ0 'A;A< AYAZ u 0 (t) (1.4) L Ž Uêp# u(t)., ë UŒUM ÜU¼ ò M. ï % Ý =, ÿ SU» 1Ui UŒU;Us b ï UŒU;Us [33] [34] k )Uq ;Us. ñ U, i :. ÌUMUc q 2 3 Uó ü ) U Uó ɛ δ, Uü )UsUb î δ AUs 9U: ï IU=U> ø, ý Õ 4 4,U3., í î )UqU;UsU% 5 œ U> +, ï, % 6 7U;Us (1.6) 8 7 9 Ua %, R LUmU;UsU% 5 œ, : VU,U5UŠUµ óu%. ; n 5 ¼, <, Ùa 6 #U'U% **( = U%U7U8UEUÏ b & ' (UßUL L, : >? @ Ä B C D Ë F G Ḧ Ï J =. K L M Ï N, Ö P Q R Ï S T Ü B V Ẅ Ẍ Ÿ @ Z [ \ Ï ] ^ _ ` ä b c d ë f g ḧ ï ë f j k. lmonqp Hrstu QR vw UB Ï x l yz, { G}~ i ƒ n I @A ˆ Š, Œ Ž ˆi ˆ., [ õ QRvw Ü B d ˆ p, Ḧ Ë š œ žÿ Q R v w Ü B š Ï n ª ˆ «N 150 D ± «c w n±² ž ³ b µ I, ¹ 1.1.

º 1» ¼¾½ 7 [ D, Q R v w U B x Q R y z, V N x S T I Q R (1.6), À N Á Â @ Ã Ä Å Æ Ç b ï Ä Å Æ È b H(t), Ö P @ É Ê I Q R GÌË Ô Õ d H(Φ; q,, H) = (1 q) L[Φ(t; q,, H) u 0 (t)] q H(t) A[Φ(t; q,, H)] (1.7) (1.6) K @ Í, Î (1.6) Ï (1.7) Ð = 1 i H(t) = 1 Ñ d Ò Ó, H(Φ; q) = H(Φ; q, 1, 1) (1.8) Q Ö, Ð q 0 Ø Ù Ú 1 Ñ, Φ(t; q,, H) Û Ü Ý Þ c u 0 (t) ß à Ú á Ü ˆ U ƒ Ï â ã c u(t). ä À, Ü ƒ H[Φ(t; q,, H)] = 0 (1.9) dåc Φ(t; q,, H) V Ïåæåç åèåâ ßåé q, Àåêåëåæåç åååæåçåb iåååæåèåb H(t). Î ì, Ð q = 1 Ñ, c í ä æ ç Å Æ Ç b ï Å Æ È b H(t). ², S T Q R (1.6) V Q, î ï Q R (1.7) Ẅ ð Ẍ Ÿ @ ñ ä b c, Ž ë f g h æ ç Å Æ Ç b ï Å Æ È b H(t), l m ò ó ô ` u. K L M Ï N, G Ẍ Y @ Z õ ö ï ø ä b c d ë f g ḧ ï ë f j k Ï [ \ ] ^. QåRåvåw UåBåNå@åÉåù @åíåiåuåb, úåûåüåý åiå å åˆåþ ÿ v Uåƒ. G }~ ûü ˆ Š, ˆ [35 39]ÿ [28,29,40 43] ÿåsåf [44, 45] ÿ n ü I ª ˆå [46] ÿ Oldroyd þåbå ª ˆ [47] ÿ [48] ÿ ˆ [49, 50] ÿ! - "# Uƒ [51] i Lane-Emden Ü ƒ [30] $. Œ Ž ˆ ï ˆ l m, š Q R v w Ü B @ A Ê Ï } ~ Ó.

c l ô l ÿ z. % 2 & ')()*)+), p, Ḧ ~ @ Ã [ Ï ˆ þ v U ƒ- /. Ó, 0 Q R v w Ü B d x 2.1 1 2 3456Ìn 789:<; œ/= I. > t? Ñ@, U( t)? j, m? é, g? L Ä j k. BC D Ú I 56 «E a U 2 ( t), a þ b. F GHIJ, Ž Û Ü ZK m du( t) d t = mg au 2 ( t) (2.1) U(0) = 0 (2.2) LMN, ;= d j ko L - ~ œp VQ Ø Ù, RST Ú @ ÃU Ï j k U., VW VXY U( t), ë W Ü ƒ (2.1) RZ[ Ú\] j k U, V U = mg a V^ > U i U /g v_? Ò` j k ï Ò` Ñ@. > p H [ Úä éb Ü ƒ Û Ü ZK (2.3) ( ) U t = t, U( t) = U V (t) (2.4) g V (t) + V 2 (t) = 1, t 0 (2.5) V (0) = 0 (2.6) å, t?å aåé båñ @,?å åú t d F. Œåä, Ð t + Ñ, V t i U( t) U Ñ, efd c Ü ƒ (2.5) i (2.6), F (2.4), p H lim V (t) = 1 (2.7) t +

º 2» gihijikil 9 Ü ƒ (2.5) i (2.6) d â ã c (2.8) m ~ V Qno c dìëp. V (t) = tanh(t) (2.8) 2.2 qsrutuvuwyxyz {y}y~ v \ ÌËp, p H } ~ ƒ Ï S T c w Ü B D d c. Ó. 2.2.1 ˆ š/ Š d/š/ n/o c, p H B I a é/b Ñ/@ t / é (Œ Š/ é ), { V (t)?t œ ä b Û Ü ZK (2.6), [ Ú α 0 = 0. N u?t ú k=0 V (t) = α 0 + α 1 t + α 2 t 2 + α 3 t 3 + (2.9) N u?t (k + 1) α k+1 + t > 0, À Ž Â Ü ƒ (2.5), k α j α k j t k = 1 j=0 α 1 = 1 (2.10) p HÌ ì[ Ú Š c α k+1 = 1 k + 1 k α j α k j, k 1 (2.11) j=0 V pert (t) = t 1 3 t3 + 2 15 t5 17 315 t7 + = α 2n+1 t 2n+1 (2.12) co Ð Ï g h 0 t < ρ 0 ë f,, ρ 0 3/2, ¹ 2.1. [ Ò _ E š Ï N, Š c (2.12) d ë f g ḧ ï ë f j k N ãï I. n=0

ò 10 š 2.1 œi iž (2.57) Ÿii iž (2.8) «ª ; «± (2.12) ; ² = 1/2 ³ µ ; = 1/5 ³ µ ; = 1/10 ³ µ 2.2.2 Lyapunov ¹º»¼½ } ~ Lyapunov ¾ Ç b B, p H Ü ƒ (2.5) ÀÁ, ɛ ¾ Ç b. ä, > V (t) + ɛ V 2 (t) = 1 (2.13) V (t) = V 0 (t) + ɛ V 1 (t) + ɛ 2 V 2 (t) + (2.14) Ž (2.14) Â Ü ƒ (2.13) i Û Ü ZK n (2.6), > ɛ QÂ dã b p 0, H [ Ú ÃÄ Ü ƒ V 0 (t) = 1, V 0 (0) = 0 æâd c N ü Ü ƒ, [ Ú V 1 (t) + V 2 0 (t) = 0, V 1 (0) = 0.. V 0 (t) = t, V 1 (t) = t3 3, V 2(t) = 2t5 15,

Ž n ò º 2» gihijikil 11, >?T (2.14) n ɛ = 1, V (t) = t 1 3 t3 + 2 15 t5 17 315 t7 + = n=0 α 2n+1 t 2n+1 (2.15) ² (2.15) Š c (2.12) ÅÆ Q, Î ì ë Ï O t Ð ] Ï g h, ¹ 2.1. /[/Ç/È I N, Lyapunov ¾ Ç b B [ Ú I c/d e f g h i e f j k ë N ãï I. 2.2.3 Adomian ÉÊ } ~ Adomian v c B ~, Ë Ž Ü ƒ (2.5) i Û Ü ZK V (t) = t t 0 (2.6). N ü Ü ƒ d Adomian c V (t) = V 0 (t) + V k (t) V 2 (t)dt (2.16) k=1 Œ Adomian üì À (2.17) Í V 1 (t) = t3 3, V 0 (t) = t V k (t) = A k (t) =. N u?t t 0 A k 1 (t) dt, k 1 k V n (t) V k n (t) n=0, p H æâ[ Ú V 2(t) = 2t5 15, V 3(t) = 17 315 t7, V (t) = t 1 3 t3 + 2 15 t5 17 315 t7 + = α 2n+1 t 2n+1 (2.17) n=0 ² Š/ c (2.12) Å/Æ Q, Î ì ë Ï O/ Ð/ I g h, ¹ 2.1. [ÇÈ Ï N, Adomian v c B [ Ú Ï c/d ë f g ḧ ï ë f j k ë N ãï I.

ä ò 12 š 2.2.4 δ ÎÏ } ~ δ ÐÑ B, p H Ü ƒ (2.5) ÀÁ, δ ž b. > V (t) + V 1+δ (t) = 1 (2.18) V (t) = V 0 (t) +, V 1+δ (t) ÐÑ œ δ d ä b n=1 V n (t) δ n (2.19) Ü ƒ (2.19) i V 1+δ = V 0 + [V 1 + V 0 ln V 0 ] δ + [V 1 (1 + ln V 0 ) + 12 ] V 0 ln 2 V 0 + V 2 δ 2 + (2.20) (2.20) Ž ÂUƒ (2.18), > δ QÂ dãb 0, [Ú œ ÃÄ V 0 + V 0 = 1, V 0 (0) = 0 V 1 + V 1 = V 0 ln V 0, V 1 (0) = 0 V 2 + V 2 = V 1 (1 + ln V 0 ) 1 2 V 0 ln 2 V 0, V 2 (0) = 0 V 3 + V 3 = V 2 (1 + ln V 0 ) V 1 (1 + 1 2 ln V 0 ) ln V 0 1 6 V 0 ln 3 V 0 V 1 2, V 3 (0) = 0 2V 0. æâd c N ü ˆ Ü ƒ, [ Ú V 0 (t) = 1 exp( t) ] V 1 (t) = exp( t) [t π2 6 + P 2 L (e t ) (1 e t ) ln(1 e t ).. P L n (z) = + k=1 z k k n

< N º 2» gihijikil 13 z I n L ú b È b V (t) 1 + exp( t) (nth polylogarithm function). Ž Ò no c ] [t π2 6 1 + P 2 L (e t ) (1 e t ) ln(1 e t ) (2.21) ÓÔÕÖ Ò no c o OØ Ã gh 0 t < +, NoqÒÙÈb Pn L (z) Ï šú, Û[Ü Ò no c ß[Ý D ÝÞß. à Ï N N, uìš Uá ÿ Lyapunov ¾ Ç bá i Adomian v cá d[ I a b c/â N/ Q I. N a b c, Ï O/ Ã Ð/ I g h 0 t < 3/2. Ì ², Š c ýo,ã Lyapunov ¾ Ç bá i Adomian v cá $ S T Š Uá d[ Ï c, O LM Ç bä ß é Ø Ù, À ˆ Øå Ñ, m Wæ. Q Ñ, Ã ž ë/è/ç/è, U/á d @/m W/é/O/ê É/ë/Ã. /[/å _ I N, S T c w Uá [ Ú Ï ä b c d e f g h i ë f j kâ N / I, À ê Š Uá i, ã Lyapunov ¾ Ç bá ÿ Adomian v cá i δ v cá $ Š Uáâ V Ẅ X Y Ã ø ï õ ö ä b c d ë f g ḧ ï ë f j k I [ \ ] ^. ìì, S T U/á/â a á/í ~ îx I V (t) O t ï a/ð Ñ I L M ˆ (2.7). Ã Ó/ñ l,, aò Š Uáó N S T Š/ Uá, â V Wô v í ~ ý õ/ I/ö /T Ú K ø n ˆ Š d<ù I. ò, [ÇÈ Ï N, ¾ Ç b ɛ i δ v_ šú Ö Ü ƒ (2.13) i n (2.18) Ï V Qúû. N, δ ÐÑ á š Ï c (2.21) Ẅ ðoø Ã g h 0 t < +, À, Ë Lyapunov ¾ Ç b/á š I c [ ü. G ô 1 u, ü p H ɛ i δ I ï è Â ß é, ýþ Ü ƒ (2.13) i (2.18) mÿ ÉÄÒ ß Uƒ. õ, K ø n Ã ˆ Š, M OP Q R ï ÄÒ ß Ü ƒ N þ L M I. 2.3 uwuv l `, p H W ~ Q I. Ó ẗ u Q R v w Uá d x l ÿ z 2.3.1 ˆ > V 0 (t)? V (t) I Û Ü Ý Þ c. c Û Ü ZK (2.6), V V 0 (0) = 0 (2.22) > q [0, 1]? I/è Â ß é. Q R v w U/á x Ö É V (t) Φ(t; q), Ð è Â ß é q 0 Ø Ù Ú 1 Ñ, Φ(t; q) Û Ü Ý Þ c V 0 (t) ß à Ú â ã c Mathematica!"#$%&' (), *+,-.. /012 % E-mail 4 3 BeyondPerturbation@126.com, 56789 http://numericaltank.sjtu.edu.cn/code.htm,!:; $=&=>=,=-=?=@=A=B=C Mathematica D=E =0=1=F=&. G=H.

c c V è 14 š V (t). žú É, J Ï, γ 1 (t) 0 i γ 2 (t) K œ/å Æ ˆ Jñ Φ(t; q) L[Φ(t; q)] = γ 1 (t) + γ 2 (t) Φ(t; q) (2.23) t Ï ž È b. Ü ƒ (2.5), I ï œ ˆ Jñ N [Φ(t; q)] = Φ(t; q) t + Φ 2 (t; q) 1 (2.24) > 0 i H(t) 0 v/_/? I/Å Æ Ç båi/ååæ Èåb. LM Á Â è Â ß é q [0, 1], Ö P œ N Ü ƒ Ž Û Ü ZK (1 q) L [Φ(t; q) V 0 (t)] = q H(t) N [Φ(t; q)] (2.25) Φ(0; q) = 0 (2.26) [å _ Ï N p, H O P Ù I ; J Ï Å Æ Ç b ÿ Å Æ È b H(t)ÿ Û Ü Ý Þ c V 0 (t) ï Å Æ ˆ Jñ L. l m ò ó ô ` N, ü I ;O Q R v w Uá nrq L M I- ~, S I Q R v w Uá ˆ ï ˆ d x T. Ð q = 0 Ñ, Ü ƒ (2.25) ß F Û Ü ZK (2.22) i L [Φ(t; 0) V 0 (t)] = 0, t 0 (2.27) Φ(0; 0) = 0 (2.28) (2.23), Ü ƒ (2.27) i (2.28) Ï c Ü N Ð q = 1 Ñ, Ü ƒ (2.25) Φ(t; 0) = V 0 (t) (2.29) Û Ü ZK H(t) N [Φ(t; 1)] = 0, t 0 (2.30) Φ(0; 1) = 0 (2.31) Φ(t; 1) = V (t) (2.32) 0ÿ H(t) 0, êf I ï (2.24), Ü ƒ (2.30) i (2.31) $ Q Ü ƒ (2.5) i (2.6). F (2.29) i (2.32), Ð è Â ß é q 0 Ø Ù Ú 1 Ñ, Φ(t; q) Û Ü Ý Þ

ä ò º 2» gihijikil 15 c V 0 (t) ß à Ú â ã c V (t). OWX n, É ß à/œ, U ƒ (2.25) i (2.26) O Q R Φ(t; q). [ \, Ü ƒ (2.25) i (2.26) Œ ˆ. Î p H O J Ÿ Å Æ Ç b ÿ Å Æ È b H(t)ÿ Û Ü Ý Þ c V 0 (t) ï Å Æ ˆ J/ñ L d ;,, m/b I G H/â JY Z, À Ð 0 q 1 Ñ Ä/Ò ß U ƒ (2.25) i (2.26) d c Φ(t; q) éo, ê ú è Â ß é q I m Ò [, V 0 (t) = m Φ(t; q) q m (2.33) q=0 V [m] ë éo, m = 1, 2, 3, [ \, V [m] 0 (t) Œ m Ò ß F \ ] ÐÑ I M (2.29) i V m (t) = V [m] 0 (t) = 1 m! m! m Φ(t; q) q m, Φ(t; q) ^ è Â ß é q ÐÑ œ ä b Φ(t; q) = Φ(t; 0) + (2.34), N u ä b m=1 Φ(t; q) = V 0 (t) + 1 m! m=1 m Φ(t; q) q m [ b. I ï (2.34) q=0 q m (2.35) q=0 V m (t) q m (2.36) B/C Å Æ Ç b ÿ Å Æ È b H(t)ÿ Û Ü Ý Þ c V 0 (t) i Å Æ ˆJ/ñ L JY Z, À ä b (2.36) O q = 1 Ñ ë f. ýþ, Ð q = 1 Ñ, ä b (2.36) ß N u?/t, F (2.32), Φ(t; 1) = V 0 (t) + V (t) = V 0 (t) + m=1 V m (t) (2.37) V m (t) (2.38) m=1 í ~_/X Ì V m (t) (m = 1, 2, 3, ) š Û Ü Ý Þ c ² V 0 (t) â ã c V (t) dëã. 2.3.2 ` ˆ I ï ä é V n = {V 0 (t), V 1 (t), V 2 (t),, V n (t)}

d c l š K q p [ ò 16 š F I ï (2.34), m ÄÒ ß Ü ƒ (2.25) i (2.26) [ Ú _X Ì V m (t) d b c Ü ƒ i Û Ü ZK. Ü ƒ (2.25) i (2.26) ú è Â ß é q d m Â, ä > q = 0, ò p e m!, [ Ú f m Ò ß g ƒ ikj prq Û Ü h (2.24) r L [V m (t) χ m V m 1 (t)] = H(t) R m (V m 1 ) (2.39) R m (V m 1 ) = (2.41), s V m (0) = 0 (2.40) 1 (m 1)! 0, m 1 χ m = 1, m 1 N [Φ(t; q)] q m 1 (2.41) q=0 i m n o (2.42) R m (V m 1 ) = V m 1 m 1 (t) + V j (t)v m 1 j (t) (1 χ m ) (2.43) [ÇÈ f t, R m (V m 1 ) u v w x j=0 V 0 (t), V 1 (t), V 2 (t),, V m 1 (t) yz e â{ LMv}~ m Ò g (2.39) r (2.40) ƒ., ˆ Š L Œ (2.23), g (2.39) Ž g, h (2.40). }, œ œ gœ (2.39) r (2.40) œ V m (t) žœÿœœ œƒ, œ tœ œ œ œ œ prq Mathematica Maple MathLab ª «. (2.38), ±, ² ³ µ (2.5) r (2.6) ¹ Ž (2.39) r (2.40) º» ¼ ½ ¾ À ŠÁÂ ŠÁÂ, ÃÄÅÆÀ ¼ rçèéê. Ë ÌÍ¼ t, ÎÏ Ð jrû Ü Ý Þ ß ¹ Ñ Ò Ó Ô Õ Ö r (Ø Ù ) Ú (à ) á â. V (t) ¼ m È ã Ž V (t) m V n (t) (2.44) Ë Ì Í ¼ t, ä p Š (2.25) ˆ L å æ ç V 0 (t) å ˆ á â r ˆ è â H(t) é Œ. ê ë ± ì p, ± í î ï ð ¼ V (t) v w x ˆ ñóò=ô=õ (2.36) ö= =ø=ù=ú=û=ü=ý ô=õ (2.25) þ (2.26), ÿ q =ú, û=ü=ý ô=õ (2.39) þ (2.40), (2.41) (2.43). n=0

p p p s q r q q q 2 17 Š L å æ ç V 0 (t) å ˆ á â r ˆ è â š H(t). }, Ó x s! " È ã # î p, ± í î ï ð ¼ $ â &% '&( )r&% '&*&+ Ó t, ¼, -. /10 Ÿ 2. 2.3.3 3 4 5 6 5 6 2.1 7 8 9 (2.38) : ;, < = V m (t) >?1@BA C D E F (2.39) G (2.40), H I J (2.42) G (2.43) K L, M N O P E F (2.5) G (2.6) Q R. S T $ â m=0 % ', { U S(t) = l V χ m Œ (2.42),» W V m (t) m=0 V m (t) lim V m(t) = 0 (2.45) m + n [V m (t) χ m V m 1 (t)] m=1 = V 1 + (V 2 V 1 ) + (V 3 V 2 ) + + (V n V n 1 ) ± = V n (t) (2.45), s ± í X Y m=1 [V m (t) χ m V m 1 (t)] = lim n + V n(t) = 0 r L Œ (2.23),» W ± í X Y m=1 L [V m (t) χ m V m 1 (t)] = L r (2.39), s m=1 [V m (t) χ m V m 1 (t)] = 0 m=1 L [V m (t) χ m V m 1 (t)] = H(t) m=1 R m (V m 1 ) = 0

š f Œ Ü q q q 0 y z š 0 % 18 Z\[\]_^ Ž 0 r H(t) 0, ` prq (2.43),» W m=1 R m (V m 1 ) = m=1 m=1 R m (V m 1 ) = 0 (2.46) Vm 1 (t) + m 1 j=0 V j (t)v m 1 j (t) (1 χ m ) = = m=0 m=0 V m (t) 1 + m=1 V m (t) 1 + j=0 m 1 j=0 m=j+1 V j (t)v m 1 j (t) V j (t)v m 1 j (t) = m=0 V m (t) 1 + V j (t) j=0 i=0 V i (t) prq (2.46) r (2.47), s = Ṡ(t) + S2 (t) 1 (2.47) prq (2.22) r (2.40),» W Ṡ(t) + S 2 (t) 1 = 0, t 0 S(0) = V m (0) = V 0 (0) + m=0 m=1 V m (0) = V 0 (0) = 0 / a ± À X Y V, S(t) Ž (2.5) r (2.6). b c. Ë ÌÍ¼ t p q Š q&e, ±íœê\d (2.23) Œ ¼ ˆ L Ž, g1d, γ 1 (t) 0 r γ 2 (t) { Ž Ó ¼ è â. h Œ ê i s j Õ k. l Ž s Î À Œê, ²³&mÔ&n\d É&oÊ&p&$â&%&'. ž&q&r, $â (2.38) ¼&%&'vwx ˆ Š á â å ˆ è â H(t) å æ ç V 0 (t) r ˆ L. s t ¼ t, u # îwvwx ï ² ³ ž à ¼zy {B} ~ Ä ³., m Õ ˆ á â ˆ è â H(t) Š æ ç V 0 (t) ˆ L ~ Ä, $ â Ü (2.38) 0 t t 0 ( ) z V ', h ( ) % ' ƒ š. }, % ' Œ ê ƒ ˆ á â ˆ è â Š H(t) æ ç V 0 (t) ˆ L ~ Ä ± ¼zyB{B, º» u # î ˆ Š Œ.

Ä å ö š Ü V Ü y e Ü V y 2 19 2.3.4 Ž ± í, ºw ä Š w, ² ³w wˆ ž à ¼ y { ~ Ä ˆ L y æ ç V 0 (t) ˆ è â H(t). ê ë ±, ± í y { Žw w, ` { ~ Äwš Ó Š ¼ ˆ è â H(t) = æ ç V 0 (t) ˆ L. r, ±, Î ÀzyB{ ã œ ž, ˆ Õ v x Ÿ Œ µ g ~ Ä. Á Â y, # È ã ¼ Œ èâ Ç È. & &ª, èâ f(x) {&«Ó&¼&ŒèâÇÈ, & & ¼&Œèâ& & &±&ˆ& &²ÇÈ. y Á Â }, Œ è â ¼ ~ Ä Ç È ³ j Õ. ˆ ² Ç È À ï Œ ¼&µ&, &~Ä& & ¼&Œèâ. s&t¼&, Š ˆ L æç V 0 (t) ˆ è â H(t) ~ Ä ± ¼zyB{, { ƒ š 1{RÓ Œ è â í ¼ V (t). ` ÎwŸ wxwy¹d, ² ³ { ~ Ä Àw w ¼, ` ±wŵ w² Ç È À ï Œ ¼ Á Â, ² ³ Å Æ ½ Ô ~ À, Ó Í U Á Â Ç. Ü š wºw»w¼, ½wž w#w¾ êw wà (Ø Ù ) Ú w (ÁwÂ ) wã ª&È É&Ê Ï ¼&Œ è â& í g Ë. Ì {e k (t) k = 0, 1, 2, } (2.48) X 2 í - Î Ì Ë ¼ Œ è â. ² ³ Ï Ë X Y» V (t) = n=0 c n e k (t) (2.49) Î Ð, c n Ž Ñ â. Ò ~ Ä Œ è â, ˆ è â H(t) æ ç Ë V 0 (t) ˆ Š Ç V Ó L Ô Û Ü ~ Ä, Õ Æ Ö ¼ Ø Ë Ù «h Œ è â Š X&Y. Î&v&x& À& &~Ä ˆèâ H(t) æç&ë V 0 (t) ˆ L ¼ Œ µ, Ú ŽÜÛ Ý Þ. Î Ú µ u # î1dbß à j Õ ¼ á. Å í, è â f(x) «Û Ü Ó ¼ Œ è â Ç È, Õ, Ï Ó ¼ Û Ý Þ, &â³&ã& & ïð & ÉÊ¼Èã&Ë. Î&, Ï&½&ž&~Ä&ä& ¼&Œèâ ƒ&ä È ã Ë. Ž& &æ &ç&è ˆèâ H(t) ~Ä±¼éyê{, ˆ Õ&vð &ë¼üì&í&î&ï&, ð Ë X Y1dR¼ ˆ Ñ â Ë, ñ (2.49) dr¼ c n, ò «ó ô, ` õ Ê p ~ ¼wŒ è â Ñw Ü. š wº w¼, ½wžwÛwÝwÞw w wwìwíwîwï, ˆ è âw «, V Ó û ü Ê Œ. ø ù, q õ ú, Ø Ã ˆ Ë. Î v x ² ³ ë ¼ÜÛ ý þ. ±í&û&ý&þ& & ÿì&í&î&ï& & &&Û&ý&þ& & &u &# î_dêß&à&jõœ¼ á, ž à + ± ¹ u # î ¼.

20 Z\[\]_^ 2.3.5 Û Ý Þ ƒ ± í îw Y ñ ¼ Ë. X Y î Ów, wu w# îwï ÓwwŒ è â ï ð Ów X 1. R Ë (2.12) Ž À µ Ö t ¼ $ â. š Ô, Ï ¼ Œ è â V (t), ð { t 2m+1 m = 0, 1, 2, 3, } V (t) = m=0 g1d, a m Ž Ñ â. Î v x h Î Ì ¼ Ð Û Ý Þ. h Û Ý Þ Ú (2.22), q r ~ Ä á Ž V (t) ¼ æ ç Ë, Õ (2.50) a m t 2m+1 (2.51) V 0 (t) = t (2.52) á Ž ˆ Š, h Š i ˆ L[Φ(t; q)] = Φ(t; q) t (2.53) g1d, C 1 Ž ¼ ñ { ñ (2.54), (2.39) Ë Ž L (C 1 ) = 0 (2.54) â. Û Ý Þ (2.51) (2.39), ˆ è â H(t) V Ó ~ Ä Ž t V m (t) = χ m V m 1 (t) + 0 H(t) = t 2κ (2.55) τ 2κ R m (V m 1 ) dτ + C 1 g1d, â C 1 {R Ú (2.40) Ê Œ. ² ³, κ 1, V m (t) t 1. Î Ó ª Û Ý Þ (2.51). Ô ù, κ 1, V m (t) Ó ˆ t š 3, Ô, ð È ã â Ö ½ ¾, t 3 ¼ Ñ â Ž ä, Ó «ó ô. Î ƒ ì í î ï Ó ª. Ž Û Ý Þ (2.51) ì í î ï V Ó, U κ = 0. ¼ ˆ è â H(t) = 1 (2.56)

a. Ü ñ ß ß 2 21 «, ² Ê Œ., Ï! V 1 (t) = 1 3 t3 ˆ " ¼, ¼ m È ã Ë Ž V 2 (t) = 1 3 (1 + )t3 + 2 15 2 t 5 V 3 (t) = 1 3 (1 + )2 t 3 + 4 15 2 (1 + )t 5 + 17 315 3 t 7. V (t) m V k (t) = k=0 m n=0 µ m,n 0 ( ) ( α 2n+1 t 2n+1) (2.57) g1d, α 2n+1 ƒ Ë (2.12) Ñ â, è â µ m,n 0 ( ) Œ ¼ µ m,n 0 ( ) = ( ) n m n j=0 ( n 1 + j j ) (1 + ) j (2.58) Ë r ˆ Ë b i # $ æ ç V 0 (t) ˆ Š L ˆ è â H(t) %RÊ Œ, ² ³ & à ¼zyB{B~ Ä ˆ á â Ë. Ë! Ì Í ¼, (2.57) ˆ á â., è â µ m,n 0 ( ) ˆ Ý Þ À ˆ è l ' â n,» W µ m,n 0 ( 1) = 1, n m (2.59) ± í À ñ (2.12),» W b)(r lim m + µm,n 0 ( ) = -. / { 1, 1 + < 1, 1 + > 1 (2.60) ï ð. Õ, = 1, { ñ (2.59) ñ (2.57) V (t) = V pert (t) (2.61) š Ô, Ë (2.12) * Ž ñ (2.57) = 1 ¼ À Ì. Lyapunov +, á â î ï ð ¼wË (2.15) Adomian wë î ï ð ¼wË (2.17), Ï wô š. Ô, ñ Ü (2.57) -. ± / î Lyapunov +, á â î Adomian Ë î! ƒ ¼ Ë, ` õ ± i 0 1. ñ3254 V 3 (t) 6 4/15 7585956 2/15, 5:5;, <575=5>5?.

0 0 V ª ß Ü ß V 0 22 Z\[\]_^ Ë! @wk ¼w, $ â (2.57) wñ â AwÖ ˆ á â. ñ (2.60), $ â (2.57) % ' ¼ Õ Ú 1 + < 1, ð 2 < < 0 % ' ( % ' ( ŵk B ¼w, $ â (2.57) w%w'w(w) C AwÖ Ë. D 2.1 w2, ( 2 < < 0) œëfefœè Öœä, $œâ (2.57) )FEœà. ²œ³FF, $œâ (2.57) ) Ž 2 0 t < ρ 0 1 g1d, ρ 0 3/2 Ë (2.12) % ' G H. š Ô, ( 2 < < 0) È ä, $ â (2.57) Ü ' À ( ) 0 t < + % ' Æ É Ê Ë V (t) = tanh(t). ƒ ˆ! " Ë # î Ó, ½ ž ~ Ä Ë, ² ³ Ï Í I Ö $ â (2.57) % ' ( ). š Ô, ˆ á â v x Ú Í I Ö $ â Ë % ' ( ) ¼ J K H. ( ) 2. L R # $ Ü ( 2 < < 0) Ö 0, {BŒ è â (2.50) ï ð ¼ $ â Ë (2.57) Ü ' À 0 t < + VwÓ ŵ, M ( 2 < < 0) N Ë ž, È ã â ž Ow! ƒ w P Ê ¼w T # $ Ü. ê ë ± è â (2.50) QR î Lyapunov +, á â îw Ë î ï ð ¼ Ë (2.12) ± ± i 0 1, M g Ç È ³ Ó š. Ô, ˆ Õ ~ Ä ± ¼ Œ è â, ` õ ± ˆ ² Ç È V (t). Å S í, ð Ó Ë T (2.5) (2.6), C Ï! U V è * + Adomian V (+ ) = 1 æ ç Ë (2.52) q r Ó Î À. ½, $ â * À ˆ è ¼ ( ) S Õ, $ â Ü (2.50) ' À ( ) 0 0 t < + Ó ž ˆ ² Ç È V (t). š Ô, {BŒ è â lim t + 1 = 0, m 1 (1 + t) m { (1 + t) m m = 0, 1, 2, 3, } % '. (2.62)

Ü V Ü X Y ¼ è â 2 23 t + Ž ˆ è Ë. Ó Í W X V (t) = m=0 V (t) Ï X Y» b m (1 + t) m (2.63) g1d, b m Ž Y Œ Ñ â. Î v x h Î Ì ¼ Z Ð Û Ý Þ. Û Ý Þ (2.63), {R Ú (2.6) V è * + (2.7), q r, ~ Ä V 0 (t) = 1 1 1 + t (2.64) á Ž V (t) ¼ æ ç Ë, Õ á Ž ¼ ˆ g1d, C 2 Ž Š L[Φ(t; q)] = (1 + t), h V m (t) = χ m V m 1 (t) + Š i ˆ Φ(t; q) t + Φ(t; q) (2.65) ( ) C2 L = 0 (2.66) 1 + t â. L Œ (2.65), Ø T (2.39) Ë Ž 1 + t t 0 H(τ) R m (V m 1 ) dτ + C 2 1 + t, m 1 g_d, œ â C 2 { œ œúœ (2.40) ÊœŒ. œ Û Ý Þ (2.63) œ œ œ Øœ FT (2.39), ˆ è â H(t) ~ Ä ¼ ñ H(t) = 1 (1 + t) κ (2.67) g1d, κ Ž ' â. ² ³, κ 0, Ø T (2.39) Ë / ln(1 + t) 1 + t, g Ó ª Û Ý Þ (2.63). κ > 1, Œ (1 + t) 2 Ó ð V m (t) d, ` õ, (1 + t) 2 ¼ Ñ â [ Ž ä, ð È ã â Ö ½ ¾ \ ½ î «ó ô. Î ] ì í î š ï& &. Ô, Ž& &Û&Ý&Þ (2.63) &ì&í&î&ï& & V&Ó, U Ç κ = 1. Î, &² Ê Œ ¼ ˆ è â H(t) = 1 1 + t (2.68)

~. ö V Ü Ü f õ 24 Z\[\]_^, Ï ^! V 1 (t) = 1 + t + 2 (1 + t) 2 (1 + t) 3 ( V 2 (t) = 1 + 7 ) 1 12 2 (1 + ) + 1 + t (1 + t) 2 (1 + 72 ) 1 (1 + t) 3 + 10 2 3(1 + t) 4 5 2 4(1 + t) 5. ², V (t) m È ã Ï X Y» V (t) 2m+1 n=0 β m,n ( ) (1 + t) n (2.69) g1d, β m,n ( ) Ž A Ö ¼ Ñ â. Ë! Ì Í ¼, ² ³ & r ˆ à ¼zyB{B~ Ä ˆ á â Ë. $ â (2.69) _ ±w ẁë. Žw ä b $ â (2.69) ¼ c d, ë f g h wÿw wµw$ â ( V (0), V (0), V (0) ) ¼w%w'. ² ³ Ý Þ, â ¼ È ãwëwãwˆ V (0) = 1, SwÕwâ Ó Ý Þ v x ˆ ¼ i j Õ k ˆ ~ Ä. M, V (0) V (0) A Ö. Ó Í U R Xw2 Swŵ w ¼ Ë wnw, gw w ¼ V (0) w$ âw%w'. Žw J, ² ³wÚ R Ž µ Ö V (0) ¼ l m n o. Œ ê Ý 2.1, À R, ¼ V (0) $ â % ' Æ T ñ. V (0) )pr R ( ) Ž Ú q r s. ² ³ Ú Î Ï Ú p Ž V (0) p. ž&q&r, â& &2&à&$â&Ë &ˆ& &(&) R. {êë (2.69) ï ð ¼ V (0) V (0) pr D 2.2 S 2. ž)(bq, $ â (2.69) ï ð ¼ V (0) V (0) Ü 3/2 1/2 &%&'. Ì, Ä&(&) 3/2 1/2 d 5 ÀÓ& Ë&, $â (2.69) ï ð¼ V (0) V (0) &$â &%&'&Æ& & ¼ÉÊË 0 2, X 2.1 &X 2.2 S&2. ˆ&ktB¼&, $â&ë &%&'&*&+tta&ö Ë. {ê$â (2.69) ïð¼ V (0) V (0) $ â = 1 % ' ä u, ú X 2.1 X 2.2. Î k v à, ² ³ Ï ½ ž ˆáâ ÍtI&$â&Ë (2.69) &%&'&*&+. Ô ù, ²³tt, $â V (0) V (0) % ', $ â Ü (2.69) ' À ( ) 0 0 t < + % ' š. Ô, Œ ê 2.1, S ˆ Á Â Î Ÿ % ' $ â Ž µ Ë. Ì, = 1, $ â Ü (2.69) ' À ( ) 0 0 t < + % ' Æ É Ê Ë, ú X 2.3., ½ ž w Ã ã ¼ pr, ñ3x5y, z5{5 5}5~.

0 2 25 ž ² ª È ¼ ˆ ( ) Ü. Ô ˆ ( ) ~ Ä À Ë Ç, Ï Ê p ¼ $ â Ë % ' š. Ô, ˆ á â Ü u # î1dbß à j Õ ¼ á. 2.2 H(t) = 1/(1 + t), ƒ (2.69) V (0) ˆ V (0) 5Š 5Œ5 Ž Ž V (0) 20 û5 5 ; 20 û5 5 V (0) 2.1 -- h, (2.69) š V (0) œ û = 1/2 = 3/4 = 1 = 5/4 = 3/2 5 0.062 500 0.001 953 0 0.001 953 0.062 500 10 0.001 953 1.9 10 6 0 1.9 10 6 0.001 953 15 0.000 061 1.9 10 9 0 1.9 10 9 0.000 061 20 1.9 10 6 1.9 10 12 0 1.9 10 12 1.9 10 6 25 6.0 10 8 1.8 10 15 0 1.8 10 15 6.0 10 8 30 1.9 10 9 1.7 10 18 0 1.7 10 18 1.9 10 9 35 5.8 10 11 1.7 10 21 0 1.7 10 21 5.8 10 11 40 1.8 10 12 1.7 10 24 0 1.7 10 24 1.9 10 12 2.2 -- h, (2.69) š V (0) œ û = 1/2 = 3/4 = 1 = 5/4 = 3/2 5 3.312 500 2.138 672 2 2.251 953 6.937 500 10 2.089 844 2.000 278 2 1.999 516 1.699 219 15 2.004 333 2.000 000 2 2.000 001 2.013 977 20 2.000 183 2.000 000 2 2.000 000 1.999 42 25 2.000 007 2.000 000 2 2.000 000 2.000 023 30 2.000 000 2.000 000 2 2.000 000 1.999 999 35 2.000 000 2.000 000 2 2.000 000 2.000 000 40 2.000 000 2.000 000 2 2.000 000 2.000 000

Ú à Ú ß a Ü 26 Z\[\]_^ 2.3 -- h= 1, m ž Ÿ (2.69) (2.8) œ t 10 û 20 û 40 û 60 û 5 5 1/4 0.244 9 0.244 9 0.244 9 0.244 9 0.244 9 1/2 0.462 1 0.462 1 0.462 1 0.462 1 0.462 1 3/4 0.634 9 0.635 1 0.635 1 0.635 1 0.635 1 1 0.751 6 0.761 6 0.761 6 0.761 6 0.761 6 3/2 0.908 2 0.905 3 0.905 1 0.905 1 0.905 1 2 0.972 0 0.964 4 0.964 0 0.964 0 0.964 0 5/2 0.998 2 0.987 0 0.986 6 0.986 6 0.986 6 3 1.008 2 0.995 0 0.995 0 0.995 1 0.995 1 4 1.011 0 0.997 9 0.999 2 0.999 3 0.999 3 5 1.008 2 0.997 3 0.999 7 0.999 9 0.999 9 10 0.998 4 0.996 8 1.000 3 1.000 1 1.000 0 100 0.998 7 0.999 8 1.000 1 1.000 0 1.000 0 Ó Ö î Lyapunov +, á â î Adomian Ë î! U ¼ Ë (2.12), 3/2 1/2, $â Ü (2.69) 'À&(&) 0 0 t < + ò&%&'&æéê Ë. ª Ô, $â&ë (2.69) Q (2.57) ±&ˆ&, #t$ êët«&ât Â&ã&Ï&Õ 't t t 0 t < + ± ² Æ ³ Ë. µ ª, Ö ¹ Î Ì õ º,» ¼ ½ (2.62) Q» ¼ ½ (2.50) ¾, ª Ô, À ¾ Á Â Ã Ä Å Æ. Ç È É Ê Ë Ì Í Î Ï Ð ½ Æ (2.69) Ñ Ò Ó Æ (2.12) Ô Õ Ö Ø. Ù = 1 Û Ü Ð ½ (2.69) Ô 10 Ý Å Þ Æ, Á ß 1 1 + t = 1 t + t2 t 3 + Ö á â ã V (t) t 1 3 t3 + 2 15 t5 17 315 t7 + 62 2835 t9 + Ò Ó Ð ½ (2.12) Ö á â ã ä å Î æ! 3. çè é ê é ë ì í î lim exp( nt) = 0, n 1 t + ª ï, ð» ¼ ½ { exp( nt) n 0 } (2.70) ßòñôóôõôöô ôøôùûú V (t) t 1 3 t3 + 2 5 t5, üôýôþôÿ 2/5 ù 2/15, þ, õ.

Y m D D / - - - - [ Ç 2 27 Öt¼t½ t tţá. "!#$ (2.7), «%t»t¼t½'&»t¼t½ **- (2.50) ¾ ß. (*) V (t) + Á*, Ï.0/ V (t) = n=0 c n exp( nt) (2.71), c n Ø ½. µ*1*2*3 ¹*4*5 Ö*6*7*8*9*:*;. <*= 9*:*; (2.71), (2.6) Ú - (2.7), >*?*@*A, B*C V (t) Ô*E*F*G*H Æ, I*J V 0 (t) = 1 exp( t) (2.72) *K*L*M*N*O*P L[Φ(t; q)] =, ¹*O*P*+ Á*N*Q Φ(t; q) t + Φ(t; q) (2.73).0/, C 3 *R*S*T ½. U*B Ö m Ý*V*W*X*Y (2.39) Ô Æ L [C 3 exp( t)] = 0 (2.74) V m (t) = χ m V m 1 (t) + exp( t).0/, R*S*T ½ C 3 L ¼ ½ H(t) *, Ï*V +C 3 exp( t), m 1 (2.40) *Z. <*= t 0 exp(τ) H(τ) R m (V m 1 ) dτ 9*:*; (2.71) Ú*[ Ý*V*W*X*Y (2.39), K H(t) = exp( κ t) (2.75) µ*\, κ Î*] ½. Ê*^*_*`, a κ 0, [ Ý*V*W*X*Y (2.39) Ö Æ*b*c t exp( t) Ç*d*e Ç*g*` ã, µ 9*:*; (2.71). a κ 2, ¼ ½» exp( 2t) F*f (2.39) Ô Æ, h*i, i*j Å Þ Ý ½****, exp( 2 t) ã Ö Ø ½*F*f Ý*V*W*X À*k*l. µno3pqrstu. vtï, 3wxy9:; (2.71) zxypqrstu, *{* κ = 1. µ*} Î Ã *Z*3*~*B Ö*K*L ¼ ½ H(t) = exp( t) (2.76) ß ø (2.8),V (+ ) ƒ5ÿ 1. ˆ Š Œ, Ž5ÿ (2.70) Ž5ÿ (2.50).

@ Ú Û ç [ Ý e 28 ' ' *, *š ï* *œ V 1 (t) = 2 e t + e 2t 2 e 3t V 2 (t) = 2 ( 1 + ) ( e t + 1 + ) e 2t (1 + ) e 3t 2 2 2 + 2 2 e 4t 2 4 e 5t ~*B Ã, V (t) Ô m Ý Å Þ*. ** V (t) 2m+1 n=0 γ m,n ( ) exp( nt) (2.77).0/, γ m,n ( ) *š*ž* Ö Ø ½. - (2.77) Ÿ Î b c K L ½ Ö Ð ½ Æ. 3 Ì Í Ü Ð ½ (2.77) ± ² N Ô, Ê ^ g V (0) Ú V (0) Ö M,, ª 2.3 h «. < = µ M, _ ` Ô Á Â, à U B Î ± â ² ³ µ ³ Ö M ±. Ô Á Â Å Þ Ý ½*¹ *¹*º,,*ª 2.3 h*«.,*»* ¹ Ô Á** * 0¼¾½ Ô*, - (2.77) g Ö V (0) Ú V (0) ÔtÐt½ÀÁ.?Â, <= ZÃ 2.1, à ^ ZSÄÀÁÅ V (0) Ú V (0) Ô*Æ*Ç*., 2.4 Ú 2.5 h*«, a = 3/2È 5/4È 1È 3/4 1/2 V (0) Ú V (0) À*Á*É*Æ*Ç*. *œ*ê*ë Ö*Ÿ, a = 1, Ð ½*À*Á œ*ì*í. ï*î, Ï*Ð V (0) Ú V (0) À*Á, ~*B V (t) Ô Ð ½ Æ (2.77) Ñ**]*Ò* * 0 t < + ¼¾À*Á*Å*Æ*Ç Æ (2.8). 5*,, a = 1, Æ (2.77) g Ö V (t) Ô Å Þ Æ Ñ*Æ*Ç Æ (2.8) Ó œ*,, 2.6 h*«. Î*Ô*? º, Õ*Ö* *Ø*Ù*Ú ¾M, À * X*Ü Ã Ì Í U Ð ½ Æ Ô*À*Á*N Ö. 2.4 -- h Þßàáâ, (2.77) ãäå V (0) æá ÿ = 1/2 = 3/4 = 1 = 5/4 = 3/2 5 0.031 250 0.000 977 0 0.000 977 0.031 250 10 0.000 977 9.5 10 7 0 9.5 10 7 0.000 977 15 0.000 031 9.3 10 10 0 9.3 10 10 0.000 031 20 9.5 10 7 9.1 10 13 0 9.1 10 13 9.5 10 7 25 3.0 10 8 8.9 10 16 0 8.8 10 16 3.0 10 8 30 9.3 10 10 8.7 10 19 0 8.7 10 19 9.3 10 10 35 2.9 10 11 8.5 10 22 0 8.5 10 22 2.9 10 11 40 9.1 10 13 8.3 10 25 0 8.3 10 25 9.1 10 13

ç õ Ý Ý 2 29 2.5 -- h Þßàáâ, (2.77) ãäå V (0) æá ÿ = 1/2 = 3/4 = 1 = 5/4 = 3/2 5 2.375 000 2.041 016 2 2.076 172 3.500 000 10 2.026 367 2.000 083 2 1.999 854 1.909 180 15 2.001 282 2.000 000 2 2.000 001 2.004 211 20 2.000 054 2.000 000 2 2.000 000 1.999 825 25 2.000 002 2.000 000 2 2.000 000 2.000 007 30 2.000 000 2.000 000 2 2.000 000 2.000 000 35 2.000 000 2.000 000 2 2.000 000 2.000 000 40 2.000 000 2.000 000 2 2.000 000 2.000 000 2.6 è -- h= 1 â, éêë (2.77) ìíîë (2.8) æïð t 5 ç ñ 10 ç ñ 15 ç ñ 20 ç òóô 1/4 0.244 9 0.244 9 0.244 9 0.244 9 0.244 9 1/2 0.461 9 0.462 1 0.462 1 0.462 1 0.462 1 3/4 0.634 2 0.635 1 0.635 1 0.635 1 0.635 1 1 0.759 6 0.761 6 0.761 6 0.761 6 0.761 6 3/2 0.902 0 0.905 1 0.905 1 0.905 1 0.905 1 2 0.961 2 0.963 9 0.964 0 0.964 0 0.964 0 5/2 0.984 5 0.986 6 0.986 6 0.986 6 0.986 6 3 0.993 7 0.995 0 0.995 1 0.995 1 0.995 1 4 0.998 8 0.999 3 0.999 3 0.999 3 0.999 3 5 0.999 7 0.999 9 0.999 9 0.999 9 0.999 9 10 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0 100 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0 2.3 H(t) = exp( t) ö, øù (2.77) ú V (0) û V (0) üý þÿ ú V (0) ç 10 ñ ; ú V (0) ç 20 ñ ; ú V (0) ç 10 ñ ; þÿ ú V (0) ç 20 ñ

à 2 e -. e _ 30 ' ' œ Ê Ë Ö Ÿ, Ð ½ Æ (2.77) ] Ò 0 t < + ¼ Á Â. S Ä Ù 2.4 2.6 Ñ 2.1 2.3 '&, Ê^_`, tð~tö, ÐtÆ (2.77) & Æ (2.69) À*Á*œ*Í. a = 1, i*j Æ (2.77) Ô 10 Ý Å Þ Æ*Ñ Ñ*Æ*Ç Æ*Ó œ*. v ï, Ð Æ (2.77) &Æ (2.69),., á*h*%, Ð Æ (2.69) & Æ (2.57). Ù, S X ¼, Õ Ö C Î Ò Ö!" Å#*M*N$% Ö Æ. Ê*^*_*` -***, V (t) Ô m Ý Å Þ Æ (2.77) *> V (t) 1 + 2 m n=1 [( 1) n exp( 2nt)] µ m,n 0 ( ) 2 a - ß [( exp( t) 1 + ) + ] m 2 2 exp( 2t) (2.78).0/ g*`, µ m,n 0 (x) (2.58) Z&. *Ë' Ö*Ÿ, µ m,n 0 ( ) ( Î) N*Q (2.59), = 2, (2.78) *. <*= V (t) 1 + 2 m ( 1) n exp( 2nt) + ( 1) m+1 exp[ (2m + 1) t] (2.79) n=1 *œ*ê*ë Ö*Ÿ, Æ*Ç Æ (2.8) +, Ð V (t) 1 + 2 ( 1) n exp( 2nt) (2.80) n=1. * * 0 < t < + ¼¾À*Á*Å*Æ*Ç Æ, -*Ÿ, t = 0. 3*?,¾*c45 ã g 1 /0 1, 1*? ( 1) m+1 exp[ (2m + 1)t] Å Þ Æ (2.79) *]*Ò* * 0 t < + ¼¾À*Á*Å*Æ*Ç Æ. 6*Å 3 Ý Å Þ Æ Ñ Ñ*Æ*Ç Æ*Ó V (t) 1 2 exp( 2 t) + 2 exp( 4 t) 2 exp( 6 t) + exp( 7 t) (2.81) œ*,,*ª 2.4 h*«. ß3ö5 5ø ñ 9 ý ù þ, õ (2.79) ý 7 8 m 2 ( 1) n exp( 2nt) n=1 m 2 ( 1) n exp( nt) n=1. ö5 5ø (2.80) : ; <ñ, õ = >.

_ L S * 3, - õ * e / ô þ - ú / / g 2 31 Å Þ Æ /JIK 2.4 3?A@ABù (2.81) CADAEù (2.8) ú F G ú ç òóô 3 ñ (2.81); H (2.8) g*` ãtö'mn IKL Ë ÇOP ln(1 + t)/(1 + t),. tòtóãq t exp( t), * 3œÉtÎRtÖtÅtÞtÆ, 1 3tÎ X t sin t, t cos t / g ` hutuvuw ã Ò Ó Æ. Ù Ú XU UXUY É 19 ZU[ ÖU\U]U^U_U`,, Lindstedt [52] È Bohlin [53] È Poincaré [54] È Gyldén S.b [55]. a'm*k ^_` Îc +,, Lighthill [56, 57] È Malkin [58] È Kuo Ú [59, 60] Tsien [61]. -*Ÿ, a t +, ln(1 + t)/(1 + t) Ú t exp( t) *d, v ï, à ^ Çe N Ò Ó*X ht ÖVW ã. h*i, 9*:*;*t*u* *kf Ù*8'M Ög&h. t exp( t) ã*çij Ò Ó*X ht ÖVW ã, Ê*^k*g V (t) l k*, Ï {t m exp( nt) m 0, n 1} (2.82). mtðtñ (2.72) ~töefghn, Ñ (2.73) ~töklmnop, -o ðj Ö*K*L Ê*^ *Õ*Ö*Ú Þ Ö*X -p m+1 V (t) 1 + 2 n=1 H(t) = 1 (2.83) œ*~*b Ö V (t) Ô m Ýq Þ,. -*** *> m+1 n k=0 σ m,n,k 0 ( ) [( 1) n ( nt)k k! ] exp( nt) (2.84)

} Ý Û * Ú - Ñ k. k e Ú à Û / D Ú - 32 ' '.0/ σ m,n,k 0 ( ) = 1 [ ] µ m,n+k 0 ( ) + µ m,n+k 1 0 ( ) (2.85) 2 g*` *Ë' Ö*Ÿ, µ m,n 0 ( ) ( Î). <*= ~*B Ö V (0) Ú V (0) Ô ¾M, Ê ^_`, a 2 < < 0, Ð (2.84) ]Ò 0 t < + ¼"ÀÁÅÆÇ n (2.8),, 2.7 h*«. 2.7 è -- h= 1 â, V (t) æéêë (2.84) ìíîë (2.8) æïð t 10 ç ñ 20 ç ñ 40 ç ñ 50 ç ñ 1/4 0.244 9 0.244 9 0.244 9 0.244 9 0.244 9 1/2 0.462 1 0.462 1 0.462 1 0.462 1 0.462 1 3/4 0.635 1 0.635 1 0.635 1 0.635 1 0.635 1 1 0.761 6 0.761 6 0.761 6 0.761 6 0.761 6 3/2 0.905 1 0.905 1 0.905 1 0.905 1 0.905 1 2 0.964 0 0.964 0 0.964 0 0.964 0 0.964 0 5/2 0.986 6 0.986 6 0.986 6 0.986 6 0.986 6 3 0.995 3 0.995 0 0.995 1 0.995 1 0.995 1 4 0.999 0 0.999 3 0.999 3 0.999 3 0.999 3 5 0.997 5 0.999 9 0.999 9 0.999 9 0.999 9 10 1.002 1 0.998 2 0.999 9 1.000 0 1.000 0 100 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0 œrs Ö Ÿ, S X ¼, V (t) l kt 8j Ö (2.50)È (2.62)È (2.70) Ú (2.82), uva45ïw}tîn. ~B!, Ê^ p œtòtð nuxyn (2.57)È (2.69)È (2.78) Ú (2.84). ÃQz, à ^{ l**]*ò* * 0 t < + ¼"ÀÁÅtÎÒÆÇn V (t) = tanh(t). 3?, - (2.57) ÙtÒn "q Ì, v ï*ÿ*ì~ Ö, v U Î*Ò Z 2 < < 0, à Î*Ò** * 0¼¾À*Á. Õ*Ö0&J 2.3È 2.6 Ú 2.7, Ê*^*_*`, * Ö Ðn (2.78) &J**S Ö Ðn (2.69) Ú *] ã Ö Ðn (2.84), v ï*ÿ*ìƒ Ö.? Ðn (2.84) &Ðn (2.69). Ù*Ò*5*P! ¾3, *S*X * ¼, Î Ò + } În Ö# M N$% n* k\]j Ö, 1?, ð Î*Ò Öl!"qn. ˆ h ð*5*p#*t Š, Â*Æ*Çn JŒ. -! ¾3Ux Õ*Ö* Ø*hTtÖ e Ú ¾M, Ç*Z*~*B Ö Ô* *, C Ö, sž Ø0J*S*X g Ö Ðn Ô*À*Á* * À*Á#*$. i*j *¹*º*À*Á* * ÖÐ, Ñ* *Õ*Ö*C e Î*Ò Ö, œ*é Î*Ò Ö Ðn. a*4*5*ñ*>*«*3*9*:*;*t*u p*q r*s*t*u**c*½*e*f*g*hn ÈK*L*M*N*O*P K*L**h *É Ö *Ð ð. òóô

z Ü à ^ þ Ž / Ú Ø * Û Ú à Ü - Z 2 33 2.3.6 *q -- h,táh%, SX - š_tötîò 'M, ÕÖ #dk g L œ htd Ý V W X Y, 1? Î 8 &Ÿžg& Ö. v ï, j **hžn*x, *S*X*1*2*3 Î* *b*c*k*l* Ö Ðn.¾ Ðn Ô*À*Á* * À*Á#*$šž*KL*, h*i, *I*Õ*Ö a*c*½ Ô* Ú Ë*¹*º. Ù*1*2*3 Î*ÒsŽ Ø Ðn Ô*À*Á* * À*Á*#*$ Ö *Ü., Ê*^ ð Î*8 ä åj Ö ß x Ð Ô*À*Á* * *Çi* *I Ú Õ Ö Î Ò K L ËsŽ. a S X Ô N*1 2 Î*Ò i È Ö*Ã*N. g` œrstöÿ, µ m,n 0 ( ) ÔZ& (2.58) Ì SXœÉ. tð n (2.57)È (2.78) Ú (2.84) /. ª Ö*Ÿ, æ Ö*Z&l 1«Ö ã Ã ±²³*œ*É. * 3 ¾Ù Î, µ Ð Z& -* g 1 1 + t = 1 t + t2 t 3 + = lim a x = 1 + + t < 1 Ú 1 + < 1, i h*i, * * m + n=0 x = 1 + + t 1 1 + t = (1 x) 1 < t < 2 1, 2 < < 0 1 1 + t = 1 x = ( 1 + x + x 2 + x 3 + ) = [ 1 1 + t = lim m + m ( 1) n t n, t < 1 (2.86) n=0 ] m (1 + + t) n n=0 1 < t < 2 1 ( 2 < < 0) (1 + + t) n ß ¹, º» = ˆ ¼ ½ ¾, À5ÿ Á Â Ã Ä Å Æ º» Ç È = ˆ É Ê Ë5ÿ Ì Í Î Ï Ð Ñ, Ò Ó ; Ô Õ Ö Ö ä å =, Ø õ, Ù Ú = Û Ü Ý 7 Þ ß. à ¹, á â ã = ˆ.

Ù - ^ Û - p Ø * * 34 ' ' ¼J. ï*î, æs*-.0/ = = k=0 m (1 + + t) n n=0 m n=0 k=0 k=0 n=k ( ) n n (1 + ) n k ( t) k k ( ) m m n (1 + ) n k k t k k ( ) m m k k + i = ( 1) k t k ( ) k+1 (1 + ) i k = = [ m ( 1) k t k k=0 i=0 ( ) k+1 m k m ( 1) n t n µ m,n 1 ( ) n=0 µ m,n 1 ( ) = ( )n+1 (2.87) Ñ*Z& (2.58) &J, ç ( ) ] k + i (1 + ) i i=0 m n j=0 i ( n + j œ*é*, Ïè Ø j ) (1 + ) j (2.87) v ï * * 1 1 + t = lim µ m,n 1 ( ) = µm+1,n+1 0 ( ) (2.88) m m + n=0 µ m+1,n+1 0 ( ) [( 1) n t n ] (2.89) 1 < t < 2 1 ( 2 < < 0) ¼. a = 1È = 1/2 Ú = 1/50,. ÀÁ SÄ 1È 1 < t < 3 Ú 1 < t < 99. é*ä!, a *d*,. À*Á* * x 1 < t < 1 < t < + Ú v ï, Ð (2.89) Ô*À*Á* * *Çil ¾K*L* sž Ÿ, ätå~töz& µ m,n 0 ( ) SXtÖ'¼. ê*ðé*ärs Ö œ, 3 ätåë!

þ ÿ Ú e - ^ * 2 35 1«Ö ã X Ô*N Z*Ã ±²³*œ*É. Ù*Òìi, 1*Ãíîz*>*«*3*S Ã*N. B ðz*%'m, ç œ*é*, Ïï*Ò g&ð*z*ã. ñò 2.2 ó ôé α (α 0, 1, 2, 3, ), ë ìõ (1 + t) α = lim m m + n=0 µ m,n α ( ) ( α n ) t n (2.90) ö Jø 1 < t < 2 1 ( 2 < < 0) ùjúû, ü ý ( α n ) = α(α 1)(α 2) (α n + 1) n! µ m,n α m n ( ) = ( )n α j=0 ( 1) j ( α n j ) (1 + ) j (2.91) x = 1 + + t. x < 1 1 + < 1, 1 < t < 2 1, 2 < < 0 žð *Ã [62], x < 1 1 + < 1, ( ) + α (1 + t) α = ( ) α (1 x) α = ( ) α n=0( 1) n n ( ) + α = ( ) α n=0( 1) n (1 + + t) n n ( ) m α = lim m + ( ) α n=0( 1) n (1 + + t) n n x n

* 36 z m m ) ( α ( ) α n=0( 1) n (1 + + t) n n ( ) m n ( ) α n = ( ) α n=0( 1) n (1 + ) n j j t j n j=0 j ( ) ( ) m m α n = ( ) α t j j=0 n=j( 1) n (1 + ) n j j n j ( ) ( ) m m j α i + j = ( ) α t j ( 1) i+j (1 + ) i j i + j j j=0 = ( ) α m j=0 j=0 i=0 ( ) ( ) α α j ( 1) i+j (1 + ) i j j i t j m j i=0 [( ) ] m m j ( ) α α j = t j ( 1) i (1 + ) i ( ) j α j i = m n=0 µ m,n α ( ) µ m,n α [( α n i=0 ) t n ] ( ) m n α n ( ) = ( )n α ( 1) j (1 + ) j j j=0. uv & (2.91) α 0, 1, 2, 3, ð,!"#$%&' < α < + ( % ). # * ' k, (2.91), + µ m,n k ( ) = ( ) n k m n j=0 ( n k 1 + j j ) (1 + ) j (2.92)!, -. µ m,n 0 ( ) / (2.58) 0 1 µ m,n 1 ( ) / (2.87). 2 0 34, # 5 6 & ' α (, + ), + 7 8, # 5 6 % 9 : * ' n, + lim m + µm,n α µ m,n α ( 1) = 1 (2.93) ( ) = 1, 1 + < 1 (2.94)

y C Q Q y C Q c ; C D 2 < =?>?@?A?B 37, E F / (2.58) (2.91), # * ' l 0, + µ m,n l ( ) = µ m+l,n+l 0 ( ) (2.95) (2.93) GHIJKL. 1+ < 1, EF / (2.91), #5M%9:*' n > 0, + lim m + µm,n α k=0 ( ) ( ) + α n = ( ) n α ( 1) k (1 + ) k k ( ) + α n = ( ) n α ( 1 ) k k k=0 = ( ) n α [1 + ( 1 )] α n = 1 3.N O P L G, RQ (2.58) / L µ m,n 0 ( ) 0 1 (2.87) / L µ m,n 1 T U V ( ) S (2.91) α = 0 α = 1 L W X. $ % Y Z, [ \ ] ^ _ 34R. ` a T b c d ' (2.57)e (2.78) (2.84) L f g h % ) h. Q n ' (2.57)e (2.78) (2.84) f g h i 2 j k l m. o $ p q, l r s r x ' L t u ' G v l L. E F h w (2.60), 1 + < 1, # 5 6 l L % 9 : * ' N, + lim N m + n=0 ( α2n+1 t 2n+1) µ m,n 0 ( ) = N α 2n+1 t 2n+1, x 5 6 % 9 : * ' N, z { ' } ~, ' (2.57) N ƒ Q (2.12) N l., ' (2.57) f t u ' v l h ˆ, Š % Œ Ž L M /. n=0 (2.12) 2 (α 1, α 3, α 5, α 7, ) S L l, Y, α k (k = 1, 3, 5, ) V ƒ Q V š l œ ž Ÿ Γ 0 (2.12) '. (α 1, 0, 0, 0, ) (α 1, α 3, 0, 0, ) ƒ Q

V r N Q ' V Q š. N N Q z 38 (α 1, α 3, α 5, 0, ). z (α 1, α 3, α 5, α 7, ) 9. I, ' ž Ÿ Γ ( ) (2.57) 2 V š l Œ / L z ` l (α 1 µ 0,0 0 ( ), 0, 0, 0, ) (α 1 µ 1,0 0 ( ), α 3 µ 1,1 ( ), 0, 0, ) 0 (α 1 µ 2,0 0 ( ), α 3 µ 2,1 0 ( ), α 5 µ 2,2 ( ), 0, ). (α 1, α 3, α 5, α 7, ) L 9. M L G, _ ž Γ ( ) ~ ª «'. E F h w (2.59), ž Ÿ Γ ( 1)( = 1 ) ` ~ œ L ž Ÿ Γ 0. 1 8 1 + < 1 C, ž Ÿ Γ ( ) ` ~ œ ž Ÿ Γ 0., E F h w (2.60),! [ z ` l (α 1, α 3, α 5, α 7, ). y C, ' (2.57) 2 l m ` L ž Ÿ Γ ( ) z ` l (α 1, α 3, α 5, α 7, ) L 9. o $ p q, Y ± 9 L ² ³ µ ~ $ j L z ž Ÿ. X, 9 x2 + y 2 lim (x,y) (0,0) L 2 L Y y V s ' / Œ % `L¹eº}z (0, 0) zžÿ. V»¼, S½¾žŸ y = βx, Y, β 5 M & '. H I J K, + x2 + y lim = 1 + β (x,y) (0,0) x À Q n V Á Â $ 0, 9 ~ z (0,0) ž Ÿ. ' (2.57) Ã Ä Å Æ ~ ª «'. µ m,n 0 ( ) ` L /. ` L ž Ÿ.ÇÈ Qn _, µ m,n α ( ) ÉÊ `L α / 9 L ` r Ë Ì z ž Ÿ. ~ Y, Í / µ m,n α ( ) VÏÎ Ð Ñ Ò Ó Ô Õ.. Ö σ m,n 0 ( ) (2.85) Ž M /, / x 0 σ m,n,k α ( ) = 1 2 [ µ m,n+k α ( ) + µ α m,n+k 1 ( ) ] (2.96)

N z Q s ' V V ; 2 < =?>?@?A?B 39 VÏÎ Ñ Ò Ó Ô Õ,, 1 + < 1, < α < +. J 3ÙØÚ# α (, + ), 0 n m + 1, + σ m,n,k α ( 1) = { 1, 0 k < m + 1 n 1/2, k = m + 1 n (2.97) Ü ', n k ( Ã lim m + σm,n,k α ( ) = { 1, 1 + < 1, 1 + > 1 (2.98) % 9 : * '. µ m,n α ( ) σm,n,k α ( ) % Û Ž L M /, y C, É Ê Ü Ý Þ ß Ä Å Æ. X, s ' f(z) L œ t u ' 2 0 / Î Ð Ñ à á â ã ä Õ Î Ñ à á â ã ä Õ lim n=0 m m + n=0 lim m m + n=0 f (n) (z 0 ) (z z 0 ) n n! [ f µ m,n (n) ] (z 0 ) α ( ) (z z 0 ) n n! [ f σα m,n,0 (n) ] (z 0 ) ( ) (z z 0 ) n n! Y α, µ m,n α ( ) σm,n,0 α ( ) T U / (2.91) (2.96) /. å æ f ç L Ä Å Æ, 2 H è Þ ß Ü Y Z / t u ' Ã. X, ' V (t) m n=0 V (t) 1 + 2 µ m,n α ( ) [ α 2n+1 t 2n+1] (2.99) m n=1 exp( t) [( 1) n exp( 2nt)] µ m,n α ( ) 2 [( 1 + ) + ] m 2 2 exp( 2t) (2.100) m+1 V (t) 1 + 2 n=1 m+1 n k=0 σα m,n,k ( ) [( 1) n ( nt)k k! ] exp( nt) (2.101)

L & Ö í Q ô Q 40 m ézé{ (2.57)e (2.78) (2.84) Œ éžémé/, Yé, 1 + < 1, α ÄéÅéÆ (, + ). α = π/4, é' (2.99) éã }é~éêééëéìéëéü, 2.5 $ î. = 1, α = ±1/2e ±π/4, ' (2.100) x ï L 20 z { ( ð ñ Q ò f, 2.8 $ î. = 1/2, α = ±1/2e ±π/4, ' (2.101) x ï Q ó ð ñ Q ò 20 z { f, 2.9 $ î. y C, s ' µ m,n α ( ) σα m,n,k ( ) ñ % Ž M /. 2.5 α = π/4 õ, ö??ø (2.99) ù?ú?û?ø (2.8) üïýÿþ ; = 1 ; = 1/2 ; = 1/5 ; = 1/10 2.8 -- h= 1, α = ±1/2 ±π/4, 20 (2.100) (2.8) t α = π/4 α = 1/2 α = 1/2 α = π/4 1/4 0.244 9 0.244 9 0.244 9 0.244 9 0.244 9 1/2 0.462 1 0.462 1 0.462 1 0.462 1 0.462 1 3/4 0.635 1 0.635 1 0.635 1 0.635 1 0.635 1 1 0.761 6 0.761 6 0.761 6 0.761 6 0.761 6 3/2 0.905 1 0.905 1 0.905 1 0.905 1 0.905 1 2 0.964 0 0.964 0 0.964 0 0.964 0 0.964 0 5/2 0.986 6 0.986 6 0.986 6 0.986 6 0.986 6 3 0.995 1 0.995 1 0.995 1 0.995 1 0.995 1 4 0.999 3 0.999 3 0.999 3 0.999 3 0.999 3 5 0.999 9 0.999 9 0.999 9 0.999 9 0.999 9 10 1.000 0 1.000 0 0.999 9 1.000 0 1.000 0 100 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0

@ Q d > N ' d Ã Ã Ä Q d Ã Ã r N X ï Ä V Ã L Ä x d s d Ã ï V @ Ý Ã r P Ý ñ Q & ; 2 < =?>?@?A?B 41 2.9 -- h= 1/2, α = ±1/2 ±π/4, 20 (2.101) (2.8)! t α = π/4 α = 1/2 α = 1/2 α = π/4 1/4 0.244 9 0.244 9 0.244 9 0.244 9 0.244 9 1/2 0.462 1 0.462 1 0.462 1 0.462 1 0.462 1 3/4 0.635 1 0.635 1 0.635 1 0.635 1 0.635 1 1 0.761 6 0.761 6 0.761 6 0.761 6 0.761 6 3/2 0.905 1 0.905 1 0.905 1 0.905 1 0.905 1 2 0.964 0 0.964 0 0.964 0 0.964 0 0.964 0 5/2 0.986 6 0.986 6 0.986 6 0.986 6 0.986 6 3 0.995 1 0.995 1 0.995 1 0.995 1 0.995 1 4 0.999 3 0.999 3 0.999 3 0.999 3 0.999 3 5 0.999 9 0.999 9 0.999 9 0.999 9 0.999 9 10 1.000 0 1.000 0 0.999 9 1.000 0 1.000 0 100 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0 "$#$% T b c d$($)$*, Í $&, ' ` a L µ m,n 0 ( ) $+ / (2.58) Ä Å Æ.9 Ä.: 2 0$, + Þ \.-./.0.1.2.+ g.3.4.5.6. Í.7.8 Ø ' Ã %$9.=$> É Ê.;.< l ª «' ì P. Í i.&, s 9 ' µ m,n α ( ) σα m,n.k ( ) ' 9 T b c d α ` + / ` L z ž Ÿ. $ % Y Z, [ ` a r.a % ) h.?. l & L¹eÿg h L.B.C. 2.3.7 D.E - F.G "JIJK Ó.H Tbcd #~, `a B~Y l T b c d. O$P L G, ` a V 0 (t)e ª «.] h.^._ Le ª «rjljm Ø Φ(t; q) ' (2.36) q = 1 L$Q$RTS 9, Í $U % Ü LWVX' å æ$y$z$[$\ ' H(t) ª «'. `! [ å æ f ç, U.c 2.a T b c d x ï ˆ O P L G, 'R` a L.b ' %.9.= N 3.b ' (2.36) q = 1 Ã. W Ä Å Æ.9 Ä.: Ý ~ ª «'. y C, ª «'?. l Ä Å Æ.9 Ä.:» ¼.e b ' Ý L Ÿ. fvjgjh cd b'ã l, ijjz{ (Padé approximation) k Jl j. b mon [m, n].i.j z { n=0 c n x n m a m,k x k k=0 n b m,k x k k=0, a m,k e b m,k ' é' c j (j = 0, 1, 2, 3,, m + n) ñ +. éµ, _qpéœé élqi Ä Å Æ g$h$m Ä$: ƒ +$b ', Ã. X, # j z { É Ü Ý _ ß Ü

Q ô z 9 V Ä N 9 * r ñ 42 rtsttou ƒ Q z.{ m}n (2.12) v j œ.i.j z {,, a m,n 0 } ~ t, (2.12) d [m, m].i.j z { 2 m n=0 m 1 n=0 m 1 n=0 m n=0 [1, 1]e [2, 2] 9 [3, 3].i.j z { 3t 3 + t 2, t(15 + t 2 ) 15 + 6t 2 a m,n 0 t n.w.x b m,n 0 t n, m a m,n 0 t n b m,n 0 t n, m b m,n 0 '. M L G, ~ t +, _.p. ê. ƒ Q d 9 í (2.12) [4, 4] [10, 10].i.j z { V.y V. ' (2.102) ' (2.103) œ.i.j z { 2.6 $ î. 2.6 - ƒ (2.106) ˆ ƒ (2.103) ù?ú?û?ø (2.8) üïýÿþ Š ; [4,4] Œ - Ž ; [4,4] Ž ; [10,10] Ž [50] ` a - i.j z { (.) _.p œ.i.j z { ` a ' (2.36) q = 1 Ã, v j œ.i.j z { T b c d. ² f. V. a.b. ~..<.. d q [m, n]

ñ { 9 a Q # s U Q Þ 9 Q ð V s x Q Q ñ z 9 s ' ; 2 < =?>?@?A?B 43 i.j z { m}n m A m,k (t) q k k=0 (2.104) n B m,k (t) q k k=0, ' A m,k (t) 9 B m,k (t) ' š.œ.1 z { +..ž, j.b ".I.K V 0 (t), V 1 (t), V 2 (t),, V m+n (t) (2.104) n Ÿ q = 1, 7. j.2 (2.32), [m, n] ` a - i.j z m A m,k (t) k=0 n B m,k (t) k=0 X, _.p ' A m,n (t) 9 B m,n (t) ~ $ å æ L V (t) d B ' (2.62), T U % [2, 2] ` a - i.j z { v L [1, 1] ` a - i.j z { t(12 + 16t + 7t 2 ) (1 + t)(12 + 4t + 7t 2 ) (2.105) t(168 000 + 362 880 t + 238 000 t 2 + 14 160 t 3 47 124 t 4 36 308 t 5 13 419 t 6 ) 3(1 + t)(56 000 + 64 960 t + 33 040 t 2 + 12 000 t 3 2 508 t 4 9 076 t 5 4 473 t 6 ) # ~ mtn ".I.K, a m,n 2 X, m [m, m] ` a - i.j z { 2 0 b m,n 2 m 2 +m+1 n=1 m 2 +m+1 n=0 a m,n 2 t n b m,n 2 t n.w.x '. µ %$ L G, Í $ $, a m,n 2. ` (2.106) b m,n 2 ~ ª «'. 2 (2.106).2 (2.102)e 2 (2.103). }, Í.., [m, m] ` - i.j z {.ª Ý _ ` ~ œ [m 2 + m + 1, m 2 + m + 1].i.j z {. œ.i.j z{j2 9 (2.102) 2 (2.103) ª t + }~ ê, I`a - ijjz{ (2.106) ª t + } ~ 1. y C, # 9 + : * ' m, [m, m] ` a - i.j z { (2.106) ð ñ œ [m, m].i.j z { (2.102) (2.103) Œ. X, í 2.6 $ î, [4,4] ` a - i.j z {} Rœ [4,4].i.j z { Œ.«,. } [10,10] œ.i.j z { Œ % ). W, ` j.b m V ' (2.70), #.v L [1, 1] ` a - i.j z {

9 Ä y # m C Ä Q Ä N n m ï y Q C Ä ï Q ñ Ä m Ÿ 44 rtsttou! :$ $ G ð ñ Q V (t) = tanh(t). % ). `é, véjé`éa - iqjézé{éé (.) V (0) Ã, 2 #.b ' 2 Φ(t; q) t 2 1 exp( 2t) 1 + exp( 2t) gq:, ` a - i$j z { bé'éã = t=0 3 Φ(t; q) t 3 = t=0 n=0 n=0. X V n (0) qn V n (0) q n jœ JiJjz{, TU [m, n] ` a - i$j z {. ' (2.69) 6 j z {$ K 2.10. ' (2.78) 6 2.11. ª _.p. m.±.², ` a - i.j z { [ H è Þ., V. (2.107) &T œ $i$j z { Œ gq: bé' V (0) 9 ~J J<J J d q [m, n] JiJjz{, Jž 9 9 L V (0) d d V (0) ` a - i L V (0) V (0) ` a - i$j z {, K g.: 9 d. V (0) V (0) Ã 2.10 ³ (2.69) µ V (0) V (0) [m, m] - ¹º [m, m] V (0) V (0) [1, 1] 0 3 [2, 2] 0 2 [3, 3] 0 2 [4, 4] 0 2 [5, 5] 0 2 [10, 10] 0 2 2.11 ³ (2.78) µ V (0) V (0) [m, m] - ¹º "qiqk [m, m] V (0) V (0) [1, 1] 0 5.571 43 [2, 2] 0 2 [3, 3] 0 2 [4, 4] 0 2 [5, 5] 0 2 [10, 10] 0 2 X, Íé q q, $é% [m, m] é`éa - iqjézé{é[é é é é~éªé«é é'.,»éöéåéæél q¼, \éiq6q½ vélqbé' éã, Íé q¾é é2é éj c d.". ` a - i.j Ã ² ³. ž.à}s Á $ î, [m, m] ` a - i.j z {.Â x ï Â ª «'. Ã.Ä L G, Í i É.ª l.å.±.².æ '.Ç 34. q = 1,

d Ã Q s ' r > 9 Ç Q Q Q s r s Ä N Q ñ 2 Q Q Ã d Ã Ä 9 d V B 8 s s Ã Q ï T ; 2 < =?>?@?A?B 45 Tb cd x ï $%YZ[JÈ 4 Ø à - ijjz{éhèþßüé' à L.b' Ä Å Æ g.h.m Ä.:, Ã Ý. ÊJË "JI»JÌJK Tbcd ".ÏJÐ ÞJÈ, ª, Í l XJÍJÎ.à. Í Qbcd Tbcd xï JÑXJÈ 4 Ø `~$%œ, à ~ª«' L e.w 'R `.B L l.ò.b '. j.y.z.[.\ e ª «.] h.^._ ª «' å æ _LÓVÔ', Í 2 Q JwJx `JB 'JÕ.ÖL.b ', y C, 2jJ J¼LJB 'Œ ï.ù.ú.û.ü.ý 9.Þ Õ.ß.à Ü.Ý % ) Þ z l.] h..ø L. Í.?,».ã 0.&.6.Y Z.[.\ e ª «.] h.^._ ª «' L å æ. Y Z.á.â.ª Ü Ý _. ` a b c d %.9.=.> Ä Å Æ.9 Ä.: L.v j. Í 34 Ø ª «' É Ê P b ' Ý. C D,.ä å ], 2.æ.ç f ç L, 0 a.b '. I, g.: Í.? c d. ` a - i.j z {, 0 b ' L Ã, Rœ.i.j z { Œ % ).

V Ÿ M s r î c r f r s s s c Q 9 s s ' ' Q T ªJð è é 3 êìëìíìîìï»jì IÉn 2, Í jl LXJ_JÍJÎ.`a T b c d l.ñ # ` a. h L.ò.p. Tb cd L.B "JÏ.Ð. "JI, µ, l.] h..ø 2 j l.ý V» ¼, ½ ¾ l 3.1 óõô öùøùúüû l.å.2 L..] h =.> c 0 1.Y$Z ( z.{ ) þ.ÿ ì.ò.p. m}n ª «N [u(r, t)] = 0 (3.1), N V.] h.^._ V 9 T U.9, u(r, t) q ', r t u 0 (r, t) ð ñ Q d u(r, t) Y$Z$[$\ ', L V ª «.] h.^._,.^._ % h w q [0, 1] f V.<.., V.., 0 V ª «', H(r, t) 0 L [f(r, t)] = 0, ` f(r, t) = 0 (3.2) ÆÏ` a H[Φ(r, t; q); u 0 (r, t), H(r, t),, q] = (1 q) {L[Φ(r, t; q) u 0 (r, t)]} q H(r, t) N [Φ(r, t; q)] (3.3) N O P L G, _.À L ` a, -. ê ª «' 9 9 ê ª «H(r, t). F Í $ q, ;.<. ê ª «' ê ª «' H(r, t) C ( ` a. $ 0, Y L ` a} Rœ L ` a Œ N 9 l$å h. ª «' ª «H(r, t) ª ` a b c d}n ˆ L j. O P L G, Í.ª å æ.y.z.[.\ u 0 (r, t)e ª «.] ' H(r, t) _.U % Û Ü L V '. h.^._ 9 Le ê ª «' ª «q [0, 1] V.<.. Ÿ. ` a (3.3) V ê,» H[Φ(r, t; q); u 0 (r, t), H(r, t),, q] = 0 Í ç ê. (1 q) {L[Φ(r, t; q) u 0 (r, t)]} = q H(r, t) N [Φ(r, t; q)] (3.4)

y Ÿ C s c ã s c c c c x Q Q c c Q d Q á y n c Q m}n ; 3 <?@?A?B 47, Φ(r, t; q) V d Q _.p 9 m, S ~.Y.Z.[.\ u 0 (r, t)e ª «.] h.^._ 8 ó Le ª «' H(r, t) ª «', I ~..<.. q [0, 1]. ~ q = 0, ê. (3.4) x V L[Φ(r, t; 0) u 0 (r, t)] = 0 (3.5) 'Rh w (3.2), J q ~ q = 1, 'R~ 0 9 H(r, t) 0, ê..` Φ(r, t; 0) = u 0 (r, t) (3.6) (3.4) ` ~.á.z N [Φ(r, t; 1)] = 0 (3.7) Φ(r, t; 1) = u(r, t) (3.8), E F.2 9 (3.6) z (3.8), ~..<.. q \ 0 ß Ü.ç 1, Φ(r, t; q) \.Y.Z.[.\ (. ) ç.á.z (3.1) L u(r, t). ª ` a g u 0 (r, t). ã V.. Y : G + / m..6 ' v jït u.õ.ö.+ g (3.4) k u [m], Φ(r, t; q) 2.Õ.Ö V ê.., Y m 0 (r, t) = m Φ(r, t; q) q m (3.9) q=0 d q b ' Φ(r, t; q) = Φ(r, t; 0) + u m (r, t) = u[m] 0 (r, t) = 1 m! m! m=1 j.2 (3.6) 9 (3.11), Φ(r, t; q) d b ' (3.10) 2 u [m] 0 (r, t) q m (3.10) m! m Φ(r, t; q) q m (3.11) q=0.w V N Φ(r, t; q) = u 0 (r, t) + m=1 O P L G, Í.U % Û Ü L V 'Rå æ.y.z.[.\ ' L.M H(r, t). [ å æ f ç, \ I Ø 9 ê ª «' ª «(1) $ % q [0, 1], ê. (3.4) L u m (r, t) q m (3.12) u 0 (r, t)eÿª «.] h.^._ Φ(r, t; q) (.ª ; (2) m = 1, 2, 3,, +,..6 u [m] 0 (r, t).ª ; Le

> ) ñ. Q A Ä Ÿ Q ç 48 rtsttou d (3) Φ(r, t; q) b (3.12) ª q = 1 Ã Â, ª _.p L.M Æ, E F.2 (3.8) 9 (3.12), Í. ç.b u(r, t) = u 0 (r, t) + 2 (3.13) xï ðñq 9 u(r, t) YJZJ[J\ x ï Æ.À L. +. m=1 u m (r, t) (3.13) u 0 (r, t) d, mén, qj1 u m (r, t) '! ", + #%$ 3.2 õô öùøùúüû u n = {u 0 (r, t), u 1 (r, t), u 2 (r, t),, u n (r, t)} & ' + # d =.> (3.11), u m (r, t) ( ) * +,. ( n-. (3.4). ( (3.4). 0.. q 1,. ( 7 m,.ž 2 3 m!, 4.ž q = 0, 5 6 ç. / +, m m}n L [u m (r, t) χ m u m 1 (r, t)] = H(r, t) R m (u m 1, r, t) (3.14), χ m 8 (2.42) 9 #, : R m (u m 1, r, t) = 1 (m 1)! m 1 N [Φ(r, t; q)] q m 1 (3.15) q=0 / ; (3.12) < 0 ; (3.15), = R m (u m 1, r, t) = 1 (m 1)! { m 1 q m 1 N [ + ]} u n (r, t) q q=0 n (3.16)?@AB AHIJK, HMIMJMK C=, D( (3.14) E=FG L, : R m (u m 1, r, t) MLMNMOM9MP N QM)M3 8.MR &M' ATUVWXYZ[ (3.15) W`. HIA 9M# (3.15), M,MSMDMM( (3.14) W ` b c u m 1. \], ^_ 1a,S D ( (3.14), u 1 (r, t), u 2 (r, t), d e, u(r, t) m, f g u(r, t) n=0 m u k (r, t) (3.17) 7ihkjkl (3.12) mknkokpkqkrkskt (3.4), u q vkwkxkykzk{ lk o, }k~kk k k kƒk k pkq r s t (3.14) (3.15), ˆ Š (3.16). Œ k=0

: é = 0 A # A A Á A A A # í B A œ A 0 Ô Ä 3 Ž 49 +, S D ( ) S ;. š A(q) œ A Ÿ B(q) q 1 a ž ( ), M MªM«; (3.18), ³ 5 6 µ A(q) B(q) A(0) = B(0) = 0, A(1) = B(1) = 1 (3.18) A(q) = AM M M M M± k=1 α k q k, B(q) = α k = 1, k=1 k=1 +, S D ¹ ( k=1 β k q k (3.19). \ A(q) B(q) œ q 1 ²aMž, 8 β k = 1 (3.20) C = º» F ¼ º%Â [1 B(q)] {L[Φ(r, t; q) u 0 (r, t)]} = A(q) H(r, t) N [Φ(r, t; q)] (3.21) L [ A ½ u m (r, t) ; Q F G m 1 k=1, 2 ¾ À β k u m k (r, t) ] R m (u m 1, r, t) = δ n (r, t) = 1 n! A Á, S D ¹ ( = H(r, t) R m (u m 1, r, t) (3.22) m α k δ m k (r, t) (3.23) k=1 n N [Φ(r, t; q)] q n (3.24) q=0 Ã Ä, +, S D ¹ ( (3.4), S D ¹ ( (3.14) (3.22) Æ Ç È É S. ^MÊ M HMIMËMÌ MÎMÏMÐ, P )MÍ ¹M(M MFM¼ÒÑMÓ IMàMáMâ (ÔMÕ ) ÖM MØMÙ AMåMæMçMè. ¾!M" M, 5M6MÚMÛMÜM ¹M( (3.1) ÝMÞMß GMã žm¹mä. Ï Ð, ¹ ( (3.1) S ; P Ê B í, ê ë Ü ) 3 Ø Ù ¹ (, ì ) 3 Ø Ù Õ (Ñ Ó ) Ö. ê ) 3 ¹ (î ðï ¹ (ñ - ï ¹ ( ò ó < ¹ (. C Ï Ð Ï Ð ¹ ( Ô Õ Ö Q ô õ g ¹ ä öm A MHMIMJMK, ø ù ú ë G ¹ ( Ñ Ó (Ô Õ ) ÖM, ûmümýmþmëmg ÑMÓMÿ Mañ ëmg Më GMõ ÏMÐ A(q) B(q). ], Më M9 B ¹M(MòMóMÑMÓ (ÔMÕ ) ÖM Â LMN. \ ], Ù a ž ¹ ä. ( ) A(q) = B(q) = q Å ¹ ( (3.21)

超越摄动 同伦分析方法导论 廖世俊著 陈晨徐航译

超越摄动同伦分析方法导论廖世俊著陈晨徐航译