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超越摄动 ——同伦分析方法导论 廖世俊 著 陈 晨 徐 航 译 北 京
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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
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·ii· 前言
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目 录 ·i·
目 录
第一部分 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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,¡
Φ(t; 1) = u(t)5U;U<H[Φ(t; q); q]|q=1 = 0UL
.UH
, .ÌUMUc q 0 U'U
1
,;U<
H[Φ(t; q); q] = 0
ALΦ(t; q)
0 .0ÌAMAc q, &'(AßAL u0(t)M0A0'A;A<
(1.4)AYAZL
u(t).§Uêp#
, ë UUMÜU¼ ò M . ï %Ý ,¸ = ÿ SU»1Ui UU;Us [33]
b ï UU;Us[34] k )Uq;Us
. ñ U , i = : .ÌUMUc q 23 ¸Uó ü ) ε¤U¥
δ,¸Uó Uü )UsUb δ
îAUs 9U: ï IU=U> ,
ø ý Õ 4 4 .,U3
, íî )UqU;UsU%5 ,U> +, ï%67U;Us
(1.6) 8 79 Ua % , R LUmU;UsU%5 , : VU,U5UUµóU% . ; n5 ¬ ¼ , < , = `Ua 6#U'U% **( = U%U7U8UEUÏ L b&'(UßUL , :>?@ABCDEFGHIJ = . KLM IN , OPQR ISTUBVWXY@Z[\I]^_`abcdefghiefjk .lmonqp Hrstu QRvw UBIx lyz , | G~i n I@A , i . , [o QRvw UBd ,
p HE¡ QRvw UB¢I£¤¥¦ n§©¨ª «¬® N 150 ¯ D° §±¨ «¬c w ® n±² ³b´µ¶·¸I , ¹ 1.1 ° .
º1 » ¼¾½ · 7 ·
[D , QRvw UBx QR yz , ¿ VNxSTI QR (1.6), À NÁÂ@Ã ÄÅÆÇb~iÄÅÆÈb H(t), OP @ÉÊI QR
H(Φ; q, ~, H) = (1 − q) L[Φ(t; q, ~, H) − u0(t)] − q ~ H(t) A[Φ(t; q, ~, H)] (1.7)
GÌË ® (1.6) K @Í , Î ® (1.6) Ï ® (1.7) Ð ~ = −1 i H(t) = 1 Ñ dÒÓ ,ÔÕ dH(Φ; q) = H(Φ; q,−1, 1) (1.8)
QÖ , Ð q × 0 ØÙÚ 1 Ñ , Φ(t; q, ~, H) ×ÛÜÝÞ c u0(t) ßàÚáÜ UIâãc u(t). äÀ , U
H[Φ(t; q, ~, H)] = 0 (1.9)
dåc Φ(t; q, ~, H) V Ïåæåç åèå ßåé q, Àåêåëåæåç åÅåÆåÇåb ~iåÅåÆåÈåb
H(t). Îì , Ð q = 1 Ñ , cí äæç ÅÆÇb ~iÅÆÈb H(t). ° ,
² ST QR (1.6) V Q , îïQR (1.7) WðXY@ñabc , efgh æç ÅÆÇb ~
iÅÆÈb H(t), lmòóô `°u . KLM IN , GXY @Zõöi÷øabcdefghiefjkI[\]^ .
QåRåvåw UåBåNå@åÉåù @åÍåIåUåB , úåûåüåý åIååååþ ÿ v Uå£ . G~ ûü , [35∼39] ÿ¥¦ [28,29,40∼43] ÿåSåF [44, 45] ÿ ü n I ª å¦ [46] ÿ Oldroyd þåb妪 ¦ [47] ÿ ¦¦ [48] ÿ [49, 50] ÿ ! - "# U [51] iLane-Emden U [30] $ . i ,
lm ¢ QRvw UB@AÊI~ Ó .
%2 & ')()*)+),
lô,p H~@ Ã [Iþ v U- /. Ó , 0QRvw UBdxlyz
.
2.1 1 23456Ìn ×789:<; ©/=I ¨ . > t ?Ñ@ , U(t) ? ¨ j , m ?
é , g ?L ¬Ajk . BC ¨ D Ú I 56 «E a U2(t), a þb . F´GHIJ ,
mdU(t)
dt= mg − aU2(t) (2.1)
ÛÜ ZK U(0) = 0 (2.2)LMN
, ; =djkO L ¬-~PVQ ØÙ , RSTÚ @ ÃU Ijk U∞. ° , VW VXY U(t), ë W × U (2.1) RZ[Ú\] jk U∞, V
U∞ =
√
mg
a(2.3)
V^ > U∞i U∞/g v_? Ò`jkiÒ` Ñ@ . >
t =
(
U∞
g
)
t, U(t) = U∞V (t) (2.4)
p H [Úaéb UV (t) + V 2(t) = 1, t > 0 (2.5)
c ÛÜ ZKV (0) = 0 (2.6)
å , t ?åaåébåÑ@ , · ?ååú t d F . åä , Ð t → +∞ Ñ , V t → ∞ iU(t) → U∞ Ñ , efd cU (2.5) i (2.6) , F ´ ® (2.4),
p H lim
t→+∞V (t) = 1 (2.7)
º2 » gihijikil · 9 ·
U (2.5) i (2.6) dâãc
V (t) = tanh(t) (2.8)
®(2.8) m ~V Qno cdÌËp .
2.2 qsrutuvuwyxyz|yy~|v \ÌËp ,
p H~ISTc w UBD d c . Ó .
2.2.1 ¢/d// n/o c ,
p H B I aé/bÑ/@ t /é ( / é ),
| V (t) ?T abV (t) = α0 + α1t + α2t
2 + α3t3 + · · · (2.9)
ÛÜ ZK (2.6), [Ú α0 = 0. | N u ?T ® ÂU (2.5), +∞∑
k=0
(k + 1) αk+1 +
k∑
j=0
αjαk−j
tk = 1
N u ?T ® ú ° t > 0 , ×Àα1 = 1 (2.10)
αk+1 = − 1
k + 1
k∑
j=0
αjαk−j , k > 1 (2.11)
p HÌ ì[Ú c
Vpert(t) = t − 1
3t3 +
2
15t5 − 17
315t7 + · · · =
+∞∑
n=0
α2n+1 t2n+1 (2.12)
cO Ð Igh 0 6 t < ρ0 ef , , ρ0 ≈ 3/2, ¹ 2.1 ° . [ Ò _ EIN , c (2.12) defghiefjkNãII .
· 10 ·
2.1 ii (2.57) i i¡i (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 ÇbB ,
p H¿U (2.5) ÀÁ V (t) + ε V 2(t) = 1 (2.13)
, ε ¾ Çb . ä ò , >V (t) = V0(t) + ε V1(t) + ε2 V2(t) + · · · (2.14)
| ® (2.14) ÂU (2.13) i ÛÜ ZK (2.6)
n, > ε QÂ dÃb 0,
p H [Ú ÃÄU
V0(t) = 1, V0(0) = 0
V1(t) + V 20 (t) = 0, V1(0) = 0
...
æÂd c N uU , [ÚV0(t) = t, V1(t) = − t3
3, V2(t) =
2t5
15, · · ·
º2 » gihijikil · 11 ·
· ò , >?T ® (2.14)n
ε = 1,
V (t) = t − 1
3t3 +
2
15t5 − 17
315t7 + · · · =
+∞∑
n=0
α2n+1 t2n+1 (2.15)
®(2.15)
² c (2.12) ÅÆ Q , ÎìëÏ O t Ð] Igh , ¹ 2.1° . /[/Ç/È IN , Lyapunov ¾ ÇbB [Ú Ic/defghiefjkë NãII .
2.2.3 Adomian ÉÊ~ Adomian v cB , ~
V (t) = t −∫ t
0
V 2(t)dt (2.16)
Ë U (2.5) i ÛÜ ZK (2.6).N uUd Adomian c
V (t) = V0(t) +
+∞∑
k=1
Vk(t)
V0(t)= t
Vk(t)=−∫ t
0
Ak−1(t) dt, k > 1
n
Ak(t) =k∑
n=0
Vn(t) Vk−n(t)
Adomian üÌ ® . N u ?T ® ,p H æÂ[Ú
V1(t) = − t3
3, V2(t) =
2t5
15, V3(t) = − 17
315t7, · · ·
×ÀV (t) = t − 1
3t3 +
2
15t5 − 17
315t7 + · · · =
+∞∑
n=0
α2n+1 t2n+1 (2.17)
®(2.17) Í ² / c (2.12) Å/Æ Q , ÎìëÏ O/ Ð/ Igh , ¹ 2.1
° . [ÇÈ IN , Adomian v cB [Ú Ic/defghiefjk ë NãII .
· 12 · 2.2.4 δ ÎÏ~ δ ÐÑ B ,
p H | U (2.5) ÀÁ V (t) + V 1+δ(t) = 1 (2.18)
, δ b . >V (t) = V0(t) +
+∞∑
n=1
Vn(t) δn (2.19)
ä ò , | V 1+δ(t) ÐÑ δ dabV 1+δ =V0 + [V1 + V0 ln V0] δ
+
[
V1(1 + ln V0) +1
2V0 ln2 V0 + V2
]
δ2 + · · · (2.20)
| ® (2.19) i ® (2.20) ÂU (2.18), > δ QÂ dÃb 0, [Ú ÃÄ
UV0 + V0 =1, V0(0) = 0
V1 + V1 =−V0 ln V0, V1(0) = 0
V2 + V2 =−V1(1 + ln V0) −1
2V0 ln2 V0, V2(0) = 0
V3 + V3 =−V2(1 + ln V0) − V1
(
1 +1
2ln V0
)
ln V0
−1
6V0 ln3 V0 −
V 21
2V0, V3(0) = 0
...
æÂd c N uU , [ÚV0(t)=1 − exp(−t)
V1(t)=exp(−t)
[
t − π2
6+ P L
2 (e−t)
]
− (1 − e−t) ln(1 − e−t)
...
P L
n (z) =
+∞∑
k=1
zk
kn
º2 » gihijikil · 13 ·
z I n Lú bÈb (nth polylogarithm function). Ò no c V (t) ≈ 1 + exp(−t)
[
t − π2
6− 1 + P L
2 (e−t)
]
− (1 − e−t) ln(1 − e−t) (2.21)
ÓÔÕÖ Ò no c o× OØ Ã gh 0 6 t < +∞ £ , ¿ No qÒÙÈb P Ln (z)
IÚ , Û[Ü Ò no c ß[Ý D ÝÞß .
à IN ,N uÌ Uá ÿ Lyapunov ¾ Çbái Adomian v cá d[
Iabc/âN/ Q I . ¿ N , abc Ï O/ Ã Ð/ Igh 0 6 t < 3/2 .
Ì ,² c ýo , ã Lyapunov ¾ Çbái Adomian v cá $ ST
Uá d[ Ic , O LM Çbä ßéØÙ , ×À ØåÑ , m Wæ . QÑ ,
à ë/È/ç/è , U/ád @/m W/é/O/êÉ/ë/à . /[/å _IN , ° STc w Uá [Ú IabcdefghiefjkâN/I , Àê , Uáiã Lyapunov ¾ Çbá ÿ Adomian v cái δ v cá $ UáâVWXY à ÷øiõöabcdefghiefjkI[\]^ . ìì , ° STU/á/â a á/í~îXI V (t) O t ï a/ðÑ I LM (2.7). à Ó/ñ ,
l N , aò UáóNST/ Uá , âVWô v í~ ýõ/ I/ö÷ /TÚK ø n d<ù I . · ò , [ÇÈ IN , ¾ Çb ε i δ v_ ÚOU (2.13) i (2.18)
n IV Qúû . ¿ N , δ ÐÑ á¢Ic (2.21) WðOØà gh 0 6 t < +∞ , ×À , Ë Lyapunov ¾ Çb/á¢Ic¸ [ü .G 1
ô °u , ü p H¿ ε i δ I ï è ßé , ýþ U (2.13) i (2.18) mÿ ÉÄÒ ß U . o , K ø n à , ¶ M OPQR iÄÒ ß UNþ LM I .
2.3 uwuvl ` ,
p H W ~ Q I . Ótu QRvw Uádx lyz .
2.3.1 > V0(t) ? V (t) I ÛÜÝÞ c . c ÛÜ ZK (2.6), V
V0(0) = 0 (2.22)
> q ∈ [0, 1] ? °I/è ßé . QRvw U/áx Ö É V (t) →Φ(t; q), Ð è ßé q × 0 ØÙÚ 1 Ñ , Φ(t; q) ×ÛÜÝÞ c V0(t) ßàÚ âãc
Mathematica !"#$%&' ´() , *+,-. . /012 % E-mail 34 ¨[email protected], 56789 ¨ http://numericaltank.sjtu.edu.cn/code.htm, !:;< $=&=>=,=-=?=@=A=B=C Mathematica D=E¥´=0=1=F=& . —— G=H
· 14 · V (t). Ú É , JI/ÅÆJñ
L[Φ(t; q)] = γ1(t)∂Φ(t; q)
∂t+ γ2(t) Φ(t; q) (2.23)
, γ1(t) 6= 0 i γ2(t) K IÈb . ©U (2.5), I ï JñN [Φ(t; q)] =
∂Φ(t; q)
∂t+ Φ2(t; q) − 1 (2.24)
> ~ 6= 0 i H(t) 6= 0 v/_/? °I/ÅÆÇbåi/ÅåÆÈåb . LM ÁÂè ßéq ∈ [0, 1], OP NU
(1 − q) L [Φ(t; q) − V0(t)] = ~ q H(t) N [Φ(t; q)] (2.25)
ÛÜ ZK Φ(0; q) = 0 (2.26)
[å _IN ,p HO PÙ I ; ©JIÅÆÇb ~
ÿ ÅÆÈb H(t) ÿ ÛÜÝÞ cV0(t)
iÅÆJñ L. lmòóô `° ,N uI ; O QRvw Uá nRQ è
LM I-~ , S I QRvw Uá idxT .
Ð q = 0 Ñ , U (2.25) ß L [Φ(t; 0) − V0(t)] = 0, t > 0 (2.27)
c ÛÜ ZKΦ(0; 0) = 0 (2.28)
F ´ ® (2.22) i ® (2.23), U (2.27) i (2.28) IcUN
Φ(t; 0) = V0(t) (2.29)
Ð q = 1 Ñ , U (2.25) ~ H(t) N [Φ(t; 1)] = 0, t > 0 (2.30)
c ÛÜ ZKΦ(0; 1) = 0 (2.31)
VΦ(t; 1) = V (t) (2.32)
©~ 6= 0 ÿ H(t) 6= 0, êF ´I ï (2.24), U (2.30) i (2.31) $ Q U (2.5) i
(2.6). F ´ ® (2.29) i (2.32), Ð è ßé q × 0 ØÙÚ 1 Ñ , Φ(t; q) ×ÛÜÝÞ
º2 » gihijikil · 15 ·
c V0(t) ßàÚ âãc V (t). OWX n , É ßà/ , U (2.25) i(2.26) O QR Φ(t; q). [\ , U (2.25) i (2.26) .
Î p HO JYÅÆÇb ~ÿ ÅÆÈb H(t) ÿ ÛÜÝÞ c V0(t)
iÅÆJ/ñ L d ; , ° , m/B IGH/âJY¶Z , ×ÀÐ 0 6 q 6 1 Ñ Ä/Ò ß U(2.25) i (2.26) dc Φ(t; q) éO , êú è ßé q I m Ò [ , V
V[m]0 (t) =
∂mΦ(t; q)
∂qm
∣
∣
∣
∣
q=0
(2.33)
ë éO , m = 1, 2, 3, · · · [\ , V[m]0 (t) m Ò ß [ b . I ï
Vm(t) =V
[m]0 (t)
m!=
1
m!
∂mΦ(t; q)
∂qm
∣
∣
∣
∣
q=0
(2.34)
F ´\] ÐÑ I M , | Φ(t; q) ^ è ßé q ÐÑ abΦ(t; q) = Φ(t; 0) +
+∞∑
m=1
1
m!
∂mΦ(t; q)
∂qm
∣
∣
∣
∣
q=0
qm (2.35)
® (2.29) i ® (2.34),N uab
Φ(t; q) = V0(t) +
+∞∑
m=1
Vm(t) qm (2.36)
B/C ÅÆÇb ~ÿ ÅÆÈb H(t) ÿ ÛÜÝÞ c V0(t)
iÅÆJ/ñ L JY¶Z ,
×À ab (2.36) O q = 1 Ñ ef . ýþ , Ð q = 1 Ñ , ab (2.36) ß Φ(t; 1) = V0(t) +
+∞∑
m=1
Vm(t) (2.37)
ä ò , F ´ ® (2.32), V (t) = V0(t) +
+∞∑
m=1
Vm(t) (2.38)
N u ?/T ® í~_/X Ì Vm(t) (m = 1, 2, 3, · · ·) ¢ ÛÜÝÞ c V0(t)² âãc
V (t) dëà .
2.3.2 ` I ïaé
V n = V0(t), V1(t), V2(t), · · · , Vn(t)
· 16 · F ´I ï (2.34), m ©ÄÒ ß U (2.25) i (2.26) [Ú _X Ì Vm(t) dbcUi ÛÜ ZK . | U (2.25) i (2.26) ú è ßé q d [ m  , ä ò > q = 0, · òd m!,
pe [Ú °f m Ò ßg L [Vm(t) − χm Vm−1(t)] = ~ H(t) Rm(V m−1) (2.39)
c ÛÜh KVm(0) = 0 (2.40)
ikj
Rm(V m−1)=1
(m − 1)!
∂m−1N [Φ(t; q)]
∂qm−1
∣
∣
∣
∣
q=0
(2.41)
l
χm =
0, m 6 1
1,imno (2.42)
pRq(2.24) r q (2.41), s
Rm(V m−1) = Vm−1(t) +m−1∑
j=0
Vj(t)Vm−1−j(t) − (1 − χm) (2.43)
[ÇÈ ft , Rm(V m−1) uvwxV0(t), V1(t), V2(t), · · · , Vm−1(t)
yz e â| LMv~ m Ò g (2.39) r (2.40) . | , L (2.23), g (2.39) g , h (2.40). , g (2.39) r (2.40) Vm(t) , ¡¢ t£¤¥¦§
Mathematica ¨ Maple ¨ MathLab ©ª«¬® .pRq
(2.38), ¯°± , ²³´µ¶ · (2.5) r (2.6) ¸¹ p · (2.39) r (2.40) º»¼½¾¿À Á , ÃÄÅÆÀ Á ¼rÇÈÉÊ . ËÌͼ t , ÎÏи¹ÑÒÓÔÕÖ×·r ( ØÙ ) Ú jRÛÜÝÞß ( à ) áâ .
V (t) ¼ m ÈãV (t) ≈
m∑
n=0
Vn(t) (2.44)
ËÌͼ t , ä · (2.25)p L ¨åæç V0(t) ¨åá
â ~ rèâ H(t) é . êë±ì ,p ±í·îïð¼ V (t) vwx
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). ——
2 · 17 ·
L ¨åæç V0(t) ¨åáâ ~ rèâ H(t).
, Ó xs ! "Èã #·î ,
p ±í·îïð¼ $â&% '&( )r&% '&*&+Ó t , ¼ , § ¯ -. /10 2 .
2.3.3 3 4 5 65 6 2.1 7 8 9 (2.38) : ; , < = Vm(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
Vm(t)
% ' , US(t) =
+∞∑
m=0
Vm(t)
l V slim
m→+∞Vm(t) = 0 (2.45)
pχm (2.42), » W
n∑
m=1
[Vm(t) − χm Vm−1(t)]
=V1 + (V2 − V1) + (V3 − V2) + · · · + (Vn − Vn−1)
=Vn(t)
± q r q (2.45), s+∞∑
m=1
[Vm(t) − χm Vm−1(t)] = limn→+∞
Vn(t) = 0
p ±í X Y q r L (2.23), » W+∞∑
m=1
L [Vm(t) − χm Vm−1(t)] = L+∞∑
m=1
[Vm(t) − χm Vm−1(t)] = 0
p ±í X Y q r· (2.39), s+∞∑
m=1
L [Vm(t) − χm Vm−1(t)] = ~ H(t)
+∞∑
m=1
Rm(V m−1) = 0
· 18 · Z\[\]_^
~ 6= 0 r H(t) 6= 0, ` y+∞∑
m=1
Rm(V m−1) = 0 (2.46)
pRq(2.43), » W
+∞∑
m=1
Rm(V m−1)=
+∞∑
m=1
Vm−1(t) +
m−1∑
j=0
Vj(t)Vm−1−j(t) − (1 − χm)
=
+∞∑
m=0
Vm(t) − 1 +
+∞∑
m=1
m−1∑
j=0
Vj(t)Vm−1−j(t)
=
+∞∑
m=0
Vm(t) − 1 +
+∞∑
j=0
+∞∑
m=j+1
Vj(t)Vm−1−j(t)
=
+∞∑
m=0
Vm(t) − 1 +
+∞∑
j=0
Vj(t)
+∞∑
i=0
Vi(t)
= S(t) + S2(t) − 1 (2.47)
pRq(2.46) r q (2.47), s
S(t) + S2(t) − 1 = 0, t > 0
pRq(2.22) r q (2.40), » W
S(0) =
+∞∑
m=0
Vm(0) = V0(0) +
+∞∑
m=1
Vm(0) = V0(0) = 0
± / a À X Y q , S(t)V · (2.5) r (2.6) . b c .
ËÌͼ t , ±íê\d p q (2.23) ¼ L q&e f, g1d , γ1(t) 6= 0 r γ2(t)
Ó ¼èâ . hê is jÕ k . l sÎÀê , ²³&mÔ&n\d É&oÊ&p&$â&%&' . &q&r , $â (2.38) ¼&%&'vwxáâ ~ ¨åèâ H(t) ¨åæç V0(t) r L. s t¼ t , u #·îwvwxï²³à¼zy|B ~Ä z ³ . | , mÕáâ ~ ¨ èâ H(t) ¨ æç V0(t) L ~Ä , ¤ $â (2.38)
Ü0 6 t 6 t0 ( )
0 %' , z V Ü h ( ) 0 % '· .
, % 'ê áâ ~ ¨èâH(t) ¨ æç V0(t) L ~ı¼zyBB , º» u #·î .
2 · 19 ·
2.3.4 § ±í , ºwä ·w , ²³wwà¼y||~Ä L ¨
æç V0(t) èâ H(t). êë± , ±íy| e ww , ` y | ~Äw¿Ó¼èâ H(t) ¨=æç V0(t) L. r y , ¥ ± , ÎÀzyBã , V Õ v x ¯µ ¡ g ~Ä .¢ ¶ Á y £ , #È㯰 ¤ ¥ ¦ ¥ § ¼ èâ ÇÈ . ¨&©&ª , èâ f(x) &« Ó&¼&èâÇÈ , ¬&&®¼&èâ&¯&°&±&&&²ÇÈ . , èâ¼ ~Ä ¢ ÇÈ ³ y £ ¶ ´ jÕ . ²ÇÈÀï ¶ Á ¼&µ&¶ , ¥&~Ä& ¢ &®¼&èâ . s&t¼&¥ , · ¥ L ¨ æç V0(t) èâ H(t) ~ı¼zyB , ¿1RÓ èâ ¸í¼ V (t) . `ÎwwXwY¹d , ²³ ~ÄÀww®¼ , ` y ±www²ÇÈÀï¼ ¶ ÁÂ
.Ü ¿wºw»w¼ , ½ww#w¾êw¿wÀ ¨ ( ØÙ ) Úw ( Áw )¶ ·wÃÄ
, ²³ ŠƽÔ~À ¶ ÁÂ Ç ¯ ª&È É&Êϼ&èâ& ¸í g Ë . ̧ , Ó Í Uek(t) | k = 0, 1, 2, · · · (2.48)
X 2 ¸í¯ - Î Ì Ë¼ § èâ . ²³ Ï´ Ë X Y»
V (t) =+∞∑
n=0
cn ek(t) (2.49)
Î Ð , cn Ñâ . Ò ~Ä èâ , èâ H(t) ¨ æç Ë V0(t) ¨ L
Ç V Ó § Ô ~Ä , Õ Æ Ö ×¼ Ø· Ë ÛÜ ¨Ù¬ ¯ « h èâX&Y . Î&v&x&À&¡&~Äèâ H(t) ¨ æç&Ë V0(t) ¨ L¼ ¯µ , ÚÜÛ Ý Þ . ÎÚµ Ü u #·î1dBß à ¶ ´ jÕ¼ á ¥ .
§ Åí , èâ f(x) ¯ ° « Ó ¼ èâÇÈ , Õ , Ï ¯ ÛÜ Ó ¼ Û ÝÞ , ¬&â³&ã&¯&°ïð&°ÉʼÈã&Ë . Î& , Ï&½&&~Ä&ä&®¼&èâ&äå Èã Ë .
&&æ&ç&è×èâ H(t) ~ı¼éyê , V Õ&vð&ë¼Üì&í&î&ï& , ð Ë X Y1dR¼ ÑâË , § ñ (2.49) dR¼ cn, ò ¯ ° « ó ô , ` õÊ p ~ö ¼wèâÑw÷ .
Ü ¿wºw¼ , ½wwÛwÝwÞwwwwìwíwîwï , èâw¯° « , Ê . ø ù , q õ ú , Ø· V Ó û ü à ¬ Ë . Î v x ²³ë¼ÜÛ ý þ .
±í&Û&Ý&Þ&& ¨ÿì&í&î&ï&&&&Û&ý&þ&& Ü &u&#·î_dêß&à&jÕ¼á ¥ , ¬ Ü à +±¹ u #·î¼ × ¥ .
· 20 · Z\[\]_^
2.3.5 Û Ý Þ±í·îw ¶ ·îÓw , wuw#·îwÏ ¥ ÓwwèâïðÓw X
Y ñ ¼ Ë .
1. R Ë (2.12) À µ Ö t ¼ $â .
Ô , Ï ¤¥§ ¼ § èâ
t2m+1 | m = 0, 1, 2, 3, · · ·
(2.50)
X Y V (t), ðV (t) =
+∞∑
m=0
am t2m+1 (2.51)
g1d , am Ñâ . Î v x h Î Ì¼Ð Û Ý Þ .
h Û Ý Þ Ú (2.22), q r × ~ÄV0(t) = t (2.52)
á V (t) ¼æç Ë , Õ
L[Φ(t; q)] =∂Φ(t; q)
∂t(2.53)
á , h i L (C1) = 0 (2.54)
g1d , C1 ´ â . Û Ý Þ (2.51) · (2.39), èâ H(t)V Ó ~ħ ¼ ñ
H(t) = t2κ (2.55)
ñ (2.54), · (2.39) ËVm(t) = χmVm−1(t) + ~
∫ t
0
τ2κ Rm(V m−1) dτ + C1
g1d , ´ â C1 RÚ (2.40) Ê . ²³ , κ 6 −1 , Vm(t) t−1
. ÎÓª Û Ý Þ (2.51). Ô ù , κ > 1 , Vm(t) Ó t3
, Ô , ð ¤ Èã
â Ö½¾ , t3 ¼ Ñâä , Ó ¯ « ó ô . Î ì í î ï Óª .
Û Ý Þ (2.51) ì í î ï ,V Ó U κ = 0.
¢ ×¼èâH(t) = 1 (2.56)
2 · 21 ·
« , ²Ê . . , Ï ! ñ
V1(t)=1
3~t3
V2(t)=1
3~(1 + ~)t3 +
2
15~
2t5
V3(t)=1
3~(1 + ~)2t3 +
4
15~
2(1 + ~)t5 +17
315~
3t7
...
"¼ ¥ ,¢ ×¼ m Èã Ë
V (t) ≈m∑
k=0
Vk(t) =
m∑
n=0
µm,n0 (~)
(
α2n+1 t2n+1)
(2.57)
g1d , α2n+1 Ë (2.12) Ñâ , ¬èâ µm,n0 (~) § ¼
µm,n0 (~) = (−~)n
m−n∑
j=0
(
n − 1 + j
j
)
(1 + ~)j (2.58)
#$ æç Ë V0(t) ¨ L èâ H(t) %RÊ , ²³& r à¼zyBB~Äáâ ~ Ë . Ë!Ìͼ ¥ , Ë (2.57) áâ ~. b ,
èâ µm,n0 (~) i °
µm,n0 (−1)=1, n 6 m (2.59)
¬ ¢ÝÞ À è l'â n, » W
limm→+∞
µm,n0 (~) =
1, |1 + ~| < 1
∞, |1 + ~| > 1(2.60)
±í a À° b)(R´ Ü ¯ - . / ïð . Õ , ~ = −1 , ñ (2.59) ¨ ñ (2.57)
ñ (2.12), » WV (t) = Vpert(t) (2.61)
Ô , Ë (2.12) * ñ (2.57) ~ = −1 ¼À¡ Ì . Lyapunov +, ß áâîïð¼wË (2.15) Adomian wËîïð¼wË (2.17), Ï §wÔ .
Ô , ñ (2.57)Ü
-. ±/ ·î¨ Lyapunov +, ß áâî Adomian Ëî!¼ Ë , `õ ± i01 .
ñ3254 V3(t) 6 4/15 7585956 2/15, 5:5; , <575=5>5? . ——
· 22 · Z\[\]_^
Ë!@wk¼w¥ , $â (2.57) wÑâAwÖáâ ~. ñ (2.60), $â (2.57)
% '¼ V ÕÚ ¥ |1 + ~| < 1, ð
−2 < ~ < 0
wkB¼w¥ , $â (2.57) w%w'w(w)CAwÖ ~ Ë . §D 2.1 w2 , ~ (−2 < ~ < 0)
ËFEFÈ Öä , $â (2.57) % ' ( )FEà . ²³FF , $â (2.57) % ' ()
0 6 t < ρ0
√
2
|~| − 1
g1d , ρ0 ≈ 3/2 ¥ Ë (2.12) % 'GH . Ô , ~ (−2 < ~ < 0) Èä , $
â (2.57)Ü 'À ( )
0 6 t < +∞0 % ' ÆÉÊ Ë V (t) = tanh(t). ! " Ë #·îÓ , ½ ~Ä ~ Ë ,
²³ ÏÍI Ö× $â (2.57) % ' ( ) . Ô , áâ ~ v x ÚÍI Ö
× $â Ë % ' ( )¼JKH .
2. L R#$Ü~ (−2 < ~ < 0) Ö 0 , Bèâ (2.50) ïð¼ $â Ë (2.57)
Ü 'À( )
0 6 t < +∞0 w , M ~ (−2 < ~ < 0) N ¢ Ë ß , Èãâ VwÓ Ow¯!w°Pʼw T .
#$Ü êë±èâ (2.50) QR·î ¨ Lyapunov +, ß áâîw Adomian
Ëîïð¼ Ë (2.12) ± ®¨ ± i01 , M gÇÈ ³Ó . Ô , V Õ ~Ä
§ ± ®¼ èâ , ` õ ± ²ÇÈ V (t).§ ÅSí , ð ¤ Ó Ë·T (2.5) (2.6), C Ï!UV è * +
V (+∞) = 1
æç Ë (2.52) q rÓÎÀ° . ½ ´ , $â* Ü À è¼ ( ) 0 % ' .
S Õ , $â (2.50)Ü 'À ( ) 0 6 t < +∞
0 Ó ¯ ²ÇÈ V (t).
ªlim
t→+∞
1
(1 + t)m= 0, m > 1
Ô , Bèâ
(1 + t)−m | m = 0, 1, 2, 3, · · ·
(2.62)
2 · 23 ·
X Y¼èâ Ü t → +∞ V èË . Ó ÍWX V (t) Ï X Y»
V (t) =
+∞∑
m=0
bm
(1 + t)m(2.63)
g1d , bm Y Ñâ . Î v x h Π̼ZÐ Û Ý Þ .
Û Ý Þ (2.63), RÚ (2.6) V è * + (2.7), q r , × ~ÄV0(t) = 1 − 1
1 + t(2.64)
á V (t) ¼æç Ë , Õ
L[Φ(t; q)] = (1 + t)∂Φ(t; q)
∂t+ Φ(t; q) (2.65)
á ×¼ , h i °
L(
C2
1 + t
)
= 0 (2.66)
g1d , C2 ´ â . L (2.65), Ø·T (2.39) Ë
Vm(t) = χmVm−1(t) +~
1 + t
∫ t
0
H(τ) Rm(V m−1) dτ +C2
1 + t, m > 1
g_d , ´ â C2 Ú (2.40) Ê . Û Ý Þ (2.63) Ø·FT(2.39), èâ H(t) × ~Ä § ¼ ñ
H(t) =1
(1 + t)κ(2.67)
g1d , κ 'â . ²³ , κ 6 0 , Ø·T (2.39) Ë/ln(1 + t)
1 + t
, gÓª Û Ý Þ (2.63). κ > 1 , (1 + t)−2 Óð Ü Vm(t) d , ` õ ,
(1 + t)−2 ¼ Ñâ[ä , ð ¤ Èãâ Ö½¾\½î « ó ô . Î] ¿ ì í î
ï&& . Ô , &&Û&Ý&Þ (2.63) &ì&í&î&ï&& ,
V&Ó U κ = 1. Î Ç , &²Ê ×¼èâ
H(t) =1
1 + t(2.68)
· 24 · Z\[\]_^
. , Ï^!
V1(t)=− ~
1 + t+
2~
(1 + t)2− ~
(1 + t)3
V2(t)=−~
(
1 +7
12~
)
1
1 + t+
2~(1 + ~)
(1 + t)2
−~
(
1 +7
2~
)
1
(1 + t)3+
10~2
3(1 + t)4− 5~
2
4(1 + t)5
...
× ² , V (t) m Èã Ï X Y»
V (t) ≈2m+1∑
n=0
βm,n(~)
(1 + t)n(2.69)
g1d , βm,n(~) A Ö ~ ¼ Ñâ .
Ë!Ìͼ ¥ , ²³& r à¼zyBB~Äáâ ~ Ë . $â (2.69) _±w¥`wË . wab ~
¢ $â (2.69) ¼cd , efghwwwµw$â ( § V ′(0),
V ′′(0), V ′′′(0) © ) ¼w%w' . ²³ ,ÝÞ â¼ÈãwËwãw V ′(0) = 1, SwÕwâÓ
¯ v x ÝÞ ¥ ¼ij Õk ~Ä ~. M ¥ , V ′′(0) V ′′′(0) A Ö ~. Ó Í U R~
Xw2Swww¼ ~ Ëwnw , gww×¼ V ′′(0) w$âw%w' . wJ , ²³wÚ R~ µ Ö V ′′(0) ¼ ~ lmno . ê 2.1,
¢Ý À ~ ∈ R~, ×¼ V ′′(0) $â% ' Æ T ñ . V ′′(0) ∼ ~ )pR Ü R~ ( )Úqrs . ²³ ÚÎÏÚp V ′′(0) ~ p . &q&r , â&¡&2&à&$â&Ë&&&(&) R~. êË (2.69) ïð¼ V ′′(0) V ′′′(0) ~ pR §D 2.2 S 2 . )(Bq , $â (2.69) ïð¼ V ′′(0) V ′′′(0)
Ü−3/2 6 ~ 6 −1/2
&%&' . Ì § , ~ Ä&(&) −3/2 6 ~ 6 −1/2 d 5 ÀÓ&Ë& , $â (2.69) ïð¼ V ′′(0) V ′′′(0) &$â¢&%&'&Æ&&×¼ÉÊË 0 −2, § X 2.1 &X 2.2
S&2 . &ktB¼&¥ , $â&Ë&%&'&*&+ttA&Ö ~ Ë . ê$â (2.69) ïð¼ V ′′(0)
V ′′′(0) $â Ü ~ = −1 % ' äu , ú X 2.1 X 2.2. Î kv à , ²³ Ï ½ ~ ö áâ ~ ÍtI&$â&Ë (2.69) &%&'&*&+ . Ô ù , ²³tt , £ $â V ′′(0) V ′′′(0) % ' , $â (2.69)
Ü 'À ( ) 0 6 t < +∞0 % ' .
Ô , ê 2.1, S Î % ' $â V µ ¶ ÁÂ Ë . Ì § , ~ = −1 , $â (2.69)
Ü 'À () 0 6 t < +∞
0 % ' ÆÉÊ Ë , ú X 2.3. f õ £ , ½ w× Ãã¼ ~ pR , ¯ °ñ3x5y
, z55|55~ . ——
2 · 25 ·
² ª È ×¼ ~ ( ) .Ü Ô ( ) 0 ~ÄÀ ~ Ë ,
Ç ÏÊ p ×¼ $â Ë % ' .
Ô , áâ ~Ü u #·î1dBß à jÕ¼ á ¥ .
2.2 H(t) = 1/(1 + t) , (2.69) V ′′(0) ∼ ~ V ′′′(0) ∼ ~ 555
V ′′(0) 20 û55 ; V ′′′(0) 20 û55
2.1 --h , (2.69) V ′′(0) 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) V ′′′(0)
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
· 26 · Z\[\]_^
2.3 --h=−1 , m ¡ (2.69) ¢£¤¡ (2.8) ¥¦
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 +, ß áâî Adomian © Ëî!U¼ Ë (2.12),
−3/2 6 ~ 6 −1/2 , $â (2.69)Ü 'À&(&) 0 6 t < +∞
0 ò&%&'&ÆÉÊË . ª Ô , $â&Ë (2.69) Q (2.57) ±&& ,
#t$ êët«&ât¬ a Â&ã&Ï&Õ Ü 'tt®t¯0 6 t < +∞ ° ±² Ƴ´ Ë . µ¶· ¥ª¸ ,
¢ Ö¹ Î Ì õº , »¼½ (2.62) Q »¼½ (2.50) ¾¿ , ª Ô , À¾ÁÂÃÄÅÆ .ÇÈÉÊ ¬ËÌÍÎÏÐ½Æ (2.69) ÑÒÓÆ (2.12) ÔÕÖ×Ø . Ù ~ = −1Ú
1
1 + t= 1 − t + t2 − t3 + · · ·
ÛÜ Ð½ (2.69) Ô 10 ÝÅÞÆ , Áß
V (t) ∼ t − 1
3t3 +
2
15t5 − 17
315t7 +
62
2835t9 + · · ·
à Öáâã Ú ÒÓн (2.12) ÖáâãäåÎæ !
3. ç èéêéëìíî
limt→+∞
exp(−nt) = 0, n > 1
ªï , 𻼽 exp(−nt) | n > 0 (2.70)
ßòñôóôõôöô÷ôøôùûú V (t) ∼ t − 13
t3 + 25
t5 · · ·, üôýôþôÿ 2/5 ù 2/15, þ , õ . ——
2 · 27 ·
Öt¼t½ t t¸tÁ . "!#$ (2.7), «%t»t¼t½'& »t¼t½(2.50) ¾¿ß . (*) V (t) +Á*,Ï **-
V (t) =
+∞∑
n=0
cn exp(−nt) (2.71)
.0/, cn ¸Ø½ . µ*1*2*3¹*4*5Ö*6*7*8*9*:*; .<*= 9*:*; - (2.71), - (2.6)
Ú -(2.7), >*?*@*A , B*C
V0(t) = 1 − exp(−t) (2.72)
D ¸ V (t) Ô*E*F*G*HÆ , I*J
L[Φ(t; q)] =∂Φ(t; q)
∂t+ Φ(t; q) (2.73)
D ¸*K*L*M*N*O*P , ¹*O*P*+Á*N*QL [C3 exp(−t)] = 0 (2.74)
.0/, C3 ¸*R*S*T½ . U*BÖ m Ý*V*W*X*Y (2.39) ÔƸ
Vm(t)=χmVm−1(t) + ~ exp(−t)
∫ t
0
exp(τ) H(τ) Rm(V m−1) dτ
+C3 exp(−t), m > 1
.0/, R*S*T½ C3
-(2.40) ´*Z .
<*= 9*:*; (2.71)Ú*[ Ý*V*W*X*Y (2.39), K
L¼½ H(t) ¸*,Ï*V -H(t) = exp(−κ t) (2.75)
µ*\ , κ ¸Î*]½ .Ê*^*_*`
, a κ 6 0 ,[ Ý*V*W*X*Y (2.39) ÖÆ*b*c
t exp(−t)
ã , µ Ç*d*e 9*:*; (2.71). a κ > 2 , ¼½» exp(−2t) F*f Ç*g*` [ Ý*V*W*XY (2.39) ÔÆ / , h*I , i*jÅÞݽ**** , exp(−2 t) ãÖؽ*F*f Ç À*k*lm
. µno3pqrstu . vtï , ¸3wxy9:; (2.71) zxypqrstu ,
**| κ = 1. µ*Îô*Z*3*~*BÖ*K*L¼½H(t) = exp(−t) (2.76)
ß ø (2.8) ,V (+∞) 5ÿ 1. , 5ÿ (2.70) 5ÿ (2.50) . ——
· 28 · ''
*, *ï**
V1(t) = −~
2e−t + ~ e−2t − ~
2e−3t
V2(t) = −~
2
(
1 +~
2
)
e−t + ~
(
1 +~
2
)
e−2t − ~
2(1 + ~) e−3t
+~
2
2e−4t − ~
2
4e−5t
...
~*BÃ , V (t) Ô m ÝÅÞ* **
V (t) ≈2m+1∑
n=0
γm,n(~) exp(−nt) (2.77)
.0/, γm,n(~) ¸*** ~ Öؽ .-
(2.77) Ρ ¡b¡c¡K¡L¡¢½ ~ ÖÐ½Æ . ¸¡3ÌÍ ~ Uн (2.77) ±²¡NÔ£¡¤,Ê¡^¡¥¡¦¡§¡g
V ′′(0)Ú
V ′′′(0) Ö ~ ¨©M , ,¡ª 2.3 h¡« .<¡= µ¡¬ ~ ¨©M , ¡®
@ _¡` ~ ÔÁ¡¯¡° ,à U¡B¡Î¡±â¡²¡³¡´¡¡µ¡³¡¶¡·¡¸Ö¡M¡± . ~ ÔÁ¡¯¡°¡
ÅÞݽ*¹ [ *¹*º , ,*ª 2.3 h*« . ,*»*¹ ~ ÔÁ**¯*°0¼¾½ ~ Ô* ,-
(2.77)¿ g Ö V ′′(0)Ú
V ′′′(0) ÔtÐt½ÀÁ . ? ,<= Zà 2.1,
à ^ ZSÄÀÁÅV ′′(0)
ÚV ′′′(0) Ô*Æ*Ç* . , 2.4
Ú 2.5 h*« , a ~ = −3/2 È −5/4 È −1 È −3/4Ú −1/2 V ′′(0)
ÚV ′′′(0) À*Á*É*Æ*Ç* . **Ê*ËÖ* , a ~ = −1 , н*À*Á
*Ì*Í . ï*Î , Ï*Ð V ′′(0)Ú
V ′′′(0) À*Á , ~*B V (t) ÔÐ½Æ (2.77) Ñ**]*Ò*¯*°0 6 t < +∞ ¼¾À*Á*Å*Æ*ÇÆ (2.8). 5*, , a ~ = −1 , Æ (2.77)
¿ g Ö V (t) ÔÅÞÆÑ*Æ*ÇÆ (2.8) Ó e *¿ , , 2.6 h*« . Î*Ô*?º , Õ*Ö*×*Ø*Ù*Ú ~ ¨¾M , ÀÛ X*ÜÃÌÍ ~ UнÆÔ*À*Á*NÖ £*¤ .
Ý2.4 --h Þßàáâ , (2.77) ãäå V ′′(0) æá
ç 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
2 · 29 ·
Ý2.5 --h Þßàáâ , (2.77) ãäå V ′′′(0) æá
ç 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
Ý2.6 è --h= −1 â , éêë (2.77) ìíîë (2.8) æïð
t 5ç ñ 10
ç ñ 15ç ñ 20
ç òóô1/4 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
õ2.3 H(t) = exp(−t) ö , ÷øù (2.77) ú V ′′(0) ∼ ~ û V ′′′(0) ∼ ~ üýþÿ ú V ′′(0) 10
ç ñ ; ú V ′′(0) 20ç ñ ; ú V ′′′(0) 10
ç ñ ; þÿ ú V ′′′(0) 20
ç ñ
· 30 · ''
Ê Ë Ö , Ð ½ Æ (2.77) ] Ò ¯ ° 0 6 t < +∞ ¼ Á Â . S Ä Ù2.4∼2.6 Ñ 2.1∼2.3 ´'& ,
Ê^_`, tð~ tÖ ~ , Ð tÆ (2.77) &
Æ (2.69) À*Á**Í . a ~ = −1 , i*jÆ (2.77) Ô 10 ÝÅÞÆ*ÑÑ*Æ*ÇÆ*Ó e* . vï , ÐÆ (2.77) & Æ (2.69) , . ,á*h*% , ÐÆ (2.69) &Æ (2.57) . Ù , ¡S¡X¼ , ¡Õ¡Ö¡CΡÒÖ !"Å#*M*N$%ÖÆ .Ê*^*_*`
, V (t) Ô m ÝÅÞÆ (2.77) *> -***
V (t)≈1 + 2
m∑
n=1
[(−1)n exp(−2nt)]µm,n0
(
~
2
)
− exp(−t)
[(
1 +~
2
)
+~
2exp(−2t)
]m
(2.78)
.0/, µm,n
0 (x) - (2.58) Z& . *Ë'Ö* , µm,n0 (~) (Î) g*` .
<*=
N*Q (2.59), a ~ = −2 ,-
(2.78)* ß
V (t) ≈ 1 + 2
m∑
n=1
(−1)n exp(−2nt) + (−1)m+1 exp[−(2m + 1) t] (2.79)
**Ê*ËÖ* , Æ*ÇÆ (2.8) +, Ð
V (t)≈1 + 2
+∞∑
n=1
(−1)n exp(−2nt) (2.80)
à *¯*° 0 < t < +∞ ¼¾À*Á*Å*Æ*ÇÆ , -* , t = 0 . ¿ g 1 /0 −1, 1*? _2. 3*? , ¾*c45ã
(−1)m+1 exp[−(2m + 1)t]
ÅÞÆ (2.79) *]*Ò*¯*° 0 6 t < +∞ ¼¾À*Á*Å*Æ*ÇÆ . 6*Å . 3 ÝÅÞÆV (t) ≈ 1 − 2 exp(−2 t) + 2 exp(−4 t) − 2 exp(−6 t) + exp(−7 t) (2.81)
ÑÑ*Æ*ÇÆ*Ó e * , ,*ª 2.4 h*« .
ß3ö5÷5ø (2.79) ý782
m∑
n=1
(−1)n exp(−2nt)
ñ9 ý ù2
m∑
n=1
(−1)n exp(−nt)
þ , õ . ö5÷5ø (2.80) :;<ñ , õ=> . ——
2 · 31 ·
õ2.4 3 ?A@ABù (2.81) CADAEù (2.8) úFG ú 3 ç ñ ô (2.81); H þ ú òóô (2.8)
ÅÞÆ /JIK g*`
ln(1 + t)/(1 + t), t exp(−t)
L ãtÖ ' M N Ç O P . tÒtÓà Q / ,* 3ÉtÎ R tÖtÅtÞtÆ , 1 g 3tάX
Ë IKL ,t sin t, t cos t
S hUTUVUW ã Ò Ó Æ / g ` . Ù Ú XU UXUY É 19 ZU[ ÖU\U]U^U_U` , ,Lindstedt [52] È Bohlin [53] È Poincare [54] È Gylden [55]
S. a'M*k .b ^_`Îc_ + , , Lighthill [56, 57] È Malkin [58] È Kuo [59, 60]
ÚTsien [61]. -* , a t → +∞ ,
ln(1 + t)/(1 + t)Ú
t exp(−t) *d , vï ,à ^ N Çe ÒÓ*X / hTÖVWã .
h*I , 9*:*;*t*u**kf * Ù*8'MÖg&h .* 3 t exp(−t) ã*Çij e ÒÓ*X / hTÖVWã ,Ê*^k*g
V (t) lk*,Ï
tm exp(−nt) | m > 0, n > 1 (2.82)
. mtðtÑ - (2.72) ~ tÖEFGH n , Ñ - (2.73) ~ tÖKLMNOP , - oðjÖ*K*L
H(t) = 1 (2.83)Ê*^ *Õ*Ö*ÚÞÖ*X -p *~*BÖ V (t) Ô m ÝqÞ ,
. *> -***
V (t) ≈ 1 + 2
m+1∑
n=1
m+1−n∑
k=0
σm,n,k0 (~)
[
(−1)n (−nt)k
k!exp(−nt)
]
(2.84)
· 32 · ''
.0/
σm,n,k0 (~) =
1
2
[
µm,n+k0 (~) + µm,n+k−1
0 (~)]
(2.85)
*Ë'Ö* , µm,n0 (~) (Î) g*` .
<*= ~*BÖ V ′′(0)Ú
V ′′′(0) Ô ~ ¨¾M ,Ê
^_`, a −2 < ~ < 0 , Ð (2.84) ]Ò¯° 0 6 t < +∞ ¼"ÀÁÅÆÇ
n (2.8), , 2.7 h*« .
Ý2.7 è --h= −1 â , V (t) æéêë (2.84) ìíîë (2.8) æïð
t 10ç ñ 20
ç ñ 40ç ñ 50
ç ñ òóô1/4 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
¡rsÖ¡ , ¡S¡X¼ , V (t) l Û kt¡8jÖ (2.50) È(2.62) È (2.70)
Ú(2.82)
, u v a45Ï wtÎ n . ~B ! ,
Ê^ p tÒtÐnUxyn (2.57) È (2.69) È (2.78)
Ú(2.84). ÃQz ,
à ^ l**]*Ò*¯*° 0 6 t < +∞¼"ÀÁÅ tÎÒÆÇ n V (t) = tanh(t). 3? ,
-(2.57) Ù tÒ n / " q |Ì
, vï**Ì~Ö , v * UÎ*Ò ¿ Z −2 < ~ < 0 ,à Î*Ò**¯*°0¼¾À*Á .
Õ*Ö0&J 2.3 È 2.6Ú
2.7,Ê*^*_*`
, * k ÖÐn (2.78) &J**S -ÖÐn (2.69)
Ú *]ã - Ñ k e ÖÐn (2.84) , vï**ÌÖ . ?Ðn (2.84) & Ðn (2.69) . Ù*Ò*5*P!¾3 , *S*X¼ , Ρҡ+¡ÎnÖ#¡M¡N$% . n*¡k\]jÖ * , 1¡? , ðÎ*ÒÖ l Û !"qn . hð*5*P#*T , Â*Æ*ÇnJ . - à !¾3Ux Õ*Ö*×Ø*h TtÖ~ ¨¾M , Ç*Z*~*BÖ ~ Ô*¯*° , C e Ö ~ , s Ú Ø0 J*S*X¿ g ÖÐnÔ*À*Á*¯*° Ú À*Á#*$ . i*j*¹*º*À*Á*¯*°ÖÐ , Ñ**Õ*Ö*CÎ*Ò e Ö ~ , *ÉÎ*ÒÖÐn . a*4*5*Ñ*>*«*3*9*:*;*t*u Ú p*qr*s*t*u**C*½*E*F*G*Hn ÈK*L*M*N*O*P Ú K*L **h*ÉÖ*Ð D ð .
2 · 33 ·
2.3.6 *q --h ,táh% , S X - _tÖtÎÒ ' M , ÕÖ Ü # dK
L¡¢ ~ ¡hTdÝ¡V¡W¡X¡Y , 1¡? ¿ g Ρ8& g&Ö . vï , j**h n*X , *S*X*1*2*3Î* *b*c*K*L*¢ ~ ÖÐn . ¾ÐnÔ*À*Á*¯*° Ú À*Á#*$*KL*¢ ~, h*I , *I*Õ*Ö a*C*½ ~ Ô*Ë*¹*º à ^ . Ù*1*2*3Î*Òs Ú ØÐnÔ*À*Á*¯*° Ú À*Á*#*$Ö*Ü¡¢ .
£ / ,Ê*^ ð¤Î*8äåjÖ¡¢ ¥ ߦx ÐÔ*À*Á*¯*°*Çi**I
Õ¡Ö Ü Î¡Ò¡K¡L¡¢Ës Ú Ø . a¥© * ¡S¡XÔ¡N*1¡2Î*Ò§ i ȨÖ*Ã*N© .
r stÖ , µm,n0 (~) ÔZ & (2.58) Ì ¦ S X É .
à g` tÐn (2.57) È (2.78)
Ú(2.84)
/. ªÖ* , æÖ*Z&l Û 1«¬Ö®¯ã - Z
ð±²³**É .* 3¾Ùδ , µ¶Ð
1
1 + t= 1 − t + t2 − t3 + · · · = lim
m→+∞
m∑
n=0
(−1)ntn, |t| < 1 (2.86)
Z&x = 1 + ~ + ~ t
z -*¿ g1
1 + t= − ~
(1 − x)
a |x| = |1 + ~ + ~ t| < 1Ú |1 + ~| < 1, i
−1 < t <2
|~| − 1, −2 < ~ < 0
,·
1
1 + t=− ~
1 − x= −~
(
1 + x + x2 + x3 + · · ·)
= −~
+∞∑
n=0
(1 + ~ + ~ t)n
h*I1
1 + t= lim
m→+∞
[
−~
m∑
n=0
(1 + ~ + ~ t)n
]
*¯*°−1 < t <
2
|~| − 1 (−2 < ~ < 0)
߸¹ , º» = ¼½¾¿ , À5ÿÁÂÃÄÅƺ»ÇÈ = ÉÊË5ÿÌÍÎÏÐÑ , ÒÓ;ÔÕÖ× ¿ =þ
, Ø õ , ÙÚ =ÛÜÝ 7Þß . à ¹ , áâã Öäå = × ¿ . ——
· 34 · ''
¼J . ï*Î ,·æS*-
−~
m∑
n=0
(1 + ~ + ~ t)n
=−~
m∑
n=0
n∑
k=0
(
n
k
)
(1 + ~)n−k (~ t)k
=−~
m∑
k=0
m∑
n=k
(
n
k
)
(1 + ~)n−k~
k tk
=
m∑
k=0
(−1)k tk(−~)k+1m−k∑
i=0
(
k + i
k
)
(1 + ~)i
=m∑
k=0
(−1)k tk
[
(−~)k+1m−k∑
i=0
(
k + i
i
)
(1 + ~)i
]
=
m∑
n=0
(−1)n tn µm,n−1 (~)
.0/
µm,n−1 (~) = (−~)n+1
m−n∑
j=0
(
n + j
j
)
(1 + ~)j (2.87)
Ù - (2.87) Ñ*Z& (2.58) &J , ç ^ *É*,ÏèØ -
µm,n−1 (~) = µm+1,n+1
0 (~) (2.88)
vï1
1 + t= lim
m→+∞
m∑
n=0
µm+1,n+10 (~) [(−1)n tn] (2.89)
*¯*°−1 < t <
2
|~| − 1 (−2 < ~ < 0)
¼ . a ~ = −1 È ~ = −1/2Ú
~ = −1/50 ,. ÀÁ¯°SÄ * x −1 < t <
1 È −1 < t < 3Ú −1 < t < 99. é*Ä! , a ~ *d* ,
. À*Á*¯*° *
−1 < t < +∞
vï , Ð (2.89) Ô*À*Á*¯*°*Çil Û ¾K*L*¢ ~ s Ú Ø . ê*Ðé*ÄrsÖ , ätå~ tÖZ & µm,n
0 (~)¥¦ S X tÖ '¼ p , 3 ätå ë · !
2 · 35 ·
1«¬Ö®¯ã - Z*ð±²³**É . Ù*Òìi , 1*Ãíîz*>*«*3*SXÔ*N Ú e Ã*N .
Bðz*%'M , ç ^ *É*,Ïï*Ò * g&ð*Z*Ã .ñò
2.2 óô é α (α 6= 0, 1, 2, 3, · · ·), ëìõ
(1 + t)α = limm→+∞
m∑
n=0
µm,nα (~)
(
α
n
)
tn (2.90)
ö÷Jø
−1 < t <2
|~| − 1 (−2 < ~ < 0)
ùJúû, üý
(
α
n
)
=α(α − 1)(α − 2) · · · (α − n + 1)
n!
þ
µm,nα (~) = (−~)n−α
m−n∑
j=0
(−1)j
(
α − n
j
)
(1 + ~)j (2.91)
ÿ x = 1 + ~ + ~ t. |x| < 1 |1 + ~| < 1,
−1 < t <2
|~| − 1, −2 < ~ < 0
ð®¯*à [62], |x| < 1 |1 + ~| < 1 , ·
(1 + t)α =(−~)−α(1 − x)α = (−~)−α+∞∑
n=0
(−1)n
(
α
n
)
xn
=(−~)−α+∞∑
n=0
(−1)n
(
α
n
)
(1 + ~ + ~ t)n
= limm→+∞
(−~)−αm∑
n=0
(−1)n
(
α
n
)
(1 + ~ + ~ t)n
· 36 · z m *
(−~)−αm∑
n=0
(−1)n
(
α
n
)
(1 + ~ + ~ t)n
=(−~)−αm∑
n=0
(−1)n
(
α
n
)
n∑
j=0
(
n
j
)
(1 + ~)n−j~
j tj
=(−~)−αm∑
j=0
tjm∑
n=j
(−1)n
(
α
n
)(
n
j
)
(1 + ~)n−j~
j
=(−~)−αm∑
j=0
tjm−j∑
i=0
(−1)i+j
(
α
i + j
)(
i + j
j
)
(1 + ~)i~
j
=(−~)−αm∑
j=0
tjm−j∑
i=0
(−1)i+j
(
α
j
)(
α − j
i
)
(1 + ~)i~
j
=m∑
j=0
[(
α
j
)
tj
]
m−j∑
i=0
(−1)i
(
α − j
i
)
(1 + ~)i (−~)j−α
=
m∑
n=0
µm,nα (~)
[(
α
n
)
tn
]
µm,n
α (~) = (−~)n−αm−n∑
j=0
(−1)j
(
α − n
j
)
(1 + ~)j
¥ .
uv& (2.91) α 6= 0, 1, 2, 3, · · · ð , !"#$%&' −∞ < α <
+∞ (%) . #*' k, (2.91), +
µm,nk (~) = (−~)n−k
m−n∑
j=0
(
n − k − 1 + j
j
)
(1 + ~)j (2.92)
!,-. µm,n0 (~) / (2.58) 01 µm,n
−1 (~) / (2.87). 2034 , #56&'α ∈ (−∞, +∞), +
µm,nα (−1) = 1 (2.93)78
, #56%9:*' n, +lim
m→+∞µm,n
α (~) = 1, |1 + ~| < 1 (2.94)
;2 < =?>?@?A?B · 37 ·
CD, EF/ (2.58) (2.91), #*' l > 0, +
µm,n−l (~) = µm+l,n+l
0 (~) (2.95)
(2.93) GHIJKL . |1+~| < 1 , EF/ (2.91), #5M%9:*' n > 0,
+lim
m→+∞µm,n
α (~)
=(−~)n−α+∞∑
k=0
(−1)k
(
α − n
k
)
(1 + ~)k
=(−~)n−α+∞∑
k=0
(
α − n
k
)
(−1 − ~)k
=(−~)n−α [1 + (−1 − ~)]α−n
=1
3 .NOPLG ,
RQ(2.58) /L µm,n
0 (~) 01 (2.87) /L µm,n−1 (~) STUV (2.91) α = 0 α = −1 LWX . $%YZ , [\]^_34R.`aTbcd ' Q (2.57) e (2.78) (2.84) Lfgh%)h .
' Q (2.57) e (2.78) (2.84) fghi2jklm c Qn . o$pq , lrs 'Ltu'GvlL . EFhw (2.60), |1 + ~| < 1 , #56l rx L%9:*' N , +
limm→+∞
N∑
n=0
(
α2n+1t2n+1
)
µm,n0 (~) =
N∑
n=0
α2n+1 t2n+1
y C,x 56%9:*' N , z|'~ , ' Q (2.57) N Q
(2.12) N l .y C
, ' (2.57) ftu'vlh , %LM/ . (α1, α3, α5, α7, · · ·)
S Ll , Y , αk (k = 1, 3, 5, · · ·)
VQ(2.12) ' .
Q(2.12) 2 V l Γ0
(α1, 0, 0, 0, · · ·)(α1, α3, 0, 0, · · ·)
· 38 · (α1, α3, α5, 0, · · ·)
...
z (α1, α3, α5, α7, · · ·) ¡9¢£ . ¤I , ' Q (2.57) 2 V l¥/L Γ (~)
(α1 µ0,00 (~), 0, 0, 0, · · ·)
(α1 µ1,00 (~), α3 µ1,1
0 (~), 0, 0, · · ·)(α1 µ2,0
0 (~), α3 µ2,10 (~), α5 µ2,2
0 (~), 0, · · ·)...
z`l r (α1, α3, α5, α7, · · ·)
L¡9¢£ .
N¦MLG , _§ Γ (~) ¨©~ª«¬' ~. EFhw (2.59),
Γ (−1)( ~ = −1 ) `~L Γ0. ~ 6= −18
|1 + ~| < 1
, Γ (~) `~ Γ0. ®¯ C , EFhw (2.60), !°[ z`l(α1, α3, α5, α7, · · ·).
y C, ' Q (2.57) 2 V lm `L Γ (~)
z`l (α1, α3, α5, α7, · · ·) L¡9¢£ . o$pq , Y±¡9¢£L²³´µ¨©~$¶jL z . X , ¡9
lim(x,y)→(0,0)
√
x2 + y2
|x|
%·¸`L¹eºz (0, 0) L z .V»¼
, S½¾ y = βx, Y ,
β 2 V 5M&' . HIJK , +lim
(x,y)→(0,0)
√
x2 + y2
|x| =√
1 + β2
$0 , ¿¡9¨©~ z (0,0) L À . Y Qn . VÁ ' Q (2.57) ÃÄÅÆ ¨©~ª«¬' ~.y V s ' µm,n
0 (~) ´¢`L ~
N/.`L z
.ÇÈ_ Qn ,
s ' µm,nα (~) ÉÊ´¢`L α ~
N/¡9¢£L`
z . ~Y rËÌ , Í°/ µm,n
α (~)VÏÎÐÑÒÓÔÕ
.V .Ö σm,n
0 (~) / (2.85) M/ , /
σm,n,kα (~) =
1
2
[
µm,n+kα (~) + µm,n+k−1
α (~)]
(2.96)
;2 < =?>?@?A?B · 39 ·
VÏÎ×ÑÒÓÔÕ,
, |1 + ~| < 1, −∞ < α < +∞. J3ÙØÚ# α ∈ (−∞, +∞),
0 6 n 6 m + 1, +σm,n,k
α (−1) =
1, 0 6 k < m + 1 − n
1/2, k = m + 1 − n(2.97)
lim
m→+∞σm,n,k
α (~) =
1, |1 + ~| < 1
∞, |1 + ~| > 1(2.98)
, n k ( V %9:*' . z s ' µm,n
α (~) σm,n,kα (~)
%Û V LM/ ,y C
, ÉÊ·Ü£ÝÞßÜ' Q à ÄÅÆ . X ,
s ' f(z) Ltu'+∞∑
n=0
f (n)(z0)
n!(z − z0)
n
20/ ÎÐÑàáâãäÕ
limm→+∞
m∑
n=0
µm,nα (~)
[
f (n)(z0)
n!(z − z0)
n
]
Î×ÑàáâãäÕ
limm→+∞
m∑
n=0
σm,n,0α (~)
[
f (n)(z0)
n!(z − z0)
n
]
Y , µm,nα (~) σm,n,0
α (~)TU / (2.91) (2.96) / . ´¢åæfçL ~
α
N, 2HèÞßÜYZ¥/tu'à ÄÅÆ . X , '
V (t)≈m∑
n=0
µm,nα (~)
[
α2n+1 t2n+1]
(2.99)
V (t)≈1 + 2
m∑
n=1
[(−1)n exp(−2nt)]µm,nα
(
~
2
)
− exp(−t)
[(
1 +~
2
)
+~
2exp(−2t)
]m
(2.100)
V (t) ≈ 1 + 2
m+1∑
n=1
m+1−n∑
k=0
σm,n,kα (~)
[
(−1)n (−nt)k
k!exp(−nt)
]
(2.101)
· 40 · Ö m |ézé Q (2.57) e (2.78) (2.84) ééMé/ , Yé , |1 + ~| < 1, α ∈(−∞, +∞). α = π/4 , é' (2.99) éà ÄéÅéÆ ~ é~éêééëéìéëéÜ ,
í
2.5 $î . ~ = −1, α = ±1/2 e ±π/4 , ' (2.100)xï L 20 |z Q (ðñQò f ,
2.8 $î . ~ = −1/2, α = ±1/2 e ±π/4 ,
' (2.101)xï
L 20 |z Qó ðñQò f ,
2.9 $î .y C
,s ' µm,n
α (~) σm,n,kα (~)
ñ& %M/ .
ô2.5 α = π/4 õ , ö?÷?ø (2.99) ù?ú?û?ø (2.8) üÏýÿþ
; ~ = −1 ; ~ = −1/2 ; ~ = −1/5 ; ~ = −1/10
2.8 --h= −1, α = ±1/2 ±π/4 , 20 (2.100) (2.8) t α = −π/4 α = −1/2 α = 1/2 α = π/4
1/4 0.244 9 0.244 9 0.244 9 0.244 9 0.244 9
1/2 0.462 1 0.462 1 0.462 1 0.462 1 0.462 1
3/4 0.635 1 0.635 1 0.635 1 0.635 1 0.635 1
1 0.761 6 0.761 6 0.761 6 0.761 6 0.761 6
3/2 0.905 1 0.905 1 0.905 1 0.905 1 0.905 1
2 0.964 0 0.964 0 0.964 0 0.964 0 0.964 0
5/2 0.986 6 0.986 6 0.986 6 0.986 6 0.986 6
3 0.995 1 0.995 1 0.995 1 0.995 1 0.995 1
4 0.999 3 0.999 3 0.999 3 0.999 3 0.999 3
5 0.999 9 0.999 9 0.999 9 0.999 9 0.999 9
10 1.000 0 1.000 0 0.999 9 1.000 0 1.000 0
100 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0
;2 < =?>?@?A?B · 41 ·
2.9 --h= −1/2, α = ±1/2 ±π/4 , 20 (2.101) (2.8)! t α = −π/4 α = −1/2 α = 1/2 α = π/4
1/4 0.244 9 0.244 9 0.244 9 0.244 9 0.244 9
1/2 0.462 1 0.462 1 0.462 1 0.462 1 0.462 1
3/4 0.635 1 0.635 1 0.635 1 0.635 1 0.635 1
1 0.761 6 0.761 6 0.761 6 0.761 6 0.761 6
3/2 0.905 1 0.905 1 0.905 1 0.905 1 0.905 1
2 0.964 0 0.964 0 0.964 0 0.964 0 0.964 0
5/2 0.986 6 0.986 6 0.986 6 0.986 6 0.986 6
3 0.995 1 0.995 1 0.995 1 0.995 1 0.995 1
4 0.999 3 0.999 3 0.999 3 0.999 3 0.999 3
5 0.999 9 0.999 9 0.999 9 0.999 9 0.999 9
10 1.000 0 1.000 0 0.999 9 1.000 0 1.000 0
100 1.000 0 1.000 0 1.000 0 1.000 0 1.000 0
"$#$%, Í°$& ï , ' `a Tbcd$($)$* L µm,n
0 (~) $+/ (2.58) 20$,+Þ\.-./.0.1.2.+g.3.4.5.6 . Í°.7.8 Ø 'à ÄÅÆ.9 à Ä.: Ý ñ &ÉÊ´¢.;.<l r ª«¬'ìP %$9.=$> . Í°i.& ï ,
s ' µm,nα (~)
9σm,n.k
α (~)
' ~9
α `N+/`L z . $%YZ , [ V `a Tbcd %)h.?@ .l r.A &L¹eÿghL.B.C .
2.3.7 D.E - F.G Ó.H#~ "JIJK X , `a Tbcd B~Yl rJLJM Ø Φ(t; q) ' (2.36) q = 1N Ã Ä . O$PLG , `a Tbcd L$Q$RTS , Í°$U%·ÜLWVX' åæ$Y$Z$[$\Q
V0(t) e ª«.]h.^._ L e ª« s ' H(t)9 ª«¬' ~. `!°[åæfç , 2.a
3.b' (2.36) q = 1N Ã Ä . W U.c OPLG , 'R`a Tbcd xï L.b' Qd à ÄÅÆ.9 à Ä.: ݨ©~ª«¬' ~.
y C, ª«¬' ~ ? @ .l r P %.9.=> b' Q d à ÄÅÆ.9 à Ä.: ÝL »¼.e .fVJgJh b'Ã Ä L cd d l , iJjz (Pade approximation) k¥Jl¶j . b
'+∞∑
n=0
cn xn
d[m, n] |.i.jz V
m∑
k=0
am,k xk
n∑
k=0
bm,k xk
mon, am,k e bm,k ' é' cj (j = 0, 1, 2, 3, · · · , m + n)
ñ + . ´éµ , _qpéééLqijzÉ·Ü£Ý_ßÜ x +$b' d à ÄÅÆ ,
g$h$m à Ä$: Ý . X , #
· 42 · rTsTtouQ
(2.12) vj.i.jz , [1, 1] e [2, 2]9
[3, 3] |.i.jzt,
3t
3 + t2,
t(15 + t2)
15 + 6t2Q(2.12)
d[m, m] |.i.jz2 .w.x
m∑
n=0
am,n0 tn
m−1∑
n=0
bm,n0 tn
, mV.y ' (2.102)
z.m−1∑
n=0
am,n0 tn
m∑
n=0
bm,n0 tn
, mV.| ' (2.103)
mn, am,n
0
9bm,n0
V ' .
N¦MLG , ~ t → +∞ , _.p. r .i.jz
~ z ê .Q
(2.12)d
[4, 4]9
[10, 10] |.i.jz í 2.6 $î .
ô2.6 - (2.106) (2.103) ù?ú?û?ø (2.8) üÏýÿþ
; [4,4] - ; [4,4] ;
[10,10]
`a - i.jz [50] _.p.i.jz 9 `a Tbcd. ²f .V . ñ a.b
' (2.36) q = 1 Ã Ä ,(.) vj.i.jz * .~..<.. q
d[m, n] |
;2 < =?>?@?A?B · 43 ·
i.jz Qm∑
k=0
Am,k(t) qk
n∑
k=0
Bm,k(t) qk
(2.104)
mn, ' Am,k(t)
9Bm,k(t) '..1z Q
V0(t), V1(t), V2(t), · · · , Vm+n(t)ñ + . . , Q (2.104)n
q = 1,7. j.2 (2.32), [m, n] |`a - i.jz
Qm∑
k=0
Am,k(t)
n∑
k=0
Bm,k(t)
(2.105)
# ".I.K X , _.p' Am,n(t)9
Bm,n(t) ¨©~$åæL V (t)d B s ' . `
¶j.B s ' (2.62),TU % vL [1, 1] |`a - i.jz
t(12 + 16t + 7t2)
(1 + t)(12 + 4t + 7t2)9[2, 2] |`a - i.jz
t(168 000 + 362 880 t + 238 000 t2 + 14 160 t3 − 47 124 t4 − 36 308 t5 − 13 419 t6)
3(1 + t)(56 000 + 64 960 t + 33 040 t2 + 12 000 t3 − 2 508 t4 − 9 076 t5 − 4 473 t6)
#~ ".I.K X ,m
[m, m] |`a - i.jz20 .w.xm2+m+1∑
n=1
am,n2 tn
m2+m+1∑
n=0
bm,n2 tn
(2.106)
mTn, am,n
2
9bm,n2
V ' . ¡µ%$¢LG , Í°$£$¤ , am,n2
9bm,n2 ¨©~ª«
¬' ~. ´¢ 2 (2.106) .2 (2.102) e¥2 (2.103) ¦.§¨© , Í°.£.¤ , [m, m] |`a - i.jz.ª ð Ý_`~ [m2 + m + 1, m2 + m + 1] |.i.jz . .i.jzJ2 (2.102)
9 2 (2.103) ª t → +∞ ~ z ê , I`a - iJjz (2.106)
ª t → +∞ ~ 1.y C
, # x +:*' m, [m, m] |`a - i.jz (2.106) ¨ [m, m] |.i.jz (2.102)
9(2.103) ðñ . X ,
í2.6 $î , [4,4] |`
a - i.jz¨R [4,4] |.i.jz Q .« ñ , ¬.¨ [10,10] |.i.jz%) . W U Þ , `¶j.B s ' (2.70) ,
m #.vL [1, 1] |`a - i.jz V
· 44 · rTsTtou1 − exp(−2t)
1 + exp(−2t)(2.107)
!:$®$¯G ðñQ V (t) = tanh(t).y C
, `a - i$jz ñ &T¨ $i$jz%) .
`é , véjé`éa - iqjézééÉ gq: bé'éÃ Ä . X ,V . gq: bé' V ′′(0)
9V ′′′(0) Ã Ä , 2 (.) #.b'
∂2Φ(t; q)
∂ t2
∣
∣
∣
∣
t=0
=+∞∑
n=0
V ′′n (0) qn
9∂3Φ(t; q)
∂ t3
∣
∣
∣
∣
t=0
=
+∞∑
n=0
V ′′′n (0) qn
¶jJiJjz ,TU m ~JJ<JJ q
d[m, n] |JiJjz , ¤J q = 1,
m [m, n] |`a - i$jz . ' Q (2.69) 6 ï L V ′′(0)9
V ′′′(0)d `a - i
jz$°K 2.10. ' Q (2.78) 6 ï L V ′′(0)9
V ′′′(0)d `a - i$jz , °K
2.11. ª_.p.m.±.² n , `a - i.jz[HèÞ g.: . V ′′(0)9
V ′′′(0)d ÃÄ
. 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
[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
# "qIqK X , Íé°q£q¤ , $é% [m, m] |é`éa - iqjézéé[éé¨é©é~éªé«é¬é'~.y C
, »éÖéåéæéL ~
Nq¼ , \éIq6q½ véLqbé' Q éÃ Ä , Íé°q¾é¤é2é¶éj
`a - i.j cd Ã Ä ²³ ..".¿ .ÀSÁ$î , [m, m] |`a - i.jz.Â
¨©ª«¬' ~. Ã.ÄLG , Í°iÉ.ªl.Å.±.².Æ xï m '.Ç34 .
;2 < =?>?@?A?B · 45 ·
$%YZ[JÈ 4 Ø `a - iJjzÉHèÞßÜÉ' `a Tbcd xï L.b' Qd à ÄÅÆ ,g.h.m à Ä.: Ý .ÊJË ÞJÈ , ª "JI , Í°´¢l r »JÌJK XJÍJÎ.`a Tbcd d B ".ÏJÐ . Í
°JÑXJÈ 4 Ø `~$% Qbcd , `a Tbcd 2 xï ¨©~ª«¬' ~ L e'R`.B s ' .w Ll.Ò.b' Q .
j.Y.Z.[.\ Q e ª«.]h.^._ 9 ª« s 'åæ_LÓVÔ' , Í°2 QJwJx
Ç`JB s 'JÕ.ÖL.b' ,
y C, 2jJ©J¼LJB s '
%)Þ zl r ¡.]h.×.ØL Q . Í°.? ï.Ù.Ú.Û.Ü.Ý 9.ÞÕ.ß.à Ü.Ý , 0.&.6.YZ.[.\ Q e ª«.]h.^._ 9 ª« s 'Låæ . YZ.á.â.ª·Ü£Ý_ ».ã .`a Tbcd L.vj . Í°34 Ø ª«¬' ~ ÉÊP %.9.=.> b' Q d à ÄÅÆ.9 à Ä.:Ý .CD
, ´¢.ä > ~ å] , 2.æ.çfçL ~
N, 0 ñ a.b' Q Ã Ä . I 8 , Í°.? ï
.`a - i.jz , 0 g.: b' Q LÃ Ä , ¿ cd ¨R.i.jz V %) .
è3 é êìëìíìîìï
ªJð 2IÉn
, Í°jl r »JÌ LXJ_JÍJÎ.`a Tbcd L.B "JÏ.Ð ."JI
, ¦l.ñ#`a Tbcd ¦.§hL.ò.p .
3.1 óõô÷öùøùúüû´µ , l r ¡.]h.×.Ø2jl.ý =.>c £01.Y$Z 9 (
z.) þ.ÿì.ò.p .V»¼
, ½¾l r l.Å.2L.¡.]h c £N [u(r, t)] = 0 (3.1)
mn, NV ¡.]h.^._ , u(r, t)
V q s ' , r9
tTU.9 V . .
u0(r, t) î ðñQ u(r, t)
d Y$Z$[$\ Q , ~ 6= 0V ª«¬' , H(r, t) 6= 0V ª« s ' , L
V ª«.]h.^._ , ¿.^._ %hwL [f(r, t)] = 0, ` f(r, t) = 0 (3.2)
q ∈ [0, 1]fV .<.. , ÆÏ`a
H[Φ(r, t; q); u0(r, t), H(r, t), ~, q]
= (1 − q) L[Φ(r, t; q) − u0(r, t)] − q ~ H(r, t) N [Φ(r, t; q)] (3.3)NOPLG , _.ÀL`a,-.¡êª«¬' ~
9 ¡êª« s ' H(r, t). FÍ°$q , ;.<.¡êª«¬' ~
9 ¡êª« s ' H(r, t)C `a ( . $0 , Y
L`a¨RL`a l$Åh . ª«¬' ~9 ª« s ' H(r, t) ª`a Tbcdn L f j .
NOPLG , Í°.ªåæ.Y.Z.[.\ Q u0(r, t) e ª«.]
h.^._ L e¡êª«¬' ~9 ª« s ' H(r, t) _.U%ÛÜL V' .M
q ∈ [0, 1]V .<.. .
`a (3.3)V ê , »
H[Φ(r, t; q); u0(r, t), H(r, t), ~, q] = 0
Í°çê|. c £
(1 − q) L[Φ(r, t; q) − u0(r, t)] = q ~ H(r, t) N [Φ(r, t; q)] (3.4)
;3 < ?@?A?B · 47 ·
mn, Φ(r, t; q)
V _.p c £ d Q ,m S¨©~.Y.Z.[.\ Q u0(r, t) e ª«.]h.^._
L e ª« s ' H(r, t)9 ª«¬' ~, I 8ó ¨©~..<.. q ∈ [0, 1]. ~ q = 0 ,
ê|. c £ (3.4)xV
L[Φ(r, t; 0) − u0(r, t)] = 0 (3.5)
'Rhw (3.2), JqΦ(r, t; 0) = u0(r, t) (3.6)
~ q = 1 , 'R~ ~ 6= 09
H(r, t) 6= 0, ê|. c £ (3.4) `~.á.Z c £N [Φ(r, t; 1)] = 0 (3.7)
.`Φ(r, t; 1) = u(r, t) (3.8)
y C, EF.2 (3.6)
9(3.8), ~..<.. q \ 0 ßÜ.ç 1 , Φ(r, t; q) \.Y.Z.[.\ Q
u0(r, t) . ã (z . ) ç.á.Z c £ (3.1) L Q u(r, t). ª`ag n , Ym
ãV . . Y:G c £ (3.4) k V ê|. c £ d á y .
+/ m |..6'u
[m]0 (r, t) =
∂mΦ(r, t; q)
∂qm
∣
∣
∣
∣
q=0
(3.9)
vjÏtu.Õ.Ö.+g , Φ(r, t; q) 2.Õ.Ö x qd b'
Φ(r, t; q) = Φ(r, t; 0) ++∞∑
m=1
u[m]0 (r, t)
m!qm (3.10)
um(r, t) =
u[m]0 (r, t)
m!=
1
m!
∂mΦ(r, t; q)
∂qm
∣
∣
∣
∣
q=0
(3.11)
j.2 (3.6)9
(3.11), Φ(r, t; q)d b' (3.10) 2 .wV
Φ(r, t; q) = u0(r, t) +
+∞∑
m=1
um(r, t) qm (3.12)
NOPLG , Í°.U%ÛÜL V'Råæ.Y.Z.[.\ Q u0(r, t) eÿª«.]h.^._ L e¡
ꪫ¬' ~9 ª« s ' H(r, t).
L.M °[åæfç , \I Ø(1) $% q ∈ [0, 1], ê|. c £ (3.4) L Q Φ(r, t; q) (.ª ;
(2) m = 1, 2, 3, · · · , +∞, ..6 u[m]0 (r, t) .ª ;
· 48 · rTsTtou(3) Φ(r, t; q)
d b (3.12) ª q = 1 Ã Ä .Â, ª_.p L.M Æ , EF.2 (3.8)
9(3.12), Í°
ç.b Q
u(r, t) = u0(r, t) ++∞∑
m=1
um(r, t) (3.13)
2 (3.13)xï ðñQ
u(r, t)9 YJZJ[J\ Q u0(r, t)
d ,mÉn
, qJ1 um(r, t) '
Æ.À xï L|.£ ñ + .
3.2 õô÷öùøùúüû !"
, +#%$un = u0(r, t), u1(r, t), u2(r, t), · · · , un(r, t)
&' +# (3.11), um(r, t)d =.> ()*+,.( (3.4)
n-. ç . /+,.( (3.4) .0.. q 1 . m
, .23 m!, 4. q = 0, 56
ç m,.(7
L [um(r, t) − χm um−1(r, t)] = ~ H(r, t) Rm(um−1, r, t) (3.14)
mn, χm 8 (2.42) 9# , :
Rm(um−1, r, t)=1
(m − 1)!
∂m−1N [Φ(r, t; q)]
∂qm−1
∣
∣
∣
∣
q=0
(3.15)
/; (3.12) <0; (3.15), =Rm(um−1, r, t) =
1
(m − 1)!
∂m−1
∂qm−1N[
+∞∑
n=0
un(r, t) qn
]∣
∣
∣
∣
∣
q=0
(3.16)
> ?@AB, C=,D( (3.14) E=FG
AHIJKL, : Rm(um−1, r, t)
MLMNMOM9MPHMIMJMK
N QM)M3 8 (3.15).MR
.&M' 9M# (3.15), M,MSMDMM(
(3.14)
ATUVWXYZ[um−1. \] , ^_
W`1a
HIA,SD( (3.14),
)W`bc
u1(r, t), u2(r, t), · · · de , u(r, t)
Am ,fg
u(r, t) ≈m∑
k=0
uk(r, t) (3.17)
7ihkjkl (3.12) mknkokpkqkrkskt (3.4), u q vkwkxkykzk lk| o , k~kkkkkkkpkqrst(3.14) (3.15), (3.16). ——
3 · 49 ·
+,SD() AS; . A(q) B(q)
|q| 6 1 a
A ¡
( ¢ ¤£ 0 ¡ ), ¥¦A(0) = B(0) = 0, A(1) = B(1) = 1 (3.18)
§A(q) =
+∞∑
k=1
αk qk, B(q) =
+∞∑
k=1
βk qk (3.19)
¨M©MªM«A(q) ¬ B(q)
AMM®M¯M°M± ¡. \ A(q) ¬ B(q)
|q| 6 1 ²aM , 8; (3.18), ³´
+∞∑
k=1
αk = 1,
+∞∑
k=1
βk = 1 (3.20)
56µ¶ · ¸ #A+,SD¹(
[1 − B(q)] L[Φ(r, t; q) − u0(r, t)] = A(q) ~ H(r, t) N [Φ(r, t; q)] (3.21)
C=º»F¼A½
;QFG , 2¾¿À · ¸ #AÁ
,SD¹(L[
um(r, t) −m−1∑
k=1
βk um−k(r, t)
]
= ~ H(r, t) Rm(um−1, r, t) (3.22)
º%ÂRm(um−1, r, t) =
m∑
k=1
αk δm−k(r, t) (3.23)
:δn(r, t) =
1
n!
∂nN [Φ(r, t; q)]
∂qn
∣
∣
∣
∣
q=0
(3.24)
ÃÄ, +,SD¹( (3.4) ¬
Á,SD¹( (3.14)
BA(q) = B(q) = q Ź( (3.21)¬ (3.22) ÆÇÈÉS .
^MÊ ,M· P
HMIMËMÌ)MÍ MÎMÏMÐ ¹M(M¬MFM¼ÒÑMÓ ( ÔMÕ ) ÖM×MØMÙ .
¾!M", 5M6MÚMÛMÜM M· ¹M( (3.1) ÝMÞMß
IMàMáMâGMã ¨ M¹Mä
AMåMæMçMè.Äé
, ¹( (3.1)
AS;PÊ , êëÜ)3ØÙ ·ÏÐ ¹( , ì)3ØÙ · ÔÕ ( ÑÓ ) Ö× . ê)3
Bí ¨ ¹(îðï ¨ ¹(ñí ¨
- ï ¨ ¹(òó< ¡ ¹( . C= ÏÐ ¹(¬ÔÕÖ×Qôõg
A¹äöM÷ , øùúëG
A ÏÐ ¹(¬ÑÓ ( ÔÕ ) ÖM× , ûMüMýMþMëMGAÑMÓMÿMañ ëMG AMHMIMJMK ¬MëGMõ A £ 0 ¡
A(q) B(q). ] , MëM9 ÏMÐ ¹M(MòMóMÑMÓ ( ÔMÕ ) ÖM× Â LMN( ) ¡ . \] , Ùa¹ä B A .
· 50 ·
3.3 ± ¡ A PMÊ .
!"# AM± ¡$$ %Më . )M3&' ,( G
ã ¨ ¹äO RA± ¡ a (3.13)
, ê9 B ü ËÌ Æa .)*
3.1 ( +, )* ) -./u0(r, t) +
+∞∑
m=1
um(r, t)
021, 354 , um(r, t) 62758:92;2<2=2> (3.22), ?2@2A (3.23) (3.24) B (2.42) C2D ,EFG @H=> (3.1) IJ .K §
s(r, t) = u0(r, t) +
+∞∑
m=1
um(r, t)
ª« ± ¡. 8 ; (3.22) ¬ (2.42), 56=M7
~ H(r, t)
+∞∑
m=1
Rm(um−1, r, t)
=
+∞∑
m=1
L[
um(r, t) −m−1∑
k=1
βk um−k(r, t)
]
=L[
+∞∑
m=1
um(r, t) −+∞∑
m=1
m−1∑
k=1
βk um−k(r, t)
]
=L[
+∞∑
m=1
um(r, t) −+∞∑
k=1
+∞∑
m=k+1
βk um−k(r, t)
]
=L[
+∞∑
m=1
um(r, t) −+∞∑
k=1
βk
+∞∑
n=1
un(r, t)
]
7MLONOPOQOROS OTOUOV |= L
[(
1 −
+∞∑
k=1
βk
)
s(r, t)
]
OWOXOY , ZO[ QO\O] . ^`_ 1 −∑+∞
k=1 βk = 0, aOXOYObOcOdOe QOfOg z ]OhOi . ——
3 · 51 ·
=L[(
1 −+∞∑
k=1
βk
)
+∞∑
m=1
um(r, t)
]
=L(
1 −+∞∑
k=1
βk
)
[s(r, t) − u0(r, t)]
\j ~ 6= 0 H(r, t) 6= 0, 8 ; (3.20) ¬ (3.2), ³´+∞∑
m=1
Rm(um−1, r, t) = 0 (3.25)
k ¹l , mn9o (3.23) ¬ (3.24), ³´+∞∑
m=1
Rm(um−1, r, t)=
+∞∑
m=1
m∑
k=1
αk δm−k(r, t)
=
+∞∑
k=1
+∞∑
m=k
αk δm−k(r, t) =
(
+∞∑
k=1
αk
)
+∞∑
n=0
δn(r, t)
mn; (3.20) (3.24) ¬ (3.25), ³´+∞∑
m=1
Rm(um−1, r, t)=
+∞∑
m=0
δm(r, t)
=
+∞∑
m=0
1
m!
∂mN [Φ(r, t; q)]
∂qm
∣
∣
∣
∣
q=0
= 0 (3.26)
^Ê , Φ(r, t; q) 륦pÓP HI ¹q (3.1).§
E(r, t; q) = N [Φ(r, t; q)]ª« ¹q (3.1)
Ar st . u ÃÄE(r, t; q) = 0
úûY pÓ¹q (3.1)
Avwa . mnl9o ,
r st E(r, t; q) ¼Y £ 0D
[qA®¯°± ¡ j
+∞∑
m=0
qm
m!
∂mE(r, t; q)
∂qm
∣
∣
∣
∣
q=0
=+∞∑
m=0
qm
m!
∂mN [Φ(r, t; q)]
∂qm
∣
∣
∣
∣
q=0
mn (3.26), x q = 1 Å , Ù ªy ;O RE(r, t; 1) =
+∞∑
m=0
1
m!
∂mE(r, t; q)
∂qm
∣
∣
∣
∣
q=0
= 0 (3.27)
· 52 · mn E(r, t; q) Æ9o , ; (3.27)
@z, x q = 1 Å , 56
bc pÓ¹q (3.1)
Avwa . \] ,
V ± ¡u0(r, t) +
+∞∑
m=1
um(r, t), ê9jpÓ¹q (3.1)
A · a . &| .)*3.2 -./
u0(r, t) +
+∞∑
m=1
um(r, t)
021, 354 , um(r, t) 62758:92;2<2=2> (3.22), ?2@2A (3.23) (3.24) B (2.42) C2D ,E
+∞∑
m=1
Rm(um−1, r, t) =
+∞∑
m=0
δm(r, t) = 0
K ~ Ù&'Â bcA ªy ; (3.26), ü9÷ ¾ Ä . &| .
¹q (3.14) Üj¹q (3.22)
A(q) = B(q) = q ÅAÇÈÉ . \] , 56=
ÀÙ9÷ .)*3.3 -./
u0(r, t) +
+∞∑
m=1
um(r, t)
01, 34 , um(r, t) 6789;<=> (3.14), ?@A (2.42) B (3.15) CD ,
EFG H=> (3.1) J , ?+∞∑
m=1
Rm(um−1, r, t) = 0
mn9÷ 3.1 ¬9÷ 3.3, 56V
Âv
ýþÑMÓMÿMa u0(r, t) H
IMJMKL £ 0 M¡ A(q) ¬ B(q)
¡ ~ M¡ H(r, t), 3wM± ¡
(3.13)
. 9M÷ 3.2 M¾ % , 8 GMã ¨ M¹MäMO R
AM± ¡ fMgMaMÆI¬v
.
3.4 ß A ¬MP ¹Mä , òMò àMåMY M . G , GMã ¨ M¹MäMìåY2
47 2 R A . ÷2¡2 , Ú =2¢Gã ¨ ¹äÆ22£ I .Ä é
, G㨠¹ä22¾2¤2 A¥:¦ ýþÑÓÿ2a u0(r, t) 2HIJK
L 2 ¡ ~
3 · 53 ·
¬ ¡
H(r, t). Ù ¥§¦ ¿]¨© , ª« Y¬ =®ô¥¦C=Ú M . Cª , ¯ B Ú ¥§¦§°± ¾Gã ¨ ¹ä=² I ¬³´ I Æ åµ .k ¹l , ¶û A· é¸ , Ú ¥§¦§¹ gº»_¨© . \] , ûü åæ p2¼Ý2½2¾ ¬2 ýþÑÓÿ2a u0(r, t) 2HIJK
L ¬2 ¡
H(r, t).¿À,
¬ÁÂà ·ÄÅ.
, ¿
2 ÆC « ,· P
HIËÌAa®ªëG Aå ¡ ªy
.
Ç,È ÉS% , ¶É÷ÊˬÔÕ ( ÑÓ ) Ö× RÌ , Í ± ýþN å ¡ ØÙO ± P HIËÌA aÎP2Ï Ä . C2ª , ú · O
±APHIËÌ
,
¬2¿À ®MýMþ MÎ å M¡ Ý ªy a . ÚM¾Mü ËMÌMA ÑÐÒÓ . ÔM¿ , M ¬ýþ Î
å ¡en(r, t) | n = 0, 1, 2, 3, · · · (3.28)
u(r, t) =
+∞∑
n=0
cn en(r, t) (3.29)
ªy ¹q (3.1) Æa u(r, t), º%Â , cn jÞ ¡ . ; (3.29) Õ¾¹q (3.1)
A ÐÒÓ. j¾Ö×ü ÐÒÓ , ÑÓÿa u0(r, t) ØÙ å ¡ ª« , Ù
u0(r, t) =
M0∑
n=0
an en(r, t) (3.30)
ÚÛ , an
BÞ ¡ , M0
B · ¯Ú ¡ .éÛ
, j¾Ö× ÐÒÓ (3.29),
HIJKLØ¥¦¿ÀÖ×Ýܹq
L[w(r, t)] = 0
ÆaØ ¦ Ù å ¡ ªy , Ùw(r, t) =
M1∑
n=0
bn en(r, t) (3.31)
º%Â , bn jï ¨ Þ ¡ , Ú ¡ M1
¦HIJKL ÆÞ Áß ¾ ¡ w± , üÞ Áß ¾ ¡ ^Ê2à2pÓ¹2q (3.1)
A Þ Á2ß ¾ ¡ FG . ÆC2ª¿] ,
B\2j Á2ß SD¹2q (3.22) Æ
^ajum(r, t) = u∗
m(r, t) + w(r, t)
º%Â , u∗m(r, t)
B¹q (3.22) ÆÇa .
k , j¾¥¦ ÐÒÓ (3.29), CýþA ¡
H(r, t) Øá Áß SD¹q (3.22)
AÇa u∗
m(r, t) ì®ü å ¡ ªy , Ùu∗
m(r, t) = L−1[~ H(r, t) Rm(um−1, r, t)] =
M2∑
n=0
dn en(r, t) (3.32)
· 54 · º%Â , dn jÞ ¡ , L−1 j HIJK L Æâ JK . \] ,
ÐÒÓ(3.29) ½¾ ¬ ýþÑÓÿa u0(r, t)
HIJKL ¬ ¡
H(r, t).~ ÐÒÓ (3.29),
¬ôãäå a% Ræ CçÑèéê . \] ,
ÐÒÓ Gã ¨ ¹ä Aëì í Aî .ÐÒÓ B ¿M]MÆ , ª« Yï R Cç A ÐÒÓðñ ÜóòôõöÐ÷øùúûüýþÐÿÐÒÓ
.¬ Ìæ,È MÉMÀ ,
M¡H(r, t) ëMô
¦ MÙ ÐÒÓðñM w± .
\M] ,
M A ¼MÝ Ð Ò¡ H(r, t) ÆMý . ¶ I· é¸ ,
¦(3.28)
± o A · å en(r, t) Qûü a ªy (3.29) Â Ræ . ¸ Æ , m
ßfga
u(r, t) ≈m∑
n=1
um(r, t) =
M3∑
n=0
cm,n en(r, t) (3.33)
ACïÞ ¡ cm,n xfg
ß ¡ ÅQôÍ . Új ¬ ¾Cç A ðñ, Ù ù ! , "#$%&' ÐÒÓ()* , + % ,-.
. üp¼/ 0 Ð ¾ ¡ Æýþ .È ÉÀ ,
~ ÐÒÓðñ ¬ ðñ, ôã w± ¡ H(r, t). \] ,
ðñ1 ã ¨ M¹Mä2 ëì 3í PÊ 2 î .mn (3.13), pÓP45 ËÌ Í67³ · ðº ÏMÐ ¹qj Áß S7M¹q (3.14) òMó (3.22)2 458 ËMÌ . 9ª , ¿:pMÓMP45 ËMÌï a , 9 ï Ú 458 ËÌ ìûü ï a . \] ,
¬ï 9ç Ð; ' ðñ : <= ð ô>?@ABC Ð ,ñ
ò2ô2õ2ö Ð u0(r, t) EDGFG?G@GHGI L÷ DGFGJ H(r, t) &GKG<GLGMGN , OGP GQC øùúûüý (3.14) RS (3.22) TUVC Ð . WXp¼/Y 0 Z[\]^_`
u0(r, t) acbd45e8 L fbdgh H(r, t)2ij
.klÐÒÓðñ a ðñ ªm Ð; ' ðñ , n\]^_ ` u0(r, t) aobd45e8 L fbdgh H(r, t)2ijp ïqr ½¾st . W u ¼v¤ rwx kyz [ 1|~2 .
3.5 YXhv rq . , h f ¡ x¢£¤ l ` ¥ gh¦§ . ¨©ª«¬ ` ~ , © |~®¯°±r³²£´ ¨© ¥ gh ¤ l YXµ§ ¶·¸¹ ` . º» , ¼ © |~ ,½¾¿ vÀX pÁÂà st
IJÆÅÇÈÉh ` . ÊËÌÍÎÏ
qÐÑÒ
, Ó¦§[\]^_ ` u0(r, t) aÔbd ¶· eÕ L fbdgh H(r, t) ij .Ö ´ ij [\]^_ ` u0(r, t) a×bd ¶· eÕ L fbdgh H(r, t),¯°Ø
ÙÚÁ± r³²£ ij bdÛh ~ Ü . ¨©ªÝ Á «¬ ` ~ , © |~
Þ3 ß àâáâãâäâå · 55 ·
æç ¾¿ Yèé Á bdÛh ~
h ` . êëì 2 íîïðÝñòfóò , bdÛh ~
ôõö h ` f ¡ x . ÷øÏ , ù iú ¥ gh¨û Ã ,bGdGÛGh ~ n GGG f G ¡ xGôGõÅ ¾ ± òýü ( þGÿGì 2 í ). iGj û~ Ü ,
± fGZGGh ` GG f ¡ x . ºG» , ¨G©GªGÝ Á «G¬ `G~G, © G|GG~GGG® [GYGXGZf Gh ` G f ¡ xG y
.
3.5.1 --h --h © |~ ¾ Yé Á bdÛh ~
h ` . iú ~ Ü , h ` v r ¾ ? ¶·¸¹ é Á Y! q ÂÃ Ç , ë ¶·"#$% a'& ·(#)*+,-.. º/ ¯G° ¾G¿ ÏGYGé Á bdGÛh ~
Gh ` , Ý , W! ÂGÃ Ç012 ª ~. ºG» , 3GbGdGÛGh ~ 4/GYGX56 ² ÅGÇ ,±7 Z8GW! ÂGÃ Ç Ñ ª
~
9¶. ðë ,
γ = u(r, t)|r=0,t=0
ÏYX Á q ÂÃ st Ç , W: , · ;<n=> t
?@.
£ ª γ / ~ gh , AÐ,½ CB 8YCD γ ∼ ~
9¶. ECF§ à 3.1 G§ à 3.3, Ý Á £ ¨© ~ ܵC8
γ hH IJ ` . Ý , ëK `L Y , Ó ° H IM ©Ü , A Ð vγ ∼ ~
9¶ îONvYDPQ ¶R , Sn ~ R~ ;< . /T y , UVW D 9¶ / ~
9¶,
M R~ / ~ . ü Ù , ëKv ~ ÁX ij
~ Ü ,
M hY . Z Ù , ëKNvS[V\ ÂÃ Ç , H ½ B 8Ó ° M ~
9¶.Ö ´
γ ] ÁÂÃ st ,ØÙç B 8 M ~
9¶. ^C_C` ¸ , VC\ ~9¶ B ¾a
,abc ëd ij ~ Ü .
¯°ef, ùghij ` u0(r, t) a bd ¶· eÕ L fbdgh H(r, t) µ§ , ¨©÷k ÂÃ Ç ~ ÁX lm W .n Io ¯°p ¨ ç µ8qr hsóò . v tuv , ¼ ÷k ÂÃ Ç ~
9¶,ë kl γ ∼ ~
9¶,ç w ¿ û ~ Ü ,
h ` u(r, t) vÀx pÁÂÃ st y >Gf=> ýG . ºG» , ~
9ý¶ G® T W x y , zGbGdGÛGh ~ nGh ` f ¡ xôõ .
3.5.2 | - ~CCC(Pade approximation) CCC ªC r h , ¡ S ¡ x
. «¬ k , u(r, t) [m, n] ½ ;m∑
k=0
Fk(r) tk
1 +
n∑
k=1
Fm+1+k(r) tk
· 56 · G
m∑
k=0
Gk(t) rk
1 +
n∑
k=1
Gm+1+k(t) rk
SÈî , Fk(r) Gk(t) /CCC . Ü ¾CC Ï , CC;CCÈî , ÕCCC CGÏ y> ÅÇ r
¡ , GÏ=> ÅÇ t
¡ . ½ G ¿ ©¢ ~G î . ë£GÝ , ©¢ G~G ¥ ª¥¤ (3.12)¦q = 1 § ©¨ V W x , º/ª (3.13) Ï« (3.12) î q = 1
¾. º» ,
(3.12)¦
q = 1 = ¬ / . ®¯ , ° (3.12)Ñ ª±²ÅÇ
q
´ «¬ ,¾¿
[m, n] ªm∑
k=0
Wk(r, t) qk
1 +
n∑
k=1
Wm+1+k(r, t) qk
Sî , Wk(r, t)
£ £ l ¡ ªuj(r, t), j = 0, 1, 2, 3, · · · , m + n § .
Ù³, EF (3.8), « q = 1,
¯° ¾¿[m, n] ©¢ -
m∑
k=0
Wk(r, t)
1 +
n∑
k=1
Wm+1+k(r, t)
¯°CeCf, [m, n] ©C¢ -
CCC ´ «¬ [m, n] CCC ¾Cµ .¦ tuv
, [m, m] ©¢ - ¨ 12 ª¶·Û ~. º» ,
Ö ´~ ¸Ü¨ZA Ð ª e¹ , G©¢ -
G¯G° 0 ç ¾ G ª . º» Ï ,¯G° Io p ¨ ç W¼ · óòOV W½ .©¢ -
½ ª ¡ ª . ðë ,½ Ó ¡
u(r, t) = u0(r, t) +
+∞∑
n=1
un(r, t)
. ®¯ ,
£ (3.12), 6∂Φ(r, t; q)
∂t= u0(r, t) +
+∞∑
n=1
un(r, t) qn
Þ3 ß àâáâãâäâå · 57 ·
° Ñ ª±² ÅÇ q
,¾¿ «¬ [m, n]
m∑
k=0
Vk(r, t) qk
1 +
n∑
k=1
Vm+1+k(r, t) qk
SÈî , Vk(r, t) Ï r t
C .¦ * ;CCÈ q = 1, ECFC (3.8),
¯° ¾¿[m, n] ©¢ -
u(r, t) ≈
m∑
k=0
Vk(r, t)
1 +
n∑
k=1
Vm+1+k(r, t)
¿ Ý , ~
9¶ ® T W x § ~ ÁX À . Á , ©¢ -Èç ÈüÃÄÈªÈ ÈÈÈ , ¡ S È ¡Å .
¦ tuv,©¢ -
´ «¬ µ ÁX , ÆÇ ¨ 12 ª¶·Û ~. º» , È ¸ WÉ û ¥ ,È ¸û à ~ ÜG ´ ©¢ -
,¯° ½ ©¢ ~ ¾ ¦ Ä ª .
3.6 ÊÌËÌÍÌËÌÎÐÏÑ vÒÓ Ô Ó Å ~Õ
[1 − B(q)] L[Φ(r, t; q) − u0(r, t)]
=A(q) ~ H(r, t) N [Φ(r, t; q)] + ~2 H2(r, t) Π [Φ(r, t; q); q] (3.34)
½ 3©¢ ~Ö W×WC¼Ø , Sî , u0(r, t) Ù L Ù H(r, t) Ù ~ Ù A(q) B(q) §Ú ëC£ÝC , ~2 /CÛCC¶C·ÛC , H2(r, t) /CÛCC¶C·CC , Π [Φ(r, t); q] / W xCÛ¶·ÜÕ , ÝÜÕ ¦ q = 0 q = 1 = . ª Ô ,Ö
Π [Φ(r, t; 0); 0] = Π [Φ(r, t; 1); 1] = 0 (3.35)
Ý Á S[ M ÑÞ H M © , ßTà Ó Å ~ÕáÁ µ W¼ Ó L[
um(r, t) −m−1∑
k=1
βk um−k(r, t)
]
=~ H(r, t) Rm(um−1, r, t) + ~2 H2(r, t) ∆m(r, t) (3.36)
· 58 · Sî
∆m(r, t)=1
m!
∂mΠ [Φ(r, t; q); q]
∂qm
∣
∣
∣
∣
q=0
(3.37)
V , â²Û¶·Û ~2 Û¶· H2(r, t),ã W xÛ¶·ÜÕ
Π [Φ(r, t; q); q],Ö W× àTG©¢äåæ çèG· . Ü ¾ Ï ,
£ (3.36) µ8 ª 12 ªéx¶·Û ~ ~2.ê ·Cë (3.35)
ÛCC¶C·CÜÕ Π [Φ(r, t; q); q] È ¸ M Z çCè . ðë ,¯°
½ È ¸Π [Φ(r, t; q); q]=A(q)[1 − B(q)]F [Φ(r, t; q)] (3.38)
Sî , F [Φ(r, t; q)] / W x ; GΠ [Φ(r, t; q); q]= [1 − A(q)]
[Φ(r, t; q)]1+q − Φ(r, t; q)
(3.39)..Ù Ð
, ° W¼ ¶·¸¹ , ëd È ¸Û¶·Û ~2 ÙìÛ¶· H2(r, t) ÛCC¶C·CÜÕ Π [Φ(r, t; q); q],pÁCíCî ²CzC .
Ô Ó Å å Õ (3.34)
CïCð <ð ,ñ Ûò 4.3 12.1
.
ó4 ô õ÷ö÷ø÷ù÷ú÷û÷üþýþÿ
í ,¯É° <É© ¢ ä å æ S [Éë Adomian ª æ Ù Lyapunov ÛCCæC δ Cæ . C# åCæ C> Ñ ,
ð ©C¢CCäCåCæC3CVC!CåCæ¬ W.
4.1 Adomian Adomian ªæ [23∼25] Ï W xà ٠À Ùª G¶G·G¸G¹G ªäá
, ï ð ª! s!" Õ " [63∼79] .¦ ì 2 í ,
¯ ° ð W x ð Õ ñ ò ,
£Adomianªæµ8 ª (2.17) #/©¢äåæµ8 ª (2.57) W x÷ð .
,¯°
óò , ©¢äåæ$é Adomian ªæ ./T À% ¤ Adomian ªæ ¥ &' , ¨()*ë v ¶·¸¹N [u(r, t)] = f(r, t) (4.1)
Sî , N / ¶· ÜÕ , u /+, , f(r, t) / , , r t ø;< y >=> ÅÇ . ¶· ÜÕ N
½ ª/N = L0 + N0 (4.2)
Sî , L0 N0 ø/ ¶· ¶· ÜÕ .¦ Ý v , -h ¶· å Õ /
L0[u(r, t)] + N0[u(r, t)] = f(r, t) (4.3)
¼ ð Adomian ªæ ,¯° 3 u(r, t) ;V W x
u(r, t) = u0(r, t) +
+∞∑
n=1
un(r, t) (4.4)
Sîu0(r, t) = L−1
0 [f(r, t)] (4.5)
Çun(r, t) = −L−1
0 [An−1(r, t)], n > 1 (4.6)
· 60 · V: , L−1
0 / L0
. ÜÕ , An(r, t) / Adomian ¡ , S§ Ú ( ÿ Cherruault[66] Babolian[75]) /
An(r, t) =1
n!
[
∂n
∂qnN0
(
u0(r, t) +
+∞∑
n=1
un(r, t)qn
)]∣
∣
∣
∣
∣
q=0
(4.7)
¨©ª Adomian ªæ ,Ö ´ (4.2) ¨6 , ©¢äåæ ØÙÁX .
¨(C« L ;C< W xC¶C· ¶· ÜÕ , u0(r, t) /CgChCiCjCª , ÇCÝCgChCiCjCª¨CY (4.5)
M © , ~ / Ô ¶·Û , H(r, t) / Ô ¶· , q ∈ [0, 1] / W x±² ÅÇ. ¼ ð ©¢äåæ ,
¯°/0 Ô Ó Å å Õ(1 − q) L [Φ(r, t; q) − u0(r, t)] = ~ q H(r, t) N [Φ(r, t; q)] − f(r, t) (4.8)
Sî , Φ(r, t; q) /+, . ü Ù , Z q = 0 q = 1 = , ø6Φ(r, t; 0) = u0(r, t) (4.9)
Φ(r, t; 1) = u(r, t) (4.10)
ºG» , Z±² ÅGÇ q A 0 Ä ¿ 1 = ,ê Ô Ó Å å Õ (4.8)
+, Φ(r, t; q)Aghijª u0(r, t)
Å Ø ¿ -hå Õ (4.1)
J ª u(r, t). ¼ ð213 G§ Ã (4.9),½ 3 Φ(r, t; q) ë v W x Ñ ª q
4 Φ(r, t; q) = u0(r, t) +
+∞∑
n=1
un(r, t) qn (4.11)
Sîun(r, t) =
1
n!
∂nΦ(r, t; q)
∂qn
∣
∣
∣
∣
q=0
(4.12)Ô Ó Å å Õ (4.8) $éghijª u0(r, t) Ù ¶· ¶· ÜÕ L Ù ¶·Û ~ ¶· H(r, t).
Ð Ç ,¯°ÚÁ ¬ Ä ³²£ È ¸Ó ° .
Ó ° H È ¸û , I ª (4.11)
¦q = 1 = , A Ð ,
£ (4.10),¯°Á ª
u(r, t) = u0(r, t) +
+∞∑
n=1
un(r, t) (4.13)
Ü ¾ Ï , (4.13)¦ Ó (4.4) 56 M © .3 Ô Ó Å å Õ (4.8) ° q @ n 7 ,
Ù³ ß n!, 8 ³ « q = 0,¯°Á W Ó Å å Õ ( Z n = 1 = )
L [u1(r, t)] = ~ H(r, t) N [u0(r, t)] − f(r, t) (4.14)
Þ4 ß 9;:âá;<;=;>;?;@;A à · 61 ·
n Ó Å å Õ ( Z n > 2 = )
L [un(r, t) − un−1(r, t)] = ~ H(r, t) Rn(r, t) (4.15)
SîRn(r, t) =
1
(n − 1)!
∂n−1N [Φ(r, t; q)]
∂qn−1
∣
∣
∣
∣
q=0
(4.16)
½ óò , Adomian ªæ#Ï©¢äåæ ¦ (4.2) 6= W x÷ð .º/ ÚÁ ¶· ¶· ÜÕ L ghijª u0(r, t)È ¸ ³²£ ,
¯° Z Ù ½ È ¸L = L0, u0(r, t) = L−1
0 [f(r, t)] (4.17)
«~ = −1, H(r, t) = 1 (4.18)
Æ3 (4.2) (4.17) B² (4.14) (4.15),¯° ø Á
L0 [u1(r, t)] = f(r, t) −L0 [u0(r, t)] −N0 [u0(r, t)] (4.19)
L0 [un(r, t)]
=L0 [un−1(r, t)] − 1
(n − 1)!
∂n−1L0 [Φ(r, t; q)]
∂qn−1
∣
∣
∣
∣
q=0
− 1
(n − 1)!
∂n−1N0 [Φ(r, t; q)]
∂qn−1
∣
∣
∣
∣
q=0
, n > 2 (4.20)£ (4.17), 6f(r, t) −L0 [u0(r, t)] = 0
º» , EF§ Ú (4.7), å Õ (4.19) /L0 [u1(r, t)] = −A0(r, t) (4.21)
Sî , A0(r, t) Ï W x Adomian ¡ . EF§ Ú (4.12), 6
L0 [un−1(r, t)] − 1
(n − 1)!
∂n−1L0 [Φ(r, t; q)]
∂qn−1
∣
∣
∣
∣
q=0
=L0 [un−1(r, t)] −L0
[
1
(n − 1)!
∂n−1Φ(r, t; q)
∂qn−1
∣
∣
∣
∣
q=0
]
=L0 [un−1(r, t)] −L0 [un−1(r, t)]
=0 (4.22)
· 62 · º» , å Õ (4.20) /
L0 [un(r, t)]=− 1
(n − 1)!
∂n−1N0 [Φ(r, t; q)]
∂qn−1
∣
∣
∣
∣
q=0
(4.23)
3 (4.11) B² (4.23), EF Adomian ¡ § Ú (4.7),
¯°ÁL0 [un(r, t)]
=− 1
(n − 1)!
[
∂n−1
∂qn−1N0
(
u0(r, t) ++∞∑
n=1
un(r, t) qn
)]∣
∣
∣
∣
∣
q=0
=−An−1(r, t) (4.24)
ü Ù , å Õ (4.21) å Õ (4.24) ª ç ¬ W ;un(r, t) = −L−1
0 [An−1(r, t)], n > 1 (4.25)
V Adomian ªæµ8 ª (4.6) 56 M © . º» , Adomian ªæ#Ï©¢äåæ ¦ (4.2) 6 , Ç ê u0(r, t) = L−1
0 [f(r, t)], L = L0, H(r, t) = 1, ~ = −1
= W x÷ð .V:C D l ½ . ®¯ ,¯°ÚÁ± Ä ³²£ È ¸ * ;¨© ghCiCjCª u0(r, t) Ù ¶C· ¶· ÜÕ L C¶C·CC H(r, t), Ý àC Ó Å å Õ (4.14) (4.15) Cª ½ ð´ Adomian CªCæÝ Ñ ð C¡ µE ¥ CC; .S7 ,
¯° 56] Á Y ¶· ÜÕ N YF ç ª (4.2). 8 ³ ,
0 Ï8 , ©C¢CCäCåCæµC8 ªé Á ¶C·ÛC ~, Ó ® T W x C CGCCª ¡Å À . º» , ©¢äåæ ´ Adomian ªæ µ á W¼ · .
4.2 HIJKLM1892 N , Lyapunov[21]
8 Ûæ .¯°¦ ì 2 íîPOðñò ,
£Lyap-
nov Ûæµ8 ª (2.15) QÏ©¢äåæµ8 ª (2.57) W x÷ð . 3óòSR Lyapunov Ûæ ¦ ë . ©ª Adomian ªæ , º» ,
0 Ï©¢äåæ W x÷ð ./T À¤ Lyapunov Ûæ ¥ &' , ¨()* W x ¶· å ÕN [u(r, t)] = f(r, t) (4.26)
Þ4 ß 9;:âá;<;=;>;?;@;A à · 63 ·
Sî , N Ï ¶· ÜÕ , u Ï+, , f(r, t) Ï , , r t øÏ y >=> ÅÇ . ¶· ÜÕ N
ç ªN = L0 + N0 (4.27)
SÈî , L0 N0 øÏ ¶· ¶· ÜÕ .
£ (4.27), âC² ÛC ε, -ChCåÕ(4.26) /
L0 [Φ(r, t; ε)] + ε N0 [Φ(r, t; ε)] = f(r, t) (4.28)
V: , Φ(r, t; ε) Ï+, . Z ε = 1 = , å Õ (4.28) ü Ù . ©Gª-hå Õ (4.26). º»Φ(r, t; 1) = u(r, t) (4.29)
3 Φ(r, t; ε) Û ε
4 ,Á
Φ(r, t; ε) = u0(r, t) +
+∞∑
n=1
un(r, t) εn (4.30)
¦ (4.30) îO« ε = 1, Ƽ ð (4.29),Á
u(r, t) = u0(r, t) +
+∞∑
n=1
un(r, t) (4.31)
Ó ¦ Ó 56 . ©ª Adomian ªæµ8 ª (4.4).3 (4.30) B² (4.28),Á
L0[u0(r, t)] − f(r, t) +
+∞∑
n=1
εn L0 [un(r, t)]
+ ε N0
[
u0(r, t) +
+∞∑
n=1
un(r, t) εn
]
= 0 (4.32)
«N0
[
u0(r, t) ++∞∑
n=1
un(r, t) εn
]
=+∞∑
n=0
wn(r, t) εn
3éTH° Û ε @ m 7 ,Ù³ « ε = 0,
Á
∂m
∂εmN0
[
u0(r, t) +
+∞∑
n=1
un(r, t) εn
]∣
∣
∣
∣
∣
ε=0
= m! wm(r, t)
¼ð§ Ú (4.7), µ8
wm(r, t) =1
m!
∂m
∂εmN0
[
u0(r, t) +
+∞∑
n=1
un(r, t) εn
]∣
∣
∣
∣
∣
ε=0
= Am(r, t)
· 64 · Sî , Am(r, t) ÏÝU Adomian
¡ . º» , 3N0
[
u0(r, t) +
+∞∑
n=1
un(r, t) εn
]
=
+∞∑
n=0
An(r, t) εn
B² (4.32),¯°Á
L0[u0(r, t)] − f(r, t) +
+∞∑
n=1
εn L0 [un(r, t)] + An−1(r, t) = 0
£ å Õ , 6L0[u0(r, t)] − f(r, t) = 0
L0 [un(r, t)] + An−1(r, t) = 0, n > 11 7ª * å Õ ,¯°Á
u0(r, t) = L−10 [f(r, t)]
un(r, t) = −L−1
0 [An−1(r, t)], n > 1
Ó ° ø Adomian ªæµ8 ª (4.5) (4.6)
M © . º» , Ûæ ¦ ë . ©ª Adomian ªæ .¦4.1
,¯° óÈò¾T Adomian CªCæ#Ï©C¢CCäCåCæ W x÷ð . º» ,
Lyapunov Ûæ 0 Ï©¢äåæ ¦ N = L0 + N0
6 , Ç ê ~ = −1, H(r, t) = 1, L = L0, u0(r, t) = L−1
0 [f(r, t)]
= W x÷ð .ëK3ÝU Û ε VW±² ÅÇ , 3å Õ (4.28) 4/ W x÷k Ô ÓÅ å Õ , XYZ ¬ à ªT .
4.3 δ [\]¦^
2 _a` ,Lbac
δ ædefªg (2.21) h ¦i ¢äåædefjk ªglmn` . op , q ð 3.6 rn`Pfs Ú Ôt Óu å Õ (3.34), vwxyzn ,
|4 9;:;~;<;=;>;?;@;A · 65 ·
δ æ i ¢äåæf . nP , h()* ^ 2 _n`Pf,
V (t) + V 2(t) = 1, V (0) = 0 (4.33)
ð i äåæ , vw LΦ =
∂Φ
∂t+ Φ − 1 (4.34)
L[V0(t)] = 0, V0(0) = 0
f ¡ V0(t), V0(t) = 1 − exp(−t) (4.35)¢£¤¥
(4.33), ¦§¨ N [Φ(t; q), q] =
∂Φ(t; q)
∂t+ [Φ(t; q)]q+1 − 1 (4.36)
©ª, ¦§«¬®
Π [Φ(t; q), q] = (1 − q)
[Φ(t; q)]q+1 − Φ(t; q)
(4.37)¯°q = 0 ± q = 1 ²³´µ . ¶ ~ · ~2
¸ g¹º» , H(t) · H2(t)¸ g¹¼» . vw½¾µ t¿ u ¤¥
(1 − q)L[Φ(t; q) − V0(t)]= q ~ H(t) N [Φ(t; q), q]
+~2 H2(t) Π [Φ(t; q), q] (4.38)À ÁÂΦ(0; q) = 0 (4.39)Ã
q = 0 ² , ÄÅ ,¤¥
(4.38) mΦ(t; 0) = V0(t) = 1 − exp(−t) (4.40)Ã
q = 1 ² ,¤¥
(4.38) ³ i ´Æ ¤¥ (4.33), ÇpΦ(t; 1) = V (t) (4.41)È
Φ(t; q) ÉÊËÌ´ q fÍλΦ(t; q) = Φ(t; 0) +
+∞∑
n=1
Vn(t) qn (4.42)
· 66 · ÏÑÐÑÒÔÓÕ
Vn(t) =1
n!
∂nΦ(t; q)
∂qn
∣
∣
∣
∣
q=0
(4.43)
Ö× Î» (4.42)°
q = 1 ²ØÙ .cPÚ
(4.40) ± Ú (4.41), ËÛV (t) = V0(t) +
+∞∑
m=1
Vm(t) (4.44)
Vm(t) fÜÝ ¤¥Þßà ¦§ (4.43) áâãä .È µ t¿ u ¤¥ (4.38) åæçuè
q éâ m ê , oëì m!, íë¶ q = 0, vwãäî t¿ u ¤¥L0[Vm(t) − χm Vm−1(t)] = ~ H(t) Rm(t) + ~2 H2(t) ∆m(t) (4.45)
ÁÂVm(0) = 0 (4.46)À ` , χm
cPÚ(2.42) ¦§ , ï
Rm(t)=1
(m − 1)!
∂m−1N [Φ(t; q), q]
∂qm−1
∣
∣
∣
∣
q=0
(4.47)
∆m(t)=1
m!
∂mΠ [Φ(t ; q), q]
∂qm
∣
∣
∣
∣
q=0
(4.48)
©ª, L0 m¦§
L0Φ =∂Φ
∂t+ Φ (4.49)¸ ÈÚ
(4.36) ± (4.37) ðç Ú (4.47) ± (4.48),b
R1(t)= V0(t) + V0(t) − 1
R2(t)= V1(t) + V1(t) + V0(t) ln V0(t)
...
±∆1(t)=V0(t) ln V0(t)
∆2(t)=−V0(t) ln V0(t) + V1(t) [1 + ln V0(t)] +1
2V0(t) ln2 V0(t)
...
|4 ñ;ò;~;ó;ô;õ;ö;÷;ø · 67 ·
° «¬ ¸ùú~ = ~2 = −1, H(t) = H2(t) = 1 (4.50)
î t¿ u ¤¥ V1 + V1 =−V0 ln V0 − R1(t), V1(0) = 0
V2 + V2 =−V1(1 + ln V0) −1
2V0 ln2 V0 − R2(t), V2(0) = 0
...û êéüýî t¿ u ¤¥ , vw bV1(t)=exp(−t)
[
t − π2
6+ P L
2 (e−t)
]
− (1 − e−t) ln(1 − e−t)
...À `P L
n (z) =
+∞∑
k=1
zk
kn
z f n þ廼» (nth polylogarithm function). ÿ , t V (t) ≈ 1 + exp(−t)
[
t − π2
6− 1 + P L
2 (e−t)
]
− (1 − e−t) ln(1 − e−t) (4.51)
À ^2 _a` c δ ÉÊdef (2.21) i . ãf ,
¤¥(4.45)± (4.46) defmgl Ú º» ~ ± ~2, © ,
¯ cδ ÉÊdefgl Ú (2.21) . ü ,
cPÚ(4.35), ËÛ R1(t) = 0. ª ,
c t¿u ¤¥, ËÛ R2(t) = 0. © , î t¿ u ¤¥ ³ i ´ ^ 2 _n`Pf δ ÉÊm ¤¥
.È
~ = ~2 = −1, H(t) = H2(t) = 1
ðçµ t¿ u ¤¥ (4.38),b
∂Φ(t; q)
∂t+ [Φ(t; q)]1+q = 1 (4.52)¯ ^
2 _n` δ ÉÊm ¤¥V (t) + V 1+δ(t) = 1
, δ ± V (t)¸ c
q ± Φ(t; q) ðf . , vw Þ È δ ÉÊn`Pf δ æç uè , f ¤¥ fµ t¿ u ¤¥ . © , δ ÉÊ! i"¤ f .
· 68 · ÏÑÐÑÒÔÓ
4.4 #%$%&%'%(]*),+«üÿ¹ , Adomian
, Lyapunov -./º»± δ ÉÊ!!0 i"¤ f21 . © , 3¨45 ¤ xy °i"¤ f768:9 ãä; . p< , ;f=>??@ A = . ÇBC , D¹ i"¤ f bE .
1:FGIHKJKLKMKNKOQPSRKTKUKVXWXYKZG[HKJK\X]XNKO^P , _K`KaKbKcKdKeKfKgKhXiKjXkXfKlXm WfKnKoKp , qKrKs HKJKLKMKNXOKF ~ = −1, H = 1 t W cKdKuKv . wKxKyKz Liao S J. Comparison
between homotopy analysis method and homotopy perturbation method. Applied Mathematics and
Computation, 2005, 169(2): 1186∼1194. —— |
~5
, "¤ « © . , vw> "¤ ± , ¡ï¢£¤¥¦ .
5.1 § ¨ 45 ¤ © « Lyapunov -./º» · δ Éʱ Adomian
³¨45 ¤ , "¤ «¬¤ª« , ¬´; "¤ , "¤ ¢®vw¯7°7²±´³¶µ7·7¸¬77¹ ¼»ºl»¦¨m . © , vw Þßà ¼½¾ ¹ ¼» , E ¿ ¨ . ÀÁ , λØÙÂñØÙÄCÀÁ¦´¸µºÅ ¹ ¼» .«Æ, ¬´; "¤ , "¤ ÇÈ¢®ÉÊ º» ~ ºÅ Ú , ·ÿé¨Ë7Ì7É21 . ÍÎ϶ÐÑØÙÂñØÙÄC³Pº» ~ Ò¦ . ÿ , º» ~ ¢®ÓÉÑÔ±ÜÝλÕØÙÂñØÙÄCÖ×ØÙ . Ú¸ÿÛ ~ ÜP , ¯ÝÄÞä ~ Õ E Âà , ÇßàãÉÑØÙλ .
©ª, á⢣ - ãä ßå ;ãä 7 7E
,Àæç °èéùú ¬êë¬ ûì ´º» ~.« 3 , ¬´45 ¤ , "¤ ¬ ûì ´íî/ ( ï° ) º» . ÿ , ð>»¦¨ÜÝ ¤¥ ±7ñò ( ) ÁÂ϶óô / ( ° ) º» , Þ ¸ "¤ .í ë , 2" ¤ °2õ2ö ü 2 Ó Lyapunov -2.2/ º »2 , δ É Ê2 ±
Adomian , ;ÉÓ¤¨45 ¤ . © , ËÉ .ã÷£ó , "¤ ¹ ´¬ø Ö× ª
(1) åíù q ∈ [0, 1], µú ¿û ¤¥ üý ° ;
(2) ÿîú ¿û ¤¥ ü ;
(3) þæç û è q ÉÊÿλ ° q = 1 ²ØÙ . ó , "¤ ¢®áâ ²±¶³ ¡ · · ¼»±º» ~. ²±¶³¶¢®Ó¯° Þ È7Òüý Ö× ËÛ . ÿ1 sd` HW . sc W , ` HW , !"#$% d` HWN . —— |
· 70 · ÏÑÐÑÒÔÓ
, üý Ö× ¡¬&'( "¤ Õ E 1 . ü , '( "¤ ÿ)ü*²± üý²±¶³ .¢£ ¦= 3.1 ±¦= 3.3, 0Á'( "¤ »£Î»+ØÙ ,
¯-, óÿé-+7¨-.-/7Õ-+ . © , á7â7!-0-1:Ï32-4-5-67½7¾7 ¡+ ·879:;< ·879¼»±7½¾7-79º» ~ Õ , =Òλ+ØÙ . ÿ= , ó56ü²±¶³>¦Ó'( "¤ Õ E ;±?@;Õ ¹A .
5.2 B C DE ß , ± ³ FGH àIJ Ó . ëK , LMNÉÑOP»Q=>÷âRSTU + ·879:;< ·879¼»±-79º» ~ Õ56 . ǸBC , á⢣Óɤ ¹ ÆV , «+ºÅÆV ·8W»XYÆV±-+7ý ° ÆV .
¯ â ° ÖZ'("[ ¸ü\äþÁ77¸ . +ºÅÆV¢®ÓÉÑ ¹-A ±-\-) ,¯ åRS TU + ·79:;< ±79¼»56ËþÁ÷⸠. W»XYÆV±+ý °ÆV ° 79¼»Ò¦Ï\]þÁ¸ , ¡Òîú ¿û [¥^_ ï+ .
W»XYÆV ¹ ´+ºÅÕ`; ; +ý ° ÆVÉba Ó E . ÿ= , W»XYÆV±-+7ý ° Æ-V7ó-c-d7½7=7 . ¶7--e-f77ó , +7º7ÅÆ-V-g ]-77É7Ñ Ö× ª^áâ-07Á7ï é ï-h7Å-i7É7¤Ì´77¥é-+7¨-:j;-.-/ Õ-k-lÅ-m . n-o Þ ÈåÉÑp ·S¥é+./klÅmq ? no Þ ÈÅiɼ ¹ ¼» Àr¹ ¼»s , ÇßÈ E ¿ ÉÑ¥é+¨:;./q ? ÿ= , =>ü , Ñg Ö× &tÓ'-( "[ Õ7-u; , vwåÉÑpx:;-.-/ , áâÇ7È-y-z7É7¤7¬77 ¹ ¼» .- 7ó , « 2 ÿ¹ , É7Ñ-x-:-;-.-/7-+7È | èé ¬ ¹ ¼»ºÅ .
« 2 ÿý , -~ÿ7Û---7-- | è7é--- ¢7£- , Lindstedt[52],
Bohlin [53], Poincare [54], Gylden [55] . +ºÅVÉZ . ó , ÉÑpS¥+./ , noÈi¤Ûq ? = , ,èé + [ g¡-]¢7-£¤ , ¥0Áï é ïh¦iÉ7¤§-7¥-+x:;./Õklm . £¤¨&tÓ¤ [ , vw&t©¤ [ª r O« ;¢¬®¯°¬±i , ²ß åå |³´ .
5.3 µ·¶·¸·¹·º¼»¾½dÓ¿À ÁÂÃÄÅ '( [ÆÇÈ ; , É ,ÊÂËÌÍÎÏÄÐ QÑÒÓ-Ô R-S T-U-Õ Ö7-9-:-;-<-×-ØjÙ-ÚjÛ ÐjÆ-ÜjÝ . Þ Í-Ð Q-Ñ-Ò-ß-à-á-â Ì ãåäæç
, èéêëìíîïðñòóô (3.34). õöïðñòóô÷øùúûüýþÿ , ÷ ïðñòóô÷øùúûüýþÿ . ——
5 · 71 ·
Ä !É#"#$#%#&#' Ä#(#)#* É#+ , ²#,#-#.#/#0#1á#2#3#"#$#4Ñ#5#6 Ä#78 * ß#9#/ . :<; , Þ#= ÎÏÄÐ#> ÑÒ#?#@#Aá#B#2¯#, .CED ÅEFEG EH JILKEMEN 9jj ÕEOEPEQERESETEU ( VEWEXEYjØ ÌjÍ Éj§Ò[Z ), [\[2[][^-Þ[_[H - Õ[` É[a[b S ãdc[e[f S-Ä[Q[R[S[T[U ?[@-É[+ .e[f[g[h[i[j[k 9 Ð[l H [m[n , o[p[W[9 Õ [H [m[n-Ä . q[r , $[=[s-Û Ð 01jÉE+j¦ Õ EtEu eEf ÕEvEwEx . yEz , E| (Norden E. Huang)[80] E~ ÂjË &EE GjÕj (the empirical mode decomposition method), J QERES Q j# a###0#1#9 S #-Û Ð (intrinsic mode functions) t#u . # Ä ? ,
qEr , .E/E\E2jE3EE$E9 S EjÛ Ð ¯E QERESETEU eEf ÕjÆjÕ t#u .
ã ó ¡¢ ý£¤ ¢¥¦§¨© , 誫 2005 ¬®¯°±² § æ³<´ Wu
W, Liao S J. Solving solitary waves with discontinuity by means of the homotopy analysis method.
Chaos, Solitons and Fractals, 2005, 26(1): 177∼185. —— ¶µ·¸ ô¹¹º¼» ·½¾¿ ¯ÀÁ ¾Âà (NASA)Goddard¾ÄÅÆÇÈ
(Norden E. Huang) ÉÊËÌͧ Å ùúÎÏÐý Å Ï , Ñ ËÒÓÔÕ Æ ýÖ× . ——
Ø Ù Ú Û Ü Ý
Þàßàáàâ,Þàãàáàä,Þæåçáàè.
—— éJê , ëJìEíJîJïñðàò
ó6 ô õ÷ö÷ø÷ù÷ú÷û÷üþýþÿ
a Duffing × ( #W#Z [11], 198 ), Q#R#S Hw′′(x) + w(x) − w3(x) = 0, w(0) = w(L) = 0 (6.1)
, x t a , w(x) tá 0 6 x 6 L Ì _Û Ð , ′ t
x Ô . ! ww(x) = 0
#H (6.1), "# , ? Ì _ Õ .w , ,
$ ÍLl
, Aá Q%Õ . # , & Ë g'() *+ G,.- .
x =
(
L
π
)
ξ, ε =
(
L
π
)2
, v(ξ) = w(x) (6.2)
H (6.1) Mv′′ + ε(v − v3) = 0, v(0) = v(π) = 0 (6.3)
, ′ t ξ Ô . "/ ε > 0, 01#H#BÉ %Õ v(ξ) = 0. 2 $ Í εl
, H (6.3) Aá Q%Õ, 3#, Ë g G, . Þ4 , .#/56 m#n H (6.3)
Q%Õ7 Aá 7 8 . ! w , 8v(ξ) ?#H (6.3)
Q%Õ, 9 −v(ξ) :; ?< Q%Õ . 2= Ì>#S ,
A = v(π/2), v(ξ) = A u(ξ) (6.4)
?@(6.4) AB#H (6.3), É
u′′ + ε(u − A2u3) = 0, u(0) = u(π) = 0 (6.5)
lCD / ? ,@
(6.5)E
A ? xF . G @ (6.4),MH
u(π/2) = 1 (6.6)
Kahn Ø Zarmi[11] Ë A I L JKLM#L = 2
∫ A
0
dz√
A2 − z2 − (A4 − z4)/2
· 76 · NPOPQSR KL ÕT
L
π
=2
π
√
1 − A2/2K
(
A2
2 − A2
)
(6.7)
U, K(ζ) ?V ÌVWVXVYVZU[]\ G . ^V_V0V1 KVL Õ , ε = (L/π)2 á |A| ` z 1
a bc . d#9e9f#Hg , Kahn Ø Zarmi[11] ËhiÕ
A ≈ ±2
√
ε − 1
3, ε > 1 (6.8)
w , , à hiÕ j ε =#+ . Xk , .#/d#9l F#Gm Hg Õ#`n 0o *+ G, Q#R#Sp#l#T#U ã.
6.1 qsrutuvuw6.1.1 xyz|d#9 p~#7 8 u(0) = u(π) = 0,
H (6.5) J Q#R#S , F , u(ξ) #9#* s#t#usin[(2m + 1)ξ] | m > 0 (6.9)
u(ξ) =
+∞∑
m=0
cm sin[(2m + 1)ξ] (6.10)
, cm
T . ' T#U J .
^_ (6.10) p~#7 8 (6.6), !#, F , u0(ξ) = sin(ξ) (6.11)
- Tu(ξ)
, 7L[Φ(ξ; q)] =
∂2Φ(ξ; q)
∂ξ2+ Φ(ξ; q) (6.12)
- T#R#S. #`n#S
L[C1 sin ξ + C2 cos ξ] = 0 (6.13)ã¡ ¡¢¡£¡¤¡¥¡¦Mathematica è 󡧡¨¡©¡ª¡« §¡¬¡ , ®«¡¯¡°¡± . ²¡³ ©¡´ ¦ © E-mail µ¶ ´
[email protected], ·¡¸ ¥¡¹¡º ´ http://numericaltank.sjtu.edu.cn/code.htm, »¡¼¡½¾ ¨¿ª¿À¿¯¿°ùúý Mathematica Á¿Â § ³ © ³ ª . ——
6 ÃÅÄÅÆÅÇÅÈÅÉÅÊÅËÍÌÍÎÅÏ · 77 ·
, C1 C2
T . GEÐ (6.5), ÑÒÓÔ
N [Φ(ξ; q), α(q)] =∂2Φ(ξ; q)
∂ξ2+ ε
[
Φ(ξ; q) − α2(q)Φ3(ξ; q)]
(6.14)
, q ∈ [0, 1]
TÕ B , α(q)TÖ× q
Ø F . Ù ~ 6= 0 Ú Û , H(ξ) Ú Û . ÜÝÞß %àá Ð
(1 − q) L[Φ(ξ; q) − u0(ξ)] = ~ q H(ξ) N [Φ(ξ; q), α(q)] (6.15)
p~âãΦ(0; q) = Φ(π; q) = 0 (6.16)
!ä , å q = 0 æ , Ð (6.15) (6.16) J TΦ(ξ; 0) = u0(ξ) (6.17)
å q = 1 æ , GEç ~ 6= 0, H(ξ) 6= 0, Ð (6.15) (6.16) èlçé Ð (6.5), 3êΦ(ξ; 1) = u(ξ), α(1) = A (6.18)
"ë# , å q 3 0 ëfëì 1 æ , Φ(ξ; q) 3 ëëíëë u0(ξ) = sin ξ ëf (á ) îëÐ
(6.5) JëïëL u(ξ) ; lëð , α(q) :ë3 ëëëíëëñ A0 ëf (á ) îëïëL ñ
A = v(π/2).ñ CëD / ëò ,%ëàëá ëÐë (6.15) ó nëë ë ~ ë ë H(ξ). ôëõ
~ H(ξ) ëëöë÷ , ëîëç %ëàëá ëÐë (6.15) (6.16) ( n
q ∈ [0, 1] ø n, ù
um(ξ) =1
m!
∂mΦ(ξ; q)
∂qm
∣
∣
∣
∣
q=0
, Am =1
m!
dmα(q)
dqm
∣
∣
∣
∣
q=0
(6.19)
m > 1 øúû . üý , dþÿ (6.17), Φ(ξ; q) α(q) Ñ q J
Φ(ξ; q) = u0(ξ) +
+∞∑
m=1
um(ξ) qm (6.20)
α(q) = A0 +
+∞∑
m=1
Am qm (6.21)
# , ôëõ ~ H(ξ) ëëöë÷ , 3ëê ë (6.20) (6.21) û q = 1 æ , G
· 78 · NPOPQSR@
(6.18),n
u(ξ) = u0(ξ) +
+∞∑
m=1
um(ξ) (6.22)
A = A0 +
+∞∑
m=1
Am (6.23)
6.1.2 yz|T *,
uk = u0(ξ), u1(ξ), u2(ξ), · · · , uk(ξ) , Ak = A0, A1, A2, · · · , Ak? %VàVá VÐV (6.15) (6.16) Õ BVV q m , V m!, VÙ q = 0,
ÜÝ nàá ÐL [um(ξ) − χmum−1(ξ)] = ~ H(ξ) Rm(um−1, Am−1) (6.24)
p~âãum(0) = um(π) = 0 (6.25)
, χm G (2.42) , ùRm(um−1, Am−1)
=1
(m − 1)!
∂m−1N [Φ(ξ; q), α(q)]
∂qm−1
∣
∣
∣
∣
q=0
= u′′m−1(ξ) + ε um−1(ξ)
−ε
m−1∑
n=0
(
n∑
i=0
AiAn−i
)
m−1−n∑
j=0
uj(ξ)
m−1−n−j∑
r=0
ur(ξ)um−1−n−j−r(ξ)
(6.26)
ñ CëD / ëò , úëû Û Ø F um(ξ) Am−1, 2 nëëÛ Mëç um(ξ)
Ð . "# ,!"#$%
, &'() Û* )AÐ +ëL Am−1. ^_ (6.10), , ÓÔ L JÔ (6.13), -.ö÷/ H(ξ), îç àá0 Ð+ (6.24) 123 Ú4 ~ H(ξ) Rm(um−1, Am−1) =
µm∑
n=0
bm,n(Am−1) sin[(2n + 1)ξ] (6.27)
57678 (6.20) 9 (6.21) :7;7<7=7>7?7@7A7B¿Á (6.15) 9 (6.16), C q D7E7F7G7H7I 87J > , »¿¨©7K7L7D7E7M7N7?7@7A7B¿Á (6.24) 9 (6.25), O7P7Q7R7S (6.26). —— T7U
V6 W ÃYXÅÄÅÆÅÇÅÈÅÉÅÊÅËÍÌÍÎÅÏ · 79 ·
Z, bm,n(Am−1) [\ , ]^ µm
Ö× ç m H(ξ) 1 á_ . `aÔ(6.13),
bbm,0(Am−1) 6= 0, c m
àá 0 Ð+ (6.24) 1 d ó Ñeξ sin ξ
f ê ,#g öhij (6.10). [klmnop , ÜÝ #q#rs
bm,0(Am−1) = 0 (6.28)
n]tuvko Û* )/xwzyç Am−1 /|Ð+ ,f ê !"$% . ~ , q Ð+ (6.24) 1
um(ξ)=χm um−1(ξ) −µm∑
n=1
bm,n
4n(n + 1)sin[(2n + 1)ξ]
+C1 sin ξ + C2 cos ξ (6.29)
Z, C1 C2 [\| . `ahij (6.10), C2 -.[ . äê , \| C1
#âã
(6.25) , [å C2 = 0 æ , âã . ò , _ (6.6),
um(π/2) = 0 (6.30)
_ ok C1 1 ñ . nð , Ö q Am−1 um(ξ).Z
Nà [
u(ξ) ≈ u0(ξ) +
N∑
m=1
um(ξ) (6.31)
A ≈ A0 +
N−1∑
m=1
Am (6.32)
6.1.3 6.1 ¡ (6.22) ¢ (6.23) £¤ , ¥§¦ , uk(ξ) ¨©ª« (6.24) ¢ (6.25),¬®
(6.26) ¢ (2.42) ¯° , ±²³´µª« (6.5) ¶¡ .· b | (6.22) , c-¸lim
m→+∞um(ξ) = 0, ξ ∈ [0,π]
¹ þ _ (6.12) _ (2.42), º, _ (6.24),
~ H(ξ)+∞∑
k=1
Rk(uk−1, Ak−1)
= limm→+∞
m∑
k=1
L[uk(ξ) − χk uk−1(ξ)]
· 80 · »½¼½¾À¿
=L
limm→+∞
m∑
k=1
[uk(ξ) − χk uk−1(ξ)]
=L[
limm→+∞
um(ξ)
]
=0
[ ~ 6= 0 H(ξ) 6= 0, Á _ÂÃ+∞∑
k=1
Rk(uk−1, Ak−1) = 0
Ä _(6.26) ÅÁ _ , ÆÇÈ . `a| (6.22) (6.23) /Ô , ¸
d2
dξ2
[
+∞∑
k=0
uk(ξ)
]
+ ε
[
+∞∑
k=0
uk(ξ)
]
−(
+∞∑
m=0
Am
)2 [+∞∑
k=0
uk(ξ)
]3
= 0
_ (6.11) (6.25),
+∞∑
k=0
uk(0) =
+∞∑
k=0
uk(π) = 0
É , '| (6.22) (6.23) , ÊÝ-ä ò Ð+ (6.5) 1 . ËÌ .
6.2 ÍÏÎÏÐÏÑ`a 6.1, ÜÝ&ÒÓö÷/ÔÕÖ| ~ ÔÕ×| H(ξ), ºØ|
(6.22) (6.23) .ñ q Ù Ú / ò , û # Û Ü h i j (6.10) ÞÝ ß à á â ã 1ä uÑ , åæþ #ç á_ /ÔÕ×| ,
H(ξ) = 1, H(ξ) = sin2(ξ), H(ξ) = cos2(ξ), H(ξ) = cos(2ξ)
è . [kÈ , ÜÝÒÓH(ξ) = 1 (6.33)
_ (6.11) (6.26), ¸~ H(ξ) R1(u0, A0) = ~
(
ε − 1 − 3
4ε A2
0
)
sin ξ +1
4~ ε A2
0 sin(3ξ) (6.34)
Ä _(6.34) è (6.27) éëê , ¸
b1,0 = ~
(
ε − 1 − 3
4ε A2
0
)
, b1,1 =1
4~ ε A2
0
V6 W ìYXYíYîYïYðYñYòYóõôõöY÷ · 81 ·
É , å m = 1 æ , _ (6.28), ÜÝ q ìo Û* )/|ø+ε − 1 − 3
4ε A2
0 = 0 (6.35)
[ ε = (L/π)2 > 0, ùº , Áú/ø+û ε < 1 æû . É , å 0 6 ε 6 1 æ ,# ú
ûÒ . å ε > 1 æ , ø+ (6.35) 1[A0 = ± 2√
3
√
1 − 1
ε, ε > 1 (6.36)
É , ε = 1 æ Ã ü ù ý / È þ ÿ .ç ø ] Â Ã k ëÒëÓëÔ ! "
Èþÿ/ âã .ñ q Ù Ú / ò ,ç ø Â Ã k # ç / w ó ¸ Ô Õ Ö | ~ / , ù ~ | (6.22) (6.23) 1 ëÔ . , A 1 | (6.23)
ò~ / | .
[k ~ | (6.23) / , Ú Â ε, ÜÝ Ã yç A 1 ~ EÓ ( Ö 24 3.5.1 ). ,
6.1 ù , ε = 10 ε = 25 æ/ A ∼ ~ ]Ó k ! / ~ 1¸ " # $ . `a 6.1, % ε = 10 & , ' −3/4 6 ~ < 0, | (6.23) -
(6.1 ε = 10 ) 25 *,+ A ∼ ~ -7ó.0/0132
ε = 10 4 A H 10 ?050607 ; 8 132 ε = 25 4 A H 10 ?050607 ; % ε = 25 & , ' −1/4 6 ~ < 0, 9| (6.23) - . ùº , `a : ; 6.1 A ∼ ~
=< , ' > ? @ # $ −3/4 6 ~ < 0 (% ε = 10 & ) A B −1/4 6 ~ < 0 ( % ε = 25 & )C ÒÓ ~ D , 9| (6.23) - E A 1 F D . G[ H , % ε = 10, ~ = −1/2 ε = 25, ~ = −1/5 & A 1 D @ I 6.1
C=JÃ. É K , L M ü , ' 9| (6.23)N O
, ç ÔÕÖ| ~ÂÃ / ! u(ξ) 1 9| (6.22) P @^ Q # $ 0 6 ξ 6 π
· 82 · »½¼½¾À¿C N O
. R , ε = 10, ~ = −1/2 & u(ξ) 1 10 S , º, ε = 25, ~ = −1/5 &u(ξ) 1 30 S , > ?è u(ξ) / F T U q t , R 6.2 ù . ùº ,
¹ V~ =< , ºø W X Y ~ 1¸ " # $ ,
f Z Ø 9| (6.22) (6.23)N O
.
[6.1 ε = 10, ~ = −1/2 \ ε = 25, ~ = −1/5 ] , A ^ 30 _,`,a,b,c,`5067? 8 ε = 10, ~ = −1/2 ε = 25, ~ = −1/5
5 0.998 33 1.010 46
10 0.995 88 1.003 13
15 0.996 24 1.001 17
20 0.996 44 1.000 49
25 0.996 44 1.000 17
30 0.996 44 1.000 00
L M ü , A / m S I4 dA ≈ ±
√
3(1 − ε−1)
m∑
k=0
βm,k(~) εk (6.37)
Zfe, βm,k g h i j ~ /\| . 6.1, % ε
0 k & , !
~ 1¸ " # $ 0 l . LM ü , A 1 9|/ N O # $ ~ m , 6.3. n o , ~ (~ < 0) D p q , 9| (6.23) / N O # $ p k . n Ú r , ~
! g ε 1×| , s~ ε t k ,Z u D !v l
. L M ü , '~ = − 1
1 + ε/3(6.38)
A / 10 S , wA≈± 1
(1 + ε/3)10
√
1 − 1
ε
(
1.180 3 + 3.907 5 ε + 5.812 8 ε2 + 5.114 9 ε3
+2.946 6 ε4 + 1.160 3 ε5 + 0.316 02 ε6 + 5.872 6 × 10−2 ε7
+ 7.129 8 × 10−3 ε8 + 5.139 6 × 10−4 ε9 + 1.700 1 × 10−5 ε10)
(6.39)
è F @ x y # $ 1 6 ε < +∞C=z T U q t , R 6.3 ù . A / 10 S
(6.39)ÂÃ
limε→+∞
|A| = 1.003 9
Z |[ 0.39%.¹ V _
(6.38), E A / 3 S , wA ≈ ±
(
7 015 ε3 + 70 251 ε2 + 220 917 ε + 226 105)
4 096√
3(ε + 3)3
√
1 − 1
ε(6.40)
V6 W ìYXYíYîYïYðYñYòYóõôõöY÷ · 83 ·
(6.2 u(ξ) +,~,,,,~.0/0132
ε=10, ~=−1/2 4 , u(ξ) H 10 ?050607 ; 800 .32 ε=10, ~=−1/2 4 , u(ξ) H 20 ?050607 ;
8 132 ε=25, ~=−1/5 4 , u(ξ) H 20 ?050607 ; 00 .32 ε=25, ~=−1/5 4 , u(ξ) H 30 ?050607
(6.3 A ñ 10 ,,,~ (6.37) ,,,~ (6.7) +0
.320 7 (6.7) ; 0 1320 7 (6.8); 0 132 ~ = −1 47H050607 (6.37);.0/0132
~ = −1/2 47H50607 (6.37) ; .0/0132 ~ = −1/4 47H050607 (6.37); 8 132 ~ = −1/(1 + ε/3) 47H050607 (6.39)
Pè F T U q t , R 6.4 ù . ùº , ÔÕÖ| ~ uvko Q m 9| 1 NO #$ NO / ÈW . Ô Õ Ö | ~ @ ç > ø / C ¡ / G V .
· 84 · »½¼½¾À¿
(6.4 ~ = −1/(1 + ε/3) * A ñ 3 ,~,,,,~ (6.40) ,,,~ (6.7) +0
.320 7 (6.7);.0/01320 7 (6.8) ; 8 132 70¢050607 (6.40)£ ¡ r / g , hij (6.10) @ ç >ø/ C P ¡ /G V . ]
g `ahij (6.10), L MÒÓ ¤ ¥ ¦ § (6.11) ÔÕ < ¨ © H (6.12). É K , ]g `ahij (6.10), L M q Yø ª (6.28),
f Z lm ξ sin ξ e « Ãü , Æ s ¬ ®¯ °. D qÙÚ « g , j ± ² ù ³ ´ « ® , ξ sin ξ µ g ¶ · Ú ¸ « ¹ ºe¼» ,
[ u(ξ) ½ R ¾o ¿ À×|ξm sin(nξ), ξm cos(nξ) | m > 0, n > 1
IÁ . nP gÂÃÄ ® Ô Õ × | H(ξ) µ g o«Å . g , ¬ V À × |(6.9) Æ u(ξ)
Ç È ¸ " .
D qÙÚ « g , (6.37) (6.40) é=É (6.8)
È t , R 6.3 6.4
ù . ÉK , E A « 3 S (6.40) P@ÊQ#$ 1 6 ε < +∞C èF
(6.7) T U q t , s Ë ÂÃ >ÿp ε = 1, º Ì « Í Q > Î .¹V ù ý ç - ÏÐ ( Ö 41 3.5.2 ), nÑ uÒ A «9 | ÓN O , I 6.2 ( % ε = 10 ε = 25 & ). L M ü , A « [m, m] S ç - Ï Ð µ h i j ÔÕÖ| ~. Ô [4, 4] S ç - Ï Ð
A ≈ 2√
3
√
1 − 1
ε
P (ε)
Q(ε)(6.41)
@ Ê Q # $ 1 6 ε < +∞C=z è F (6.7) T U q t , Ô e
»ÖÕ0× (6.5) Ø0Ù0Ú0Û0Ü 0 6 ξ 6 π. Ý0Þ0Ù0Ú0Û0ß , ξ sin ξ Ü0à0á0â . —— ã0ä
å6 æ ì,çYíYîYïYðYñYòYóõôõöY÷ · 85 ·
P (ε)=8 665 210 296 046 039 923 + 2 500 964 782 519 057 396 ε
+604 034 298 653 768 562 ε2 + 62 408 285 303 687 028 ε3
+3 874 319 809 940 915 ε4
Q(ε)=25 430 938 337 575 455 089 + 7 921 677 254 280 814 588 ε
+1 930 521 704 826 790 758 ε2 + 213 027 971 364 041 596 ε3
+13 310 678 950 379 441 ε4
± ² n Q H èf ,ç > ø é¸Èþ >ÿ « < ¨ ® g ¸ " « .
[6.2 ε = 10 \ ε = 25 ] , A ^ [m, m] _,ê,ë - ì,í,b,c,`
[m,m] ε = 10 ε = 25
[2, 2] 0.999 14 1.011 67
[4, 4] 0.996 51 1.001 13
[6, 6] 0.996 44 1.000 12
[8, 8] 0.996 44 0.999 96
[10, 10] 0.996 44 0.999 94
[12, 12] 0.996 44 0.999 94
[15, 15] 0.996 44 0.999 94
î7 ï ðòñòóòôòõòöò÷ùøùúùû
üý <¨®þ@ ýÿ . G  H , µLM ³´ Duffing H® , ª
v′′ + ε(v − v3) = 0 (7.1)
v(0) = v(π) = 0 (7.2)
, ′ I ξ . 6 ² , L M V > Ë !"# >$ «
ε = 1, % V R ¾ À&'sin[(2m + 1)ξ] | m > 0 (7.3)
I Á Ô ÿ . D()* « g , þ @+, ý Q R ¾ À&'sin[(2m + 1)κ ξ] | m > 0, κ > 1 (7.4)
Ô e , κ > 1 g- Q Ë Ê' ,z. V/ I Á (7.2) « &' .
* r0 ,
ª (7.1)
(7.2).12 ý ÿ
. 3 R4 . ± ² , 5 V À&' (7.4), L M V6 7( ª (7.1)
(7.2) 8 2 « 9 ÿ » .
µ: -; ¨ , < ¸A = v(π/2κ), v(ξ) = A u(ξ) (7.5)
=, ª (7.1) > d
u′′ + ε(u − A2u3) = 0, u(0) = u(π) = 0 (7.6)
? ()* « g , @ (7.6)e
A AB . CD@ (7.5), dEu(π/2κ) = 1 (7.7)
»GFIHIJIKILIMINIO Mathematica P ÕIQIRISITIU ØIVIW , XIYIZI[I\ . ]I^I_I` OIS E-mail [email protected], eIfIg ¹Ihdc http://numericaltank.sjtu.edu.cn/code.htm, PIiIjk RlTlm Zl[lnlolplq0à Mathematica ×lr Øl^l_ls T . —— ã0ä
å7 æ t,çvuvwvxvyvzvv|~ · 87 ·
7.1 7.1.1 À&' (7.4)
u(0) = u(π) = 0, % (7.6)
, u(ξ). Á
u(ξ) =
+∞∑
m=0
cm sin[(2m + 1)κξ] (7.8)
Ô , cm ' . ¡¢£¤¥¦§
.¨© ¥¦§(7.8), % @ (7.7), ª«¬
u0(ξ) = sin(κ ξ) (7.9)
® u(ξ) °¯±²³ ÿ , ´ , κ > 1 µ¶ ' . ·¸¥¦§ (7.8), ¬L[Φ(ξ; q)] =
∂2Φ(ξ; q)
∂ξ2+ κ2 Φ(ξ; q) (7.10)
® ¹º »¼ ,¡ »¼½ 2¾¿ À
L[C1 sin(κ ξ) + C2 cos(κ ξ)] = 0 (7.11)
´ , C1
C2 ' . 4Á , CD (7.6), < ¾¿ »¼N [Φ(ξ; q), α(q)] =
∂2Φ(ξ; q)
∂ξ2+ ε
[
Φ(ξ; q) − α2(q)Φ3(ξ; q)]
(7.12)
´ , q ∈ [0, 1] ÃÄ >Å , Φ(ξ; q) ÆÇ ξ
q¤ AB&' , α(q) ÆÇ q
¤ AB&' . È ~ 6= 0
¹ºÉ ' , H(ξ) 6= 0 ¹º &' , ÊË 9ÌÍ >
(1 − q)L[Φ(ξ; q) − u0(ξ)] = ~ q H(ξ) N [Φ(ξ; q), α(q)] (7.13)
ÎÏ Φ(0; q) = Φ(π; q) = 0 (7.14)
Ðq = 0 Ñ , (7.13)
(7.14)
¤ ÿ Φ(ξ; 0) = u0(ξ), ξ ∈ [0,π] (7.15)
Ðq = 1 Ñ , (7.13)
(7.14) Ò Ó (7.6), ÔÕΦ(ξ; 1) = u(ξ), α(1) = A (7.16)
· 88 · ÖØ×ØÙÛÚÜ 4 ,
Ðq Ô 0 >ÞÝÞß 1 Ñ , Φ(ξ; q) ÔÞ¯Þ±Þ²Þ³ ÿ u0(ξ) = sin(κξ) >ÞÝ (
Í > )à (7.6)¤á ÿ u(ξ) ;
â, α(q) ãÔ¯±²³ ? A0 >Ý (
Í > )à á ?
A = v(π/2κ).? ()*ä ,
9ÌÍ > (7.13) å 2 ¹ºÉ ' ~ ¹º &
' H(ξ). æç ~
H(ξ) ¬ ¸è ,Î à Ó9ÌÍ > (7.13)
(7.14) 8 2
q ∈ [0, 1] é 2 ÿ , um(ξ) =
1
m!
∂mΦ(ξ; q)
∂qm
∣
∣
∣
∣
q=0
, Am =1
m!
dmα(q)
dqm
∣
∣
∣
∣
q=0
(7.17)
m > 1 é þê . ëì ,°íîïð <ñ @ (7.15),
.òΦ(ξ; q)
α(q)
ï𠾿q¤óô '
Φ(ξ; q) = u0(ξ) +
+∞∑
m=1
um(ξ) qm (7.18)
α(q) = A0 ++∞∑
m=1
Am qm (7.19)
4Á , æç ~
H(ξ) ¬ ¸è , ÔÕ óô ' (7.18)
(7.19) ê q = 1 Ñõö ,=
, C@ (7.16),
2 ô ' ÿ
u(ξ) = u0(ξ) +
+∞∑
m=1
um(ξ) (7.20)
A = A0 +
+∞∑
m=1
Am (7.21)
7.1.2 ÷ "ø , <ÂùDÅ
uk = u0(ξ), u1(ξ), u2(ξ), · · · , uk(ξ) , Ak = A0, A1, A2, · · · , Akò 9ÌÍ > (7.13)
(7.14) q m ú , ûü Î m!, ýþÈ q = 0, ª«(
ß mÌÍ >ÿ
L [um(ξ) − χmum−1(ξ)] = ~ H(ξ) Rm(um−1, Am−1) (7.22)
ÎÏ um(0) = um(π) = 0 (7.23)
ÿ (7.18) (7.19) (7.13) (7.14), q , P RS lp (7.22) (7.23), (7.24). —— !
"7 # t%$vuvwvxvyvzvv|~ · 89 ·
´ , χm C (2.42) <Â , Rm(um−1, Am−1)
=1
(m − 1)!
∂m−1N [Φ(ξ; q), α(q)]
∂qm−1
∣
∣
∣
∣
q=0
=u′′m−1(ξ) + ε um−1(ξ)
−ε
m−1∑
n=0
(
n∑
i=0
Ai An−i
)
m−1−n∑
j=0
uj(ξ)
m−1−n−j∑
r=0
ur(ξ)um−1−n−j−r(ξ)
(7.24)
? ()*ä , um(ξ)
Am−1 éAB . &ª«' 2 -() Ó um(ξ) * 6 .Ü 4 ,
¡¢£,+,-,., /,0,1,2 -,(,3 2,4' Î < Am−1. æç H(ξ) ¬
¸è , ÔÕ5 ÌÍ > (7.22)¤67 .8
~ H(ξ) Rm(um−1, Am−1) =
µm∑
n=0
bm,n(Am−1) sin[(2n + 1)κ ξ] (7.25)
´ , bm,n(Am−1) ' , µ¶ ' µm ÆÇ Ó H(ξ)
m.¨©
L¤ À (7.11), 9
bm,0(Am−1) 6= 0, mÌÍ > (7.22)
¤ ÿ: å ¾¿ ξ sin(κ ξ)
¡ +·¸¥¦§(7.8). ;< -= ,
+ ( +>?
bm,0(Am−1) = 0 (7.26)
µ@ -(3 2BA ) Ó Am−1 4' , ÔÕC ¡¢£-. . Dþ , EF7( (7.22)
¤ ÿ
um(ξ)=χm um−1(ξ) +
µm∑
n=1
bm,n
[1 − (2n + 1)2κ2]sin[(2n + 1)κ ξ]
+C1 sin(κξ) + C2 cos(κξ) (7.27)
´Ø , C1
C2 ' .
¨© ¥¦§(7.8), C2 GIH 9 .
? ()*ä , ' C1+ 1 C (7.23) < ,Ü C2 = 0 Ñ (7.23) JLK . MÕ , C
@ (7.7), Eum(π/2κ) = 0 (7.28)
@ (7.28).N - < C1.
â, ª« . Æ ú7( Am−1
um(ξ).
· 90 · ÖØ×ØÙÛÚFB , N
ÌOP ÿ u(ξ) ≈ u0(ξ) +
N∑
m=1
um(ξ) (7.29)
A ≈ A0 +N−1∑
m=1
Am (7.30)
7.1.3 QRSTS,T 7.1 U,V,W,X,Y (7.20) Z (7.21) [,\ , ]_^ , uk(ξ) `,a,b,c (7.22) Z (7.23),def
(7.24) Z (2.42) gh , ijklmnbc (7.6) oY .p 9 ô ' ÿ (7.20) õö ,= G 2
limm→+∞
um(ξ) = 0, ξ ∈ [0,π]
CD@ (2.42) Al@ (7.10) @ (7.22),
q @ , E~ H(ξ)
+∞∑
k=1
Rk(uk−1, Ak−1)
= limm→+∞
m∑
k=1
L[uk(ξ) − χk uk−1(ξ)]
=L
limm→+∞
m∑
k=1
[uk(ξ) − χk uk−1(ξ)]
=L[
limm→+∞
um(ξ)
]
=0
Ü ~ 6= 0
H(ξ) 6= 0, 8 Î ,2
+∞∑
k=1
Rk(uk−1, Ak−1) = 0
ò(7.24) 4 Ä q @ , %Ý" .
¨© ô ' (7.20)
(7.21)¤ õö , E
d2
dξ2
[
+∞∑
k=0
uk(ξ)
]
+ ε
[
+∞∑
k=0
uk(ξ)
]
−(
+∞∑
m=0
Am
)2 [+∞∑
k=0
uk(ξ)
]3
= 0
CD@ (7.9)
(7.23), E+∞∑
k=0
uk(0) =
+∞∑
k=0
uk(π) = 0
"7 # t%$vuvwvxvyvzvv|~ · 91 ·
8 Î , r0 ô ' (7.20)
(7.21) õö , s« G <ä (7.6)¤ ÿ
. tu .
7.2 vxw ¨© <ñ 7.1, ª«'/¬ ¸è ¹º &' H(ξ)
¹ºÉ ' ~,Î y ô '
(7.20)
(7.21) õö . µ ¾ ê 6 z|ë â , ê +~ ¿,. ¬ + Í @ ¹º H(ξ). ø , ª«¬
H(ξ) = 1 (7.31)
=, CD@ (7.9) @ (7.24), E
~ H(ξ) R1(u0, A0)
=~
(
ε − κ2 − 3
4ε A2
0
)
sin(κ ξ) +1
4~ ε A2
0 sin(3κ ξ) (7.32)
¨© @ (7.25),2
b1,0 = ~
(
ε − κ2 − 3
4ε A2
0
)
, b1,1 =1
4~ ε A2
0
CD@ (7.26), ª«ß4 ε − κ2 − 3
4ε A2
0 = 0 (7.33)
´ Ð ε > κ2 Ñ 2 9A0 = ± 2√
3
√
1 − κ2
ε(7.34)
Ü 4 , µ¶ κ > 1,Ð
ε = κ2 (7.35)
Ñ 6 . ( , ε?
, ê ( 6 = .? ä ,
(7.4) < Ó κ¤?
.Ü 4 , ε
?, ê .+ :¡ ; , ª«' κ = 2 κ = 3 ¢£¤¥ .
? ä ,ô (7.20)
(7.21) õö¦§õö¨©C ¹ºÉ ~ ª< . «¬ ε µ¶ κ, ,
ε > κ2 > 1, ®¯°±²³ A ∼ ~ ´D ( ɵ 24 ¶ 3.5.1 · ), ¸ß¡ ( ~¤¹º ¦
§ ,Î ªy ô (7.21) õö . » ¾ , κ = 2, ε = 10 A 25 A 100 κ = 3, ε = 40 A 90 A 225
Ñ A ∼ ~ ´D , ¼½ ¾¾ 7.1 ¾ 7.2 ¿À .¨© Á ~ ´D , ÂM ,
Ð~ = −1,
ε = 10, κ = 2 A 3 Ñ , ÃÄ ~ = −1/2, ε = 40, κ = 2, ÃÄ ~ = −1/2, ε = 90, κ = 3, ÃÄ ~ = −1/5, ε = 100, κ = 2,
ÎÏ~ = −1/5, ε = 225, κ = 3 Ñ ,
ô (7.21) õö ,¾
7.1 7.2 ¿À .
· 92 · ÖØ×ØÙÛÚÅ
7.1 --h= −1, H(ξ) = 1, ε = 10 Æ κ = 2, 3 Ç A È%É%Ê%Ë%Ì%ÉÍÎ κ = 2 κ = 3
2 0.869 414 205 4 0.364 317 573 1
4 0.869 693 253 2 0.364 310 089 9
6 0.869 685 765 6 0.364 310 019 4
8 0.869 686 026 5 0.364 310 018 7
10 0.869 686 016 4 0.364 310 018 7
12 0.869 686 016 8 0.364 310 018 7
14 0.869 686 016 8 0.364 310 018 7
16 0.869 686 016 8 0.364 310 018 7
18 0.869 686 016 8 0.364 310 018 7
20 0.869 686 016 8 0.364 310 018 7
Å7.2 --h= −1/2, H(ξ) = 1, ε = 40, κ = 2 Æ ε = 90, κ = 3 Ç A È%É%Ê%Ë%Ì%ÉÍÎ ε = 40, κ = 2 ε = 90, κ = 3
2 0.980 70 0.981 71
4 0.999 12 0.998 54
6 0.996 13 0.996 39
8 0.996 34 0.996 25
10 0.996 56 0.996 58
12 0.996 35 0.996 35
14 0.996 49 0.996 48
16 0.996 41 0.996 42
18 0.996 45 0.996 44
20 0.996 43 0.996 44
22 0.996 44 0.996 44
24 0.996 44 0.996 44
26 0.996 44 0.996 44
28 0.996 44 0.996 44
30 0.996 44 0.996 44¾¾7.1 7.2 ¿À , A ~
¤¹º ¦§D«¬ κ A ε 1ÕÏÐ .ÜÑ
, ~
ÒÓÔD ε 1ÕÏÐ . ª«Õ , 9È~ = −
(
1 +ε
3κ2
)−1
(7.36)
Ö×Ø ε > κ2 > 1 «¬ ε κ,ô (7.21) ®ê¦§
κ26 ε < +∞
Ù õö , ´ÚÔ 10ÌOP
A≈±(
1 +ε
3κ2
)−10√
1 − κ2
ε
(
1.180 3 + 3.907 5ε
κ2+ 5.812 8
ε2
κ4
"7 # Û%$%Ü%Ý%Þ%ß%àvv|~ · 93 ·
+5.114 9ε3
κ6+ 2.946 6
ε4
κ8+ 1.160 3
ε5
κ10+ 0.316 02
ε6
κ12
+5.872 6 × 10−2 ε7
κ14+ 7.129 8 × 10−3 ε8
κ16+ 5.139 6 × 10−4 ε9
κ18
+ 1.700 1 × 10−5 ε10
κ20
)
(7.37)
á7.1 H(ξ) = 1, κ = 2 â ε = 10, 40, 100 ã A ∼ ~ äàåæçèêé
ε = 10 ë A 20 ÍÎì ;æçèêé
ε = 40 ë A 20 ÍÎì ; í èêé ε = 100 ë A 20
ÍÎì
á7.2 H(ξ) = 1, κ = 3 â ε = 10, 90, 225 ã A ∼ ~ äàåæçèêé
ε = 10 ë A 20 ÍÎì ;æçèêé
ε = 90 ë A 20 ÍÎì ;í èêé ε = 225 ë A 20 ÍÎì
· 94 · ÖØ×ØÙÛÚ
ê ¶( ¦§ κ2 6 ε < +∞Ù|îðï|ñòóò « á ª ôõöε =
8κ2
π2(2 − A2)
K
(
A2
2 − A2
)
(7.38)
÷¸ E @ , ´ , K Àø¡ùúûüðý|þ¼ ,
¾¾7.3 ¿À . ÿ q ,
¾7.3
«ª«¡ ( Duffing ¼ ¢£ ú ¶ ¼ ¾ .
á7.3 A Þ 10 %Ý (7.37) â%Ý (7.38) æêé ì
; è 1é
κ = 1 ; è 2é
κ = 2; è 3é
κ = 3; è 4é
κ = 4 ; è 5é
κ = 5; è 6é
κ = 6 - OP ( ɵ 41 ¶ 3.5.2 · ), ¯2¨ ô (7.21)
¤ õö¨© ,¾
7.3 7.4 ¿À . ª«Õ , [m, m]Ì
- OP + ÆǹºÉ ~.[4, 4]
Ì - OP
A ≈ 2√
3
√
1 − κ2
ε
P (ε)
Q(ε)(7.39)
ê ¶( ¦§ 1 6 ε/κ2 < +∞Ù ¹º
, ´P (ε)=8 665 210 296 046 039 923 + 2 500 964 782 519 057 396
( ε
κ2
)
+604 034 298 653 768 562( ε
κ2
)2
+ 62 408 285 303 687 028( ε
κ2
)3
+3 874 319 809 940 915( ε
κ2
)4
Q(ε)=25 430 938 337 575 455 089 + 7 921 677 254 280 814 588( ε
κ2
)
+1 930 521 704 826 790 758( ε
κ2
)2
+ 213 027 971 364 041 596( ε
κ2
)3
+13 310 678 950 379 441( ε
κ2
)4
"7 # Û%$%Ü%Ý%Þ%ß%à "! · 95 ·
Å7.3 H(ξ) = 1, ε = 10 Æ κ = 2, 3 Ç A È [m, m] #$% - &'%Ë%Ì%É[m, m] κ = 2 κ = 3
[1, 1] 0.869 402 945 7 0.364 310 463 6
[2, 2] 0.869 690 237 7 0.364 310 017 8
[3, 3] 0.869 685 956 9 0.364 310 018 7
[4, 4] 0.869 686 017 6 0.364 310 018 7
[5, 5] 0.869 686 016 8 0.364 310 018 7
[6, 6] 0.869 686 016 8 0.364 310 018 7
[7, 7] 0.869 686 016 8 0.364 310 018 7
[8, 8] 0.869 686 016 8 0.364 310 018 7
[9, 9] 0.869 686 016 8 0.364 310 018 7
[10, 10] 0.869 686 016 8 0.364 310 018 7
Å7.4 H(ξ)=1, ε=40, κ=2 Æ ε=90, κ=3 Ç A È [m, m] #$% - &'%Ë%Ì%É
[m, m] ε = 40, κ = 2 ε = 90, κ = 3
[1, 1] 0.974 744 985 5 0.975 317 974 5
[2, 2] 0.998 880 376 6 0.998 825 089 5
[3, 3] 0.996 055 184 0 0.996 076 135 0
[4, 4] 0.996 495 776 6 0.996 492 182 9
[5, 5] 0.996 430 442 0 0.996 430 557 1
[6, 6] 0.996 437 076 6 0.996 436 833 6
[7, 7] 0.996 435 286 0 0.996 435 270 9
[8, 8] 0.996 435 370 7 0.996 435 361 4
[9, 9] 0.996 435 331 4 0.996 435 331 4
[10, 10] 0.996 435 335 2 0.996 435 335 5
[11, 11] 0.996 435 335 1 0.996 435 336 3
[12, 12] 0.996 435 336 2 0.996 435 336 3
[13, 13] 0.996 435 336 3 0.996 435 336 3
[14, 14] 0.996 435 336 3 0.996 435 336 3
[15, 15] 0.996 435 336 3 0.996 435 336 3
ª«Õ , r0 ô (7.21) õö , ( ï Ú ~ Ó« u(ξ)¤ô (7.20) ê ¶
( ¦§ ξ ∈ [0,π]Ù*) õö ,
¾¾7.4 ¾ 7.5 ¿À .
¨© + (7.6),-./
, 9u(ξ) ä , ëì −u(ξ) ã G ¬ä . 0 ø , ª«ê ¾ 7.4 ¾ 7.5 *1 ¹ «ù
.-./ Duffing ¼23 ε Ó ¹ . » ¾ ,
Ðε = 10 Ñ , ¼½ ¹
) Ó κ = 1 ¢ ( -4 , κ = 2
¢ ( -4 , κ = 3 ¢ ( -4 , ÔÕ ¹5 (-4
. °6 , «¬ ε > 1, Duffing ¼23 ¹ 2[√
ε ] ( -4 , ,
[x] À x ¶ . ¿ Î , ε Ó7 ,
-4 ( 7 ,¾¾
7.3 ¿À .Ð
ε 8 Ó9: Ñ , ê 9 : ( -;4 .ÜIÑ
,-;.;/ ;+ (7.1) ;<;= I (7.2) >;?IIÚ;@AB C õD EF ¼ .
· 96 · GIHIJLK
á7.4 H(ξ) = 1, κ = 2 M ε = 10, 40, 100 ã v(ξ) = Au(ξ) NO%ÝP%Ýçèêé
~ = −1, ε = 10 ë v(ξ) Q 5 R ÍÎì ;æçèêé
~ = −1/2, ε = 40 ë v(ξ) Q 10 R ÍÎì ; íèêé~ = −1/5, ε = 100 ë v(ξ) Q 20 R ÍÎì
á7.5 H(ξ) = 1, κ = 3 M ε = 10, 90, 225 ã v(ξ) = Au(ξ) NO%ÝP%ÝS èêé
~ = −1, ε = 10 ë v(ξ) Q 5 R ÍÎì ;æçèêé
~ = −1/2, ε = 90 ë v(ξ) Q 10 R ÍÎì ; íèêé~ = −1/5, ε = 225 ë v(ξ) Q 20 R ÍÎìTVU ¸VWVX, ± + , YVZV[ (7.8) \,V]V^ V_ . V`,»VaVbVcVdVegf ,
Ô ; ¼ ô ;h , i ;j ; I , ¯;kI -;.;/ 2;3I¿ ¹; .; ¼ ô
h YZ[lmªnopqr -./ 23 ¡`ûstu .
v8 w xzyzz|zz~z
6 -./ 23IÓ;; . »ef , ÖÔ ¼ ô h k -./ + Ó ._ 0»a ,j ` -./ + Ó23
u′′(x) + λ u(x) + ε u3(x) = 0 (8.1)
×Ø<=u(0) = u(1) = 0 (8.2)
, ′
x ¡ , ε 0`¢ . ¤£¦¥§©¨ Uª « λn
un(x), ¬u′′
n(x) + λn un(x) + ε u3n(x) = 0 (8.3)
® <=un(0) = un(1) = 0 (8.4)
¯°∫ 1
0
u2n(x)dx = 1 (8.5)
± , ¥ n > 1 0`² . ³´ [12](A. H. Nayfeh) µXp¶ ·¸ h WX«23 , ¹º»¼ ·¸½¾ un(x) =
√2 sin(nπx) − ε
√2
16n2π
2sin(3nπx) + O(ε2) (8.6)
λn = n2π
2 − 3
2ε + O(ε2) (8.7)
¿ ·¸ À Á ε ÂÃ .
ÅÄÇÆÇÈÇÉÇÊÇËÇÌÇÍ Mathematica ÎÇÏÇÐÇÑÇÒÇÓÇÔÇQÇÕÇÖ , ×ÇØÇÙÇÚÇÛ . ÜÇÝÇÞÇß Í Ò E-mail àáãâ[email protected], äÇåÇæÇçÇè â http://numericaltank.sjtu.edu.cn/code.htm, ÎÇéÇêë ÑÓìÙÚíîïðñ Mathematica òóQÝÞôÓ . —— õö
· 98 · GIHIJLK
8.1 ÷ùøûúûüûý8.1.1 þÿ
<= (8.4), (8.3) , , isin[(2k + 1)nπx] | n > 1, k = 0, 1, 2, 3, · · · (8.8)
un(x),
un(x) =+∞∑
k=0
an,k sin[(2k + 1)nπx] (8.9)
±, an,k . (8.9) nop un(x) YZ[ . YZ[ (8.9), * (8.4) (8.5), ,
un,0(x) =√
2 sin(nπx) (8.10)
_ un(x) !"# . $% , YZ[ (8.9) (8.3), `&' (a
LΦ =∂2Φ
∂x2+ (nπ)2Φ (8.11)¿ (a)Â*
L [C1 sin(nπx) + C2 cos(nπx)] = 0 (8.12)±, C1 C2 + . (8.3), ,-(a
N [Φ(x; q),Λ(q)] =∂2Φ(x; q)
∂x2+ Λ(q) Φ(x; q) + ε Φ3(x; q) (8.13)
±., q ∈ [0, 1]
§ V` /101213 , Φ(x; q) 141516 x q VV , Λ(q) 141516 q V , 7 1819V 6 un(x) λn.jV1:
~ 6= 0 V`11; & ' ¢V , H(x) 6= 0
`; & ' .D< ;¼=2
(1 − q) L [Φ(x; q) − un,0(x)] = q ~ H(x) N [Φ(x; q),Λ(q)] (8.14)
® <=Φ(0; q) = Φ(1; q) = 0 (8.15)
, @ q = 0 > , ;¼=2 (8.14) (8.15) ÂΦ(x; 0) = un,0(x) (8.16)
?8 @ ACBCDCECFCGCHCI · 99 ·
@ q = 1 > , ;V¼1=12 1 (8.14) (8.15)8191J 61K ! 1 (8.3) (8.4), L1M
Φ(x; 1) = un(x), Λ(1) = λn (8.17)
N $ , @1/101213 q L 0 O1P1Q 1 > , Φ(x; q) L11!1"1#V un,0(x) 2V°1Q1R1SVV un(x), Λ(q) L!"#« λn,0 2°QRS« λn. «TU § , ;¼=2 >?& ' ¢ ~ & ' H(x),
N $ , Φ(x; q) Λ(q) 456 ~ H(x).VW~ H(x) XY , Z[ 6 ;¼=2 (8.14) (8.15)
\ Â q ∈ [0, 1] ]Âun,k(x) =
1
k!
∂kΦ(x; q)
∂qk
∣
∣
∣
∣
q=0
, λn,k =1
k!
∂kΛ(q)
∂qk
∣
∣
∣
∣
q=0
(8.18)
k > 1 ]^ T , _`a
Φ(x; q) =+∞∑
k=0
un,k(x) qk (8.19)
Λ(q) =
+∞∑
k=0
λn,k qk (8.20)
Tq = 1 >bc . de (8.16) (8.17), fg
un(x) = un,0(x) ++∞∑
k=1
un,k(x) (8.21)
λn = λn,0 +
+∞∑
k=1
λn,k (8.22)
WVX 1 1h1i un,k(x) λn,k
819 ºV»VVVVkj VV«l17 !1"#Vmn .
8.1.2 oÿpq , ,-re3
un,m = un,0(x), un,1(x), un,2(x), · · · , un,m(x)
λn,m = λn,0, λn,1, λn,2, · · · , λn,m
;¼=2 (8.14) (8.15)
q ¡ k s , tuZ k!, vw : q = 0, Âx¼=
· 100 · GIHIJLK2
L [un,k(x) − χk un,k−1(x)] = ~ H(x) Rn,k(un,k−1, λn,k−1) (8.23)®y =un,k(0) = un,k(1) = 0 (8.24)
, χk e (2.42) ,- , Rn,k(un,k−1, λn,k−1)
=u′′n,k−1(x) +
k−1∑
m=0
λn,mun,k−1−m(x)
+ε
k−1∑
m=0
un,k−1−m(x)
m∑
j=0
un,j(x)un,m−j(x) (8.25)
«TU § , z Uº,² n > 1, @ k > 1 > , un,k(x) λn,k−1 ] .|§
, Â`n 6 un,k(x) ~ 8 (8.23),
N $ ,¿ j
, ^O;`; , ZS, λn,k−1. § ,; ÂP& ' ;
H(x). *YZ[ (8.9), Rn,k(un,k−1, λn,k−1) f
Rn,k(un,k−1, λn,k−1) =
Mn,k∑
m=0
dn,m sin[(2m + 1)nπx]
±, dn,m , Mn,k 456 n k ² .
YZ[ (8.9), e (8.11) (8.23), H(x) =
H(x) = sin2[(2m − 1) nπx] (m > 1) (8.26)
H(x) = sin[(2m)nπx] (m > 1) (8.27)
H(x) = cos2[(2m − 1) nπx] (m > 1) (8.28)
H(x) = cos[(2m)nπx] (m > 1) (8.29)
(8.19) (8.20) RÏò (8.14) (8.15), q ¡Q¢ £ , ÎÑÒ¤¥¦§ï¨RÏò (8.23) (8.24), ©ª«¬ (8.25). —— õö
?8 @ ACBCDCECFCGCHCI · 101 ·
[H(x) = 1 (8.30)±
, m > 1 `² .N $ , Â
H(x) Rn,k(un,k−1, λn,k−1) =
µn,k∑
m=0
bn,km (λk−1) sin[(2m + 1)nπx] (8.31)
, bn,k
m (λk−1) , µn,k ` H(x) n j k S,®² . ¯bn,k0 (λk−1) 6= 0
x¼=2 (8.23) °±?Â~ bn,k
0 (λk−1) sin(nπx)
.N $ ,
* (8.12), un,k(x) ?Âx sin(nπx)
.|¿ j² XYZ[ (8.9). p³´ u , µ¶·¸
bn,k0 (λk−1) = 0 (8.32)
®¹º»`ºjS, λn,k−1 , LM¬ ¿ .T W» k λn,k−1 w , Qun,k(x) = χk un,k−1(x) +
µn,k∑
m=1
~ bn,km (λk−1)
n2π
2 [1 − (2m + 1)2]sin[(2m + 1)nπx]
+C1 sin(nπx) + C2 cos(nπx) (8.33)
±, C1 C2 ] .
YZ[ (8.9), fgC2 = 0
«TU § , W»¼ ¸ ®y½ ; (8.24),N $ , C1
j¾ S, . ¿M ,
*¯° (8.5), fg∫ 1
0
(
k∑
m=0
un,m(x)
)2
dx = 1 (8.34)
· 102 · GIHIJLK± º»
C21 + 4αC1 + 2β = 1 (8.35)±
α =
∫ 1
0
wn,k(x) sin(nπx)dx, β =
∫ 1
0
w2n,k(x)dx (8.36)
wn,k(x) =
k−1∑
j=0
un,j(x) + χk un,k−1(x)
+
µn,k∑
m=1
~ bn,km (λk−1)
n2π
2 [1 − (2m + 1)2]sin[(2m + 1)nπx] (8.37)
(8.35), QÀ
C1 = −2α +√
4α2 − 2β (8.38)
C1 = −2α −
√
4α2 − 2β (8.39)
7 k8k9 6 À ` j .kÁ
, 4 s k λn,0 j un,1(x) j λn,1 jun,2(x)
J. z Uº,® n,
« m ¼ ½¾ 89 un(x) ≈ un,0 +
m∑
k=1
un,k(x) (8.40)
λn ≈ λn,0 +
m∑
k=1
λn,k (8.41)
8.1.3 ÂÃÄÅÄÅ 8.1 ÆÇÈÉ
un,0(x) ++∞∑
k=1
un,k(x)
Êλn,0 +
+∞∑
k=1
λn,k
Ë1Ì, Í.Î , un,k(x) Ï1Ð1Ñ1Ò (8.23) j (8.24)
Ê(8.34), Ó1Ô1Õ (8.11) j (8.25)
Ê(2.42)Ö×
, ØÙÚÛÜÝÞÑÒ (8.1)Ê
(8.2) ßàáâÉ Ê àáã .
?8 @ ACBCDCECFCGCHCI · 103 ·
ä ¯abc , µ,fglim
m→+∞un,m(x) = 0
d , e (8.11) j (8.23) (2.42), fg
~ H(x)
+∞∑
k=1
Rk(un,k−1, λn,k−1)
= limm→+∞
m∑
k=1
L[un,k(x) − χkun,k−1(x)]
=L
limm→+∞
m∑
k=1
[un,k(x) − χkun,k−1(x)]
=L[
limm→+∞
un,m(x)
]
=0
å N ~ 6= 0 H(x) 6= 0,\ Z+∞∑
k=1
Rk(un,k−1, λn,k−1) = 0
(8.25) 0æ , ¹° p . «abc , fg
d2
dx2
[
+∞∑
k=0
un,k(x)
]
+
(
+∞∑
m=0
λn,m
)[
+∞∑
k=0
un,k(x)
]
+ ε
[
+∞∑
k=0
un,k(x)
]3
= 0
e (8.10) (8.24), fg+∞∑
k=0
un,k(0) =
+∞∑
k=0
un,k(1) = 0
$% , e (8.34), ¯° (8.5) ç ® .N $ ,
è Àéabc , 7 µ,§ (8.3) j (8.4) (8.5) un(x) « λn. êë .
8.2 ìîí ú ü«;TU § , a; (8.21) (8.22) ïð& ' ¢; ~ & ' ; H(x).
zUVº1,1 n ε « , VV«V1aV (8.22)
§~ 1ñ1aV , L1M , 711b1c1ò1óV1b1c
· 104 · ôöõö÷ùøúû 456 ~.
,ü 8.1, ý Rþ& ' ¢ ~ & ' H(x)
ZSÿÀéabc .
(8.9) , & ' H(x) = , (8.26)∼(8.30). pq , H(x) = 1. º, n ε « ,
hi λn ∼ ~ e ( ¢ 24 3.5 ) ~ «a (8.22) bcò
ó . , ε = 5 j 25 j −50 >« λ1 ∼ ~ e , 8.1\
. ~ e , ! ª Q ~ "ÂÃòó , ZSÿa# (8.22) bc . 8.1
\, ε = −50
> , λ1 "a# (8.22) $ ~ = −1/2
~ = −2/5 >bc . M , %& - '( ½¾ ( ¢41 ) 3.5.2 ) *+, ú bc , 8.2
\. -./0 ,
è «"a#(8.22) bc , 12#"a# (8.21) 3$²éòó 0 6 x 6 1 4ebc . , ε = −50 > , 2# u1(x) ½¾5 8.2
\.N $ ,
z Uº, n ) ε « , 6 Qbc«)72# .
8.3 8»9 «" 5:½;¾ . 9 2# 5:5 , 8.3 ) 8.4
\.
;8.1 H(x) = 1 < λ1/π
2∼ ~ =B
>@?@A âε = 5 B 20 C@D@E@F ; G >@?@A â ε = 25 B 20 C@D@E@F ; H A â ε = −50 B 30 C@D@E@FI
8.1 H(x) = 1, ε = −50 J λ1/π2 KMLMNMOMPMLD@E@C ~ = −1/2 ~ = −2/5
5 7.537 534 384 2 7.539 945 705 1
10 7.538 448 834 1 7.538 460 057 8
15 7.538 447 119 8 7.538 447 307 8
20 7.538 447 114 1 7.538 447 126 1
25 7.538 447 114 1 7.538 447 114 6
30 7.538 447 114 1 7.538 447 114 1
35 7.538 447 114 1 7.538 447 114 1
40 7.538 447 114 1 7.538 447 114 1
?8 @ ACBCDCECFCGCHCI · 105 ·
I8.2 H(x) = 1, ε = −50 J λ1/π2 K [m, m] QMRMS - TMU OMP
[m,m] ~ = −1/2 ~ = −2/5
[2, 2] 7.540 753 911 1 7.541 081 021 1
[4, 4] 7.538 447 432 1 7.538 448 528 2
[6, 6] 7.538 447 364 4 7.538 448 039 4
[8, 8] 7.538 447 114 1 7.538 447 114 1
[10, 10] 7.538 447 114 1 7.538 447 114 1
[12, 12] 7.538 447 114 1 7.538 447 114 1
[14, 14] 7.538 447 114 1 7.538 447 114 1
[16, 16] 7.538 447 114 1 7.538 447 114 1
[18, 18] 7.538 447 114 1 7.538 447 114 1
[20, 20] 7.538 447 114 1 7.538 447 114 1I8.3 --h= −1, H(x) = 1 J λn/(nπ)2 KMLMNMV
ε n = 1 n = 2 n = 3
−25 4.432 77 1.917 46 1.415 24
−20 3.785 08 1.738 57 1.333 24
−15 3.123 28 1.557 58 1.250 74
−10 2.443 17 1.374 30 1.167 71
−5 1.738 57 1.188 52 1.084 14
0 1 1 1
5 0.212 582 0.808 470 0.915 264
10 −0.647 567 0.613 626 0.829 909
15 −1.618 38 0.415 125 0.743 906
20 −2.756 08 0.212 582 0.657 228
25 −4.130 61 0.005 561 0.569 843
;8.2 ~ = −1/2, H(x) = 1, ε = −50 <CECFMWMX u1(x) YMZM[M\M]MZ^@A â @C@D@E@F ; H A â 5 C@D@E@F ; _ > â 10 C@D@E@F
· 106 · ôöõö÷ùø
;8.3 H(x) = 1 < , ECFMWMX u1(x) YMZM[MZ
H A â ε = 50, ~ = −1/2 B 30 C@D@E@F ; G >@?@A â ε = 25, ~ = −1/2 B 10 C@D@E@F ; ` ^@A â ε = 5,
~ = −1 B 5 C@D@E@F ;>@?@A â
ε = −25, ~ = −1/2 B 20 C@D@E@F ; a ^@A â ε = −50 ~ = −1/2 B20 C@D@E@F
;8.4 ~ = −1, H(x) = 1 < , ECFMWMX u2(x) YMZM[MZ
H A â ε = 100 B 10 C@D@E@F ; G >@?@A â ε = 50 B 5 C@D@E@F ;>@?@A â
ε = −50 B 10 C@D@E@F ;^
A âε = −100 B 20 C@D@E@F
-./0 , bcO· , ~ "ÂÃòó2d , 8.1\e
.N $ , ε fgh
?8 @ ACBCDCECFCGCHCI · 107 ·
i P , ~ "h¶ ijk ; . g ε < 0, l~ = − 1
√
1 + |ε|(8.42)
h + m 6 bc 5 . æn , $2 6 + ·>o 5 h)2# . gp;!qr ε, sh$ 8.4 8t , s2# 8.5∼ 8.7\e
.u 5:vw
, fg
limε→−∞
λn
ε= −1 (8.43)
I8.4 H(x) = 1 J λn/ε KMLMNMV
ε λ1/ε λ2/ε λ3/ε
−200 −1.221 −1.488 −1.810
−400 −1.152 −1.325 −1.524
−600 −1.122 −1.272 −1.412
−1 000 −1.093 −1.196 −1.310
−2 000 −1.065 −1.135 −1.215
−5 000 −1.041 −1.083 −1.133
−10 000 −1.029 −1.059 −1.090
;8.5 H(x) = 1 < , ECFMWMX u2(x) YMZM[MZ
H Ayx ε = −5 000, ~ = −1/50 B 100 C@D@E@F ; G >@?@Ayxz ε = −1 000, ~ = −1/10 B 20 C@D@E@F ;>@?@Ayxε = −400, ~ = −1/4 B 20 C@D@E@F ;
^@Ayxε = −100, ~ = −1 B 20 C@D@E@F
· 108 · ôöõö÷ùø
;8.6 H(x) = 1, ~ = −1/50, ε = −10 000 <CECFMWMX u3(x) YMZM[MZ
H Ayx 70 C@D@E@F ; _ >yx 90 C@D@E@F
;8.7 H(x) = 1, ~ = −1/20, ε = −10 000 <CECFMWMX u4(x) YMZM[MZ
H Ayx 40 C@D@E@F ; _ >yx 60 C@D@E@F)
limε→−∞
un(x) =
1, 2k/n < x < (2k + 1)/n
−1, (2k + 1)/n < x < (2k + 2)/n(8.44)
?8 @ ACBCDCECFCGCHCI · 109 ·
u|, n > 2, k = 0, 1, 2, · · · , [(n−1)/2], [x]
e g x ~ . h 6 T1U1 , ε = −10 000
> , 6 9éfgh177r ~ h , ZSÿa# (8.21) ) (8.22) bc .
uå 9sê , & ' # ~ $%& 8 : 4c è .
-./0 , (8.38) # C1, 1 2#$~éòó0 < x < 1/n
4 .
u+n (x) eu 2# . w (8.39) C1, 12#$% òó4r ,
u−
n (x) e .u 2# x g¡ . ¢£ , g¤¥
ε ) n, ¦$§9h λn, ¨©¦$ é2# u+n (x) ) u−
n (x), ª«u−
n (x) = −u+n (x)
¬7, 7727# u−
n (x) "7®7#°¯±7727# u+n (x) "7®7#7²7³ 67´ ,
u7µè 7¢7 ,
u+n (x) ¶· k¸¹º» 5 (8.10). w¼ ¸¹º» 5
un,0(x) = −√
2 sin(nπx)
½ 7¾7 (8.39), 77727# u−n (x) "7®7#7²7³ 6 ¶7¿ . ¢7£ , À7Á7%7&7Â :7
, Ãm 6ÄÅÆÇÈÉ ËÊÌ 2# .ÍÎ ÊÌ vw, ÏÑÐÓÒÔ2# H(x) = 1 ¤t . -./0 , Ð (8.26)∼
(8.29)¥Õ Ö ÒÔ2#3פt²³ vw . l Ø % ÒÔ2# , Ãm 6 1%
h)2# .¬
, ÐÓsÙ Ö ÒÔ2#¤t ®# 5 ¯ÐÓÒÔ2# H(x) = 1 ¤t ®#²³ ´ . ¢£ , H(x) = 1 Ã× ÄÈÉÚÛ ÒÔ72# , ÜÝ-. Ø ×Þ u 9ß .uà áâÑ , %& : Ã Í ã m 6äÅÆÇå h ÈÉ ÊÌæç hèæçéê
.
ë9 ì íïîïð - ñïòïóïôïõïö
÷ùø oùúËûùüùý - þùÿ (Thomas-Feimi) ùá [81, 82] ûùüùý - þùÿu′′(x) =
√
u3(x)
x(9.1)
ª« åu(0) = 1, u(+∞) = 0 (9.2)
ûüý - þÿáá á , ½ è átá v .
(9.1) !"# à$ áá%&¡ ' Â( .
Ð (9.1) è (9.2), )* u′′(0) → +∞. ¢£ , u(x)
x = 0 ¦ # à+,
ß . - $./01 2 Â [83, 84] 3 δ 45 [85∼87] 3 Adomian Âú [88∼91],Í
6 #78Ù [92∼97], 9úûüý - þÿ : ú;<=ú . ¨> , ?7 @A êB 9C u′(0)
: B, ¢ , Ï>Dú; 3 D êB .
Ú < , EFG [51] HI Â; , J éê
(1 + x)−n | n > 1
(9.3)
KL ¤M"ûüý - þÿ # àN O 3QP úR; ® ê ú . ¨> , ÜÝ Ä ®ê ú [51]S àTU V ÌW , &X x, 8²³YZ ´ .
÷ø, [\ HI Â;]
, ÀÁ^_#`¶ Û éê , ¤Mûüý - þÿ # ¶ ÌW ® ê ú .
9.1 acbedefeg9.1.1 hijk
(9.1) Ãlmnx [u′′(x)]2 − u3(x) = 0 (9.4)2o
τ = 1 + λ x (9.5)
]qpsrstsusvswsxsyMathematica zss|ss~sssss , ssss . sss y ~ E-mail
[email protected], ssss http://numericaltank.sjtu.edu.cn/code.htm, zss Mathematica ¡¢ . —— £¤
¥9 ¦ §©¨©ª - «©¬©©®©¯©° · 111 ·
8 , λ > 0 >± ¥² ê , ³ (9.4)2 n
λ3 (τ − 1)
(
d2u
dτ2
)2
− u3(τ) = 0 (9.6)
ª« åu(1) = 1, u(+∞) = 0 (9.7)
B Cµ´µ¶ > , ³ (9.6)ص·µ¸µ¹µº Æǵ» èµ¼ ( X ) ½ ê .
Ê Í, 8 ÅÆÇÅ ² ä .
(9.7), ¾ τ → +∞, u(τ) ¿À êÁÂÃÄ , Å 0 ¿Æ êÁÂÃÄ .¬
, г(9.6) è (9.7), ÇÈÉ ¥ u(x)
ÊËÌ Í < ÇÎ . ¢£ ,Ø ÏÐÑ
u(τ) ¿À êÁÂÃÄ . , u(τ)
ÌÍ <O u(τ) ∼ τκ, τ → +∞
8 , κ >Ò* ² ê . Ó Î ÀÔ³ (9.6), ÕÖ µ× » ,Ì
κ = −3 (9.8)
¢£ , u(τ) ÃØÙÚ éê O
τ−m | m > 3
(9.9)
Û u(τ) =
+∞∑
m=3
cm τ−m (9.10)
8 , cm ê .
(9.10) ÜÝ" ÄÈÉ :ßÞàá .
9.1.2 âãäåæç Þàá(9.10), J å (9.7), )* , ^_
u0(τ) = τ−3 (9.11)
u(τ)¸¹º» ú . Ð (9.6),
½ Þàá(9.10), [\^_
L[Φ(τ ; q)] =(τ
4
) ∂2Φ(τ ; q)
∂τ2+
∂Φ(τ ; q)
∂τ(9.12)
ÒÔ ÆÇ á , 8è Ì ÇÎL(
C1
τ3+ C2
)
= 0 (9.13)
· 112 · éëêëìîí8 , C1 è C2 ê . г (9.6),
¥Õ ÙÚ ÅÆÇ áN [Φ(τ ; q)] = λ3 (τ − 1)
[
∂2Φ(τ ; q)
∂τ2
]2
− Φ3(τ ; q) (9.14)
8 , Φ(τ ; q) ï @A τ è q: Ò* éùê .
Ø ù~ ï# à Å Ä ÒùÔ½ ê ,
H(τ) ï# à Å Ä ÒÔ éê .ð Äñò2 ³
(1 − q) L [Φ(τ ; q) − u0(τ)] = ~ H(τ) q N [Φ(τ ; q)] (9.15)
ª« åΦ(1; q) = 1, Φ(+∞; q) = 0 (9.16)
8 , q ∈ [0, 1] # àßó Ô 2ô .
Ðõ (9.11), ¾ q = 0 ö , ³ (9.15) è (9.16): ú÷
Φ(τ ; 0) = u0(τ) (9.17)
Ð ~ 6= 0 è H(τ) 6= 0, ¾ q = 1 ö , ³ (9.15) è (9.16) Âøù Hú ³ (9.6) è(9.7),
ÛûΦ(τ ; 1) = u(τ) (9.18)
üý, ¾ q
Û0 þXÿ 1, Φ(τ ; q) Û ú u0(τ)
2 ÿ³ (9.6) è (9.7): ú u(τ).
J45èõ (9.17), ÃÓ Φ(τ ; q) 45n q: ê
Φ(τ ; q) = u0(τ) +
+∞∑
k=1
uk(τ) qk (9.19)
8 uk(τ) =
1
k!
∂kΦ(τ ; q)
∂qk
∣
∣
∣
∣
q=0
(9.20)
)* , Φ(τ ; q) @A ú ÒÔ½ ê ~ èÒÔ éê H(x).ÐÑ
~ è H(x) ^_ ,Í
Ãú ê(9.19)
q = 1 ö , Jõ (9.18),
Ì
u(τ) = u0(τ) +
+∞∑
k=1
uk(τ) (9.21)
ÜÝ" ú u0(x) Éú u(x):
.
¥9 ¦ §©¨©ª - «©¬©©®©¯©° · 113 ·
9.1.3 ãäåæç÷ , Õ
un = u0(τ), u1(τ), u2(τ), · · · , un(τ)
Ó Äñò2 ³ (9.15) è (9.16) & q 9 kL
, q = 0,Ú Í
k!,Ì ñò
2 ³ ]L [uk(τ) − χkuk−1(τ)] = ~ H(τ) Rk(uk−1, τ) (9.22)
!" åuk(1) = 0, uk(+∞) = 0 (9.23)
8 , χk Ðõ (2.42) Õ , #Rk(uk−1, τ)
=
k−1∑
j=0
[
λ3 (τ − 1) u′′j (τ) u′′
k−1−j(τ) − uk−1−j(τ)
j∑
i=0
ui(τ) uj−i(τ)
]
(9.24)
B CR´R¶%$R> , uk(τ)(k > 1)!%" Æ7Ç ³ (9.22) è Æ7Ç7åRRR (9.23).
üý
,R õ (9.21),
HRI ÂR;R³%& ÷ Î Î Ó !%" ³ (9.6) è (9.7): Å7Æ7Ç7ÈÉ' 2 n ÊË $à !" ³ (9.22) è (9.23)
: ÆÇ á ÈÉ .B C´¶$> , ?( '2)*Ø+×¹º ¼ ( X ) ½ ê .Ø, u∗
k(τ) -ï³
L[u∗k(τ)] = ~ H(τ) Rk(uk−1, τ)
$# àæ ú . ./ , Jõ (9.13), ³ (9.22):0 ú÷
uk(τ) = χk uk−1(τ) + u∗k(τ) + C1 τ−3 + C2 (9.25)
8 , ê
C1 è C2 Ð å (9.23) É . ?1 , 23 H(τ) 4* , Ã @ L 9ú ñò2 ³ (9.22) è (9.23). Þàá
(9.10), H(τ) 5 Ä è Ì ÙÚ ò õH(τ) = τσ (9.26)
8 , σ ># à ± S ê . [\67 , ¾σ > 4
]98;:;<(9.19) =;>;?;@;A;B;C;D (9.15) E (9.16), F q G;H;I;J;K <;L A , z~;M;N;G;H;O;B;C;D (9.22) E (9.23), P;Q;R;S;T (9.24). —— £¤
· 114 · éëêëìîíö , UCÿ$ú ·¸ τ ln τ
»,VW Þàá (9.10). ¾
σ < 4
ö , τ−4» $ ê X ÷ Ä , YZ<= ñê[ú ÊË Ê &l\ . ? VW ]^_`a%b
.üRý
, ÷R"%c W ÞRàRá (9.10) d W %]%^%_%` a%b , e%fR^R_ σ = 4. ?%g#É"hi éê
H(τ) = τ4 (9.27)
j , k @ L 9ú ñò2 ³ (9.22) è (9.23).
9.1.4 lm àáno%p ÿ7û7ü7ý - þ7ÿR%qRR%$%r ×%s , t%e ×%u MR8 ê ú : N õ%-RORõ .
[\67 , uk(τ) k-Onuk(τ) =
2k∑
n=0
αk,n
τn+3(9.28)
8 , αk,n ÷ ê . Óv!-OõÀÔ ñò2 ³ (9.22) è (9.23), kCÿÙÚwxõ
αk,j =χkχ2k−j αk−1,j
+4~
[
χ2k+1−j
(
λ3βk,j+1 − γk,j+1
)
− χj λ3βk,j
]
j(j + 3)(9.29)
βk,i =
k−1∑
j=0
min2j,i−2∑
n=max0,i+2j−2k
(n + 3)(n + 4)(i + 1 − n)
×(i + 2 − n) αj,n αk−1−j,i−n−2 (9.30)
γk,i =
k−1∑
j=0
min2j,i−2∑
n=max0,i+2j−2k
δj,n αk−1−j,i−n−2 (9.31)
8
δj,n =
j∑
i=0
min2i,n∑
r=max0,n+2i−2j
αi,r αj−i,n−r (9.32)
y õ (9.23) tαk,0 = −
2k∑
n=1
αk,n (9.33)
¥9 ¦ §©¨©ª - «©¬©©®©¯©° · 115 ·
y õ (9.11), )* , z# ê ÷α0,0 = 1 (9.34)
üý, Jv|$wxõ ,
yα0,0 = 1, k @ L~ MUt8 ê αk,n. U , [
\Cûüý - þÿq$# N õ ê úu(x) =
+∞∑
k=0
2k∑
n=0
αk,n
(1 + λ x)n+3(9.35)
5$ mñ <=ú÷
u(x) ≈m∑
k=0
2k∑
n=0
αk,n
(1 + λ x)n+3(9.36)
y õ (9.36), tu′(0) ≈ −λ
m∑
k=0
2k∑
n=0
(n + 3)αk,n (9.37)
èu′′(0) ≈ λ2
m∑
k=0
2k∑
n=0
(n + 3)(n + 4)αk,n (9.38)
9.1.5 9.1
u0(τ) +
+∞∑
k=1
uk(τ)
, , uk(τ) (9.22) (9.23), (9.12) 3 (9.24) (2.42) , - ¡ ¢£¤ .¥ 3¦ § , et
limm→+∞
um(τ) = 0
s(τ) = u0(τ) +
+∞∑
k=1
uk(τ)
· 116 · éëêëìîí-O¦ § . ¨Jõ (9.12) 3 (9.22) © (2.42) t
~ H(τ)
+∞∑
k=1
Rk(uk−1, τ)= limm→+∞
m∑
k=1
L[uk(τ) − χkuk−1(τ)]
=L
limm→+∞
m∑
k=1
[uk(τ) − χkuk−1(τ)]
=L[
limm→+∞
um(τ)
]
=0
y ú~ 6= 0 © H(τ) = τ4, & ¹ ¶ τ > 1, võ u M
+∞∑
k=1
Rk(uk−1, τ) = 0
Óõ (9.24) ÀÔvõ , , t+∞∑
k=1
Rk(uk−1, τ)
=
+∞∑
k=1
k−1∑
j=0
[
λ3(τ − 1) u′′j (τ) u′′
k−1−j(τ) − uk−1−j(τ)
j∑
i=0
ui(τ) uj−i(τ)
]
=λ3 (τ − 1)
[
+∞∑
k=0
u′′k(τ)
]2
−[
+∞∑
k=0
uk(τ)
]3
=λ3 (τ − 1)
[
d2s(τ)
dτ2
]2
− s3(τ)
=0
y õ (9.11) © (9.23), nªs(1) = 1, s(+∞) = 0N%«
, s(τ)!%" ³ (9.6) © (9.7),
üRý, >7û7ü7ý - þ7ÿR³ (9.1) © (9.2)
: ú .¬.
9.2 ®°¯ d f 9.1, [\5ɱ § ú (9.35) .
B C´¶$> , ¦ § ·¸ hi½ § ~ © λ,
\²³´ §: TU ©Y .üý
, 5µ ¶^_ ~ © λ.
¥9 ¦ §©¨©ª - «©¬©©®©¯©° · 117 ·
ûüý - þÿ , q$· ô y
E =6
7
(
4π
3
)2/3
Z7/3 u′(0)
É , 8 , Z >Ò* ' .üý
,¸¹
u′(0) ètr × $º¶» . [\ KÏ ./~ © λ & u′(0)
: s $²³ . Ǽ N , u′(0)@A ú
~ © λ. & ¹ ¶ u $ ~,
kRÓ u′(0) ½%¾ λ $R#%%¿ § , À%Á%ÂRM 5%$ u′(0) ∼ λ ÃÅÄ ,ÛRû .R/
λ & §u′(0) s : ²³ , ÙÆ 9.1 Uï (~ = −1, ~ = −3/4 © ~ = −1/2). ÇÈ
9.1, u′(0) 5ÿ É $Ê , Ë5 ú Æ 9.1 Ì ÍÎÏÐÑÒ$ÄÓ . ÇÈÆ 9.1, Ô«, Õ 0.2 < λ < 0.3 © −1 6 ~ 6 −1/2 ö ,
§u′(0) .
üý, Ö×ØÙ
λ = 1/4
÷ÚÛÜ ~ Ý λ = 1/4 ÞË § u′(0) ßàá©âã$²³ , Ö×ÁÂä 5$ u′(0) ∼ ~ Ã Ä ( åæ 24 ç© 3.5.1 è ), éÆ 9.2 Uê . Õ λ = 1/4 Þ , u′(0)
Ýàá −2 < ~ < 0 ë .ýì
, Ö×67 , íî u′(0) ß § , 5 u(x) ß%§ %Ý%ï%%à%á 0 6 x < +∞ ëÅ% .
üRý, Õ λ = 1/4 © −2 < ~ < 0 Þ ,
§
(9.35) Ýïàá 0 6 x < +∞ ë . ðé , ~ = −1 © λ = 1/4 Þ , ñ (9.36)
ß 10 òóôõ 100 òóôö÷øù , éÆ 9.3 Uê . úÔ « -¼ , 5 §û
% . ~ = −1 © λ = 1/4 Þ , u(x) %%$ û%ü%ý%þ é%- 9.1 U%ê . Ç%È%% 9.1, e%%ÿ - û . ~ = −1 © λ = 1/4 Þ , § (9.35) k%½ - ß û Í » .
9.1 30 u′(0) ∼ λ
~ = −1 ; ~ = −3/4; ~ = −1/2
· 118 · "!"#%$
9.2 λ = 1/4 & , 30 u′(0) ∼ ~
9.3 ~ = −1 ' λ = 1/4 & , ()* - +,-./00 (9.36)1 10 2345 ;
100 2345
Kobayashi[98] 6 ä § Ê ûu′(0) = −1.588 071 (9.39)
~ = −1 © λ = 1/4 Þ , 7 (9.36) 6 ä 8 9 ¸¹ u′(0) ß ûü óô û , é : 9.2 ;
<9 = ()* - +,>?@A · 119 ·
ê . ÔB , CD%ó%ô%òEFG , HIJK . BL , u′(0) ßMN%â%ã"O u(x) ßMN%âãP%øQ , úRS%ÿ"7UT%Ý x = 0 VWXYZ[\ . ]^ É_ - `a%ó%ô ( æ 41
ç b 3.5.2 è ), ÷ c 8 9 d e u′(0) f g ûü óô û , é : 9.3 ;ê . ~ = −1 bλ = 1/4 Þ , u′′(0) ûü óô û é : 9.4 ;ê . u′′(0) ß É _ - ` aóôé : 9.5 ;ê . Ô B , 7 (9.35) 6 ä u′′(0) h T i j . ú k l D ,
É _ m ü n R o û p W XY Z q r s Z t u .
v9.1 --h= −1 w λ = 1/4 x , yz (9.36) | u(x) ~
x u(x) x u(x)
0.25 0.755 202 4.25 0.099 697 9
0.50 0.606 987 4.50 0.091 948 2
0.75 0.502 347 4.75 0.085 021 8
1.00 0.424 008 5.00 0.078 807 8
1.25 0.363 202 6.00 0.059 423 0
1.50 0.314 778 7.00 0.046 097 8
1.75 0.275 451 8.00 0.036 587 3
2.00 0.243 009 9.00 0.029 590 9
2.25 0.215 895 10.0 0.024 314 3
2.50 0.192 984 15.0 0.010 805 4
2.75 0.173 441 20.0 0.005 784 94
3.00 0.156 633 25.0 0.003 473 75
3.25 0.142 070 50.0 0.000 632 255
3.50 0.129 370 75.0 0.000 218 210
3.75 0.118 229 100 0.000 100 243
4.00 0.108 404 1 000 1.351 3×10−7
v9.2 --h= −1 w λ = 1/4 x , yz (9.37) | u′(0) Kobayashi ~
342 u′(0) (%)
10 −1.285 90 19.03
20 −1.409 32 11.26
30 −1.463 06 7.87
40 −1.492 36 6.03
50 −1.510 63 4.88
60 −1.523 09 4.09
70 −1.532 11 3.52
80 −1.538 95 3.09
90 −1.544 30 2.76
100 −1.548 60 2.49
110 −1.552 14 2.26
120 −1.555 09 2.07
· 120 · "!"#%$v
9.3 --h= −1 w λ = 1/4 x , yz (9.37) | u′(0) ~ [m, m] - Kobayashi ~[m,m] u′(0) (%)
[5, 5] −1.504 19 5.28
[10, 10] −1.546 00 2.65
[15, 15] −1.564 37 1.49
[20, 20] −1.564 74 1.47
[25, 25] −1.576 66 0.72
[30, 30] −1.558 032 0.49
[35, 35] −1.581 87 0.39
[40, 40] −1.583 01 0.32
[45, 45] −1.583 88 0.26
[50, 50] −1.584 69 0.21
[55, 55] −1.585 38 0.17
[60, 60] −1.586 05 0.13
v9.4 --h= −1 w λ = 1/4 x , yz (9.38) | u′′(0) ~
342 u′′(0)
10 3.79
20 6.41
30 8.96
40 11.49
50 14.01
60 16.52
70 19.03
80 21.54
90 24.04
100 26.55
110 29.05
120 31.56
v9.5 --h= −1 w λ = 1/4 x , yz (9.38)| u′′(0) ~ [m, m] -
[m, m] u′′(0)
[5, 5] 122.7
[15, 15] 6 087.7
[30, 30] 168 917
[40, 40] 643 063
[50, 50] 2.157 07 ×106
[60, 60] 8.783 29 ×106
Ý , Ö× x → +∞ Þ u(x) E ñ h T . Ýú ¡ ¢ , Ö× £÷Ú - Ýï ¡àá 0 6 x < +∞ ë¤W ¥ §¦¨M N E û . ;
<9 = ()* - +,>?@A · 121 ·
©, ú ¡ ÿ ª « . ¬ , Ö× W «®7¤¯ ° , - ß û Ý i j ± V E ² J . ³÷ ´ k ÿ , ø µ ^ E ³ n÷ cú ¡ ý ¶ . ú ¡ð · ¸ :®¹ , º ]» Ú û½¼ û t½u½½¾½¿½Z½À , ¸½S½Á½Â½^½Ã _½m ü ½n , ĽŽ½ ͽƽǽȽE , ÷½cr s Z t u É Á g ûü óô û .
Ý Ê W k Ë c i j ± V Ìó Z Àß Íî Z Î Ï ¢ , Liao[51] Ð u(x) : Ñ Òé¢ Óñ
u(x) =
+∞∑
n=1
an
(1 + x)n
Ô ]½^½Ã _½m ü ½n , ÷½c½½½ - ½½½½ ͽ¡ Ô ñ½½E û . Õ ÿ , Ö½½E û½×G x MN%÷ØP . ¬ ,
× Í¡ij%á ëUrsZtu , ÙÚÛÜÝÞ û Ýij±V Ìó Z À . ú RÔ ß à á â E û ß M Nâã . ³÷ ´ k ÿ , - ½ û S½Á m½ã ^½Ç½È E (9.3) b (9.10) :½Ñ , L½ä , 7æå½ç 6 ä½½½E û O½7¤è ç6 äE û MN%÷fé . ú:"¹ , êë - %ß û ô%Ð%ÿì%Í , Õÿ , í R î ï Ã Ç È E : Ñ , äÝ í×ß®ð¤R S ñÝ D òù Ç È E .
ó10 ô Volterra õ÷ö÷ø÷ù÷ú
ûüýþÿ ë¡¡E Volterra [99] É%é¢r s Z m - m
βdu(t)
dt= u(t) − u2(t) − u(t)
∫ t
0
u(x)dx (10.1)
b 8 9u(0) = α (10.2)
õ®ð , u(t) :ê à E ( i ¬ ), t :êÞ , β = c/(ab) :ê iåE , a > 0 ÿ%âe ÿ E , b > 0 :%ê ÿ E , c > 0 :%êZ ÿ E (%æ [99]∼[102]).
10.1 !#"%$%&%'10.1.1 ()*+,-.
λ > 0 :/012 ¬ · . 34τ = λ t, w(τ) = u(t) (10.3)
(10.1) Ò(
β λ2) dw(τ)
dτ= λ
[
w(τ) − w2(τ)]
− w(τ)
∫ τ
0
w(x)dx (10.4)
É 8 9w(0) = α (10.5)
Small[100] 56 , Ö t u78 Ü9:; ò â< , C å 5 E ² J . ; © , R ^= ¢> Æ Ç È Eexp(−nτ) | n > 1 (10.6)
: Ñ w(τ), ºw(τ) =
+∞∑
n=1
an exp(−nτ) (10.7)
<10 = Volterra ?A@AB@A · 123 ·
C ð , an ÿ E . D áEF Ö t u7GHI . 7JGHI (10.7), K ^L (10.5), RMN8 9OP8
w0(τ) = α exp(−τ) + γ [exp(−τ) − exp(−2τ)] (10.8)
C ð , γ Q > ¡ ¼ R E . STGHI (10.7), 7¤ (10.4), RMNLf =
d f
dτ+ f (10.9)
3 VUW s ZX · , ÖX · p W Z ÀL[e−τ ] = 0 (10.10)
7¤ (10.4), Y= ¢ r s Z m - m X ·N [Φ(τ ; q),Λ(q)] =βΛ2(q)
∂Φ(τ ; q)
∂τ− Λ(q)
[
Φ(τ ; q) − Φ2(τ ; q)]
+Φ(τ ; q)
∫ τ
0
Φ(x; q)dx (10.11)
C ð , q ∈ [0, 1] Q > ¡Z[ , Φ(τ ; q) Q > ¡\] T τ b q È E , Λ(q) Q > ¡\] T q È E^ . ï_ . ~ 6= 0 :/ > ¡ r VUWR E , H(τ) :/ > ¡ r VUWÈ E . `a b Óc
(1 − q) L [Φ(τ ; q) − w0(τ)] = q ~ H(τ) N [Φ(τ ; q),Λ(q)] (10.12)
É 8 9Φ(0; q) = α (10.13)
C ð , q ∈ [0, 1] Q > ¡VZ[ .dq = 0 0 , Òe
Φ(τ ; 0) = w0(τ) (10.14)
dq = 1 0 , ¬ ~ 6= 0 b H(τ) 6= 0, ; © , b Óc (10.12) b (10.13)
m ãfà Tc (10.4) b (10.5), g L
Φ(τ ; 1) = w(τ), Λ(1) = λ (10.15)
¬ ,d
q g 0 F G c 1, Φ(τ ; q) g 8 9OP8 w0(τ) cc (10.4) b (10.5)
78 w(τ), Ãh , Λ(q) ¸g 8 9OP ³Λ(0) = λ0 (10.16)
^jilk Λ(q) mlnlolplq λ(q), rlsltlu , vlm ilwlx . —— ylt
· 124 · "!"#%$z½cz0zz1z2½¬½· λ. ³z½´½k½zQ , zb½Ózzcz| (10.12) z~zUzWzR½E ~ bzUW È E H(τ), 8 9OP8 w0(τ) ~UWR E γ. íMN ª ,
© ; T b Óc| (10.12) b (10.13) 78 Φ(τ ; q) ¦ Λ(q) ¡ q ∈ [0, 1] ¤ñ , ä
wn(τ) =1
n!
∂nΦ(τ ; q)
∂qn
∣
∣
∣
∣
q=0
(10.17)
λn =1
n!
∂nΛ(q)
∂qn
∣
∣
∣
∣
q=0
(10.18)
×n > 1 ñ . , ST « , K ^L (10.14) b (10.16), R Φ(τ ; q)
b Λ(q) Ò= ¢ EΦ(τ ; q) = w0(τ) +
+∞∑
n=1
wn(τ) qn (10.19)
Λ(q) = λ0 +
+∞∑
n=1
λn qn (10.20)
UWRE ~ ¦ γ bUWÈE H(τ) MNª ,© ;T9 ü E q = 1 0MN .
, 7JL (10.15), W E8w(τ) = w0(τ) +
+∞∑
n=1
wn(τ) (10.21)
λ = λ0 +
+∞∑
n=1
λn (10.22)
CM b8
w(τ) ≈ w0(τ) +M∑
n=1
wn(τ) (10.23)
λ ≈ λ0 +
M∑
n=1
λn (10.24)
10.1.2 )*+,- , YJ
wn = w0(τ), w1(τ), · · · , wn(τ) , λn = λ0, λ1, · · · , λn
<10 = Volterra ?A@AB@A · 125 ·
½ zb½Ózzcz| (10.12) b (10.13)× Zz[zz q oz n , z © n!, ò½å . q = 0,
c âb Óc| ^L [wn(τ) − χn wn−1(τ)] = ~ H(τ) Rn(wn−1, λn−1) (10.25)
É 8 9wn(0) = 0 (10.26)
C ð , χn 7 (2.42) Y , äRn(wn−1, λn−1)
=1
(n − 1)!
∂n−1N [Φ(τ ; q),Λ(q)]
∂qn−1
∣
∣
∣
∣
q=0
=β
n−1∑
j=0
w′n−1−j(τ)
j∑
i=0
λiλj−i −n−1∑
j=0
λj wn−1−j(τ)
+n−1∑
j=0
λn−1−j
j∑
i=0
wi(τ)wj−i(τ)
+
n−1∑
j=0
wn−1−j(τ)
∫ τ
0
wj(x)dx (10.27)
ñ¡¡ λn−1 b wn(τ). BL ,» W > ¡¢T wn(τ) £c| (10.25).
¬ , Ötuï ýþ , ¤¥F¦ > ¡Ec| ©§ λn−1. 7L (10.8) b (10.27),¨ R1(w0, λ0) =
4∑
m=1
a1,m exp(−m τ) (10.28)
C ða1,1 = (α + γ)
(
α +γ
2− λ0 − β λ2
0
)
b a1,j (j = 2, 3, 4) ÿ E . ³ ´ k£Q , UW È E H(τ) . STGHI (10.7)
bc| (10.25), UW È E Â p W= ¢ ÓLH(τ) = exp(κ τ)
C ð , κ Q > ¡ E . ©ª« ,d
κ > 1, c| (10.25) 78 wn(τ) ~ W > ¡¬ E ,
Ö i j ± V ﮯ , ¬ , ï° ªGHI (10.7).d
k 6 −2 0 , c| (10.25) 7^±l² (10.19) ³ (10.20) ´lµl¶l·l¸2l¹lºl»l¼ (10.12) ³ (10.13), ½ q ¾l¿lÀlÁlÂlrlql¸ ,ÃlÄlÅlÆlÇ ¾l¿lÈlÉ2l¹lºl»l¼ (10.25) ³ (10.26), ÊlËlÌlÍlÎ (10.27). —— ylt
· 126 · ÏÑÐÑÒÔÓ
8 wn(τ) ï~ exp(−2τ) , D ï° ª ;Õ£Ö×ØÙÚÛ . ¬ , κ Â Ö 0 Ü1.d
κ = 1 0 , ï SÝ 6 ¢ T λ0 7 Ec| , t uÞ B ï ý þ , D ï° ªGßàÚÛ . ; © , á W κ = 0 Q R S£ . D ì > § FUW È E
H(τ) = 1 (10.29)
0 , > b Óc| (10.25) £âã~ exp(−τ) . , ST Z À (10.10), w1(τ)
~ τ exp(−τ) , g L ï° ªGHI (10.7). ¬ , F ÉGHI (10.7), äå qæa1,1 = 0, º
(α + γ)(
α +γ
2− λ0 − β λ2
0
)
= 0 (10.30)
Dçè áEF > ¡¢é λ0 £ êc| ,C pë rì8
λ0 =
√
1 + 2β(γ + 2α) − 1
2β(10.31)
í å , î ¨ ï8
w1(τ) = ~
4∑
m=2
(
a1,m
m − 1
)
(
exp−τ − exp−mτ)
(10.32)
©ª« , Rn(wn−1, λn−1) ð îñ Ñ ÒRn(wn−1, λn−1) =
2(n+1)∑
m=1
an,m exp(−m τ) (10.33)
C ð , an,m ÿ ê . q æan,1 = 0 (10.34)
o8 Öc| ºðï λn−1. Dh , ð\ £ âb Óc| (10.25) ò (10.26) 78
wn(τ) = χn−1 wn−1(τ) + ~
2(n+1)∑
m=2
(
an,m
m − 1
)
(
e−τ − e−mτ)
(10.35)
10.1.3 óôHIõö÷ ï Volterra £ Í¥ø ,
ë ä¥Ý 6 8£ùL8úñûL . ©ª« , wn(τ) ðñûü
wn(τ) =
2(n+1)∑
m=1
bn,m exp(−mτ) (10.36)
ý10 þ Volterra ?A@ABAÿ · 127 ·
C, bn,m ê . [c| (10.25) ò (10.26), ð=L (n > 1)
λn−1 =
∆n,1 −n−2∑
j=0
(λj + βδj) bn−1−j,1 − βb0,1
n−2∑
i=1
λiλn−1−i
(1 + 2βλ0)b0,1(10.37)
bn,i = χnχ2n+2−ibn−1,i +~ (Πn,i + ∆n,i − χ2n+2−iΓn,i)
(1 − i), i > 2 (10.38)
bn,1 = −2(n+1)∑
i=2
bn,i (10.39)
D Πn,i =
n−1∑
j=max0,[(i+1)/2]−2
λn−1−j dj,i, 2 6 i 6 2(n + 1)
∆n,i =
n−1∑
j=0
min2(n−j),i∑
s=max1,i−2(j+1)
bn−1−j,s cj,i−s, 1 6 i 6 2(n + 1)
Γn,i =
minn−1,n−[(i+1)/2]∑
j=0
(iβδj + λj) bn−1−j,i, 1 6 i 6 2n
C
dn,m =
n∑
i=0
min2(i+1),m−1∑
j=max1,m−2(n−i+1)
bi,j bn−i,m−j , 2 6 m 6 2(n + 1)
cn,m = −bn,m
m
cn,0 =
2(n+1)∑
m=1
bn,m
m
δn =n∑
i=0
λi λn−i
1 6 m 6 2(n + 1)
9ñûL , [x] ñ/N x £ê . JL (10.8), ï=êb0,1 = α + γ, b0,2 = −γ (10.40)
· 128 · ÏÑÐÑÒÔÓ
Dzzê , Kz9z£zL , ðz\zzï ë C zê bn,j .¨ , u(t)
7 M bu(t) ≈
M∑
n=0
2(n+1)∑
m=1
bn,m exp(−m λ t) (10.41)
Cλ ≈
M−1∑
n=0
λn (10.42)
dM → +∞ 0 ,
ë ùLê8
u(t) =
+∞∑
n=0
2(n+1)∑
m=1
bn,m exp(−m λ t) (10.43)
Cλ =
+∞∑
n=0
λn (10.44)
10.1.4 10.1 !"# (10.21) $ (10.22) %& , '( , wn(τ) )*+, (10.25) $
(10.26), -./ (10.27) $ (2.42) 01 , 2345.6+, (10.4) $ (10.5) 7# .8 9 ê8 (10.21) ò (10.22) :; , ä ë
limm→+∞
wm(τ) = 0 (10.45)
, KL (10.9) < (10.25) ò (2.42),ë
~ H(τ)
+∞∑
n=1
Rn(wn−1, λn−1)
= limm→+∞
L [wm(τ)] = L[
limm→+∞
wm(τ)
]
= 0 (10.46)
Jé ~ 6= 0 ò H(τ) = 1, L (10.46) Ý 6+∞∑
n=1
Rn(wn−1, λn−1) = 0 (10.47)
L (10.27) [L (10.47), => ,ë
ý10 þ Volterra ?A@ABAÿ · 129 ·
β
(
+∞∑
n=0
λn
)2d
dτ
[
+∞∑
n=0
wn(τ)
]
=
(
+∞∑
n=0
λn
)
[
+∞∑
n=0
wn(τ)
]
−[
+∞∑
n=0
wn(τ)
]2
−[
+∞∑
n=0
wn(τ)
]
∫ τ
0
[
+∞∑
n=0
wn(x)
]
dx (10.48)
JL (10.8) ò (10.26), üe+∞∑
n=0
wn(0) = α (10.49)
L (10.48) ò (10.49) ?c| (10.4) ò (10.5) @BA ,¨ , :C;£ê (10.21) ò
(10.22) äQ Volterra DEFGHIJ . KL .
10.2 MONOPRQST U 10.1, ©á¤V XWY §Z ê (10.21) ò (10.22) :; . [\]^
£_ , `a (10.1)Xbcd e
∫ t
0
u(x)dx
µ =
∫ +∞
0
u(x)dx (10.50)
ñfghiêj , klm ëno ^p . qrs (10.3), t (10.50) rüλ µ =
∫ +∞
0
w(ξ)dξ (10.51)
10.2.1 uvwxyz|~α ò β, ë ê ~ ò γ IX , ê êJ (10.21) ò (10.22)
:;ò:; . l , γ I[ ,
ðCCCCCC ∫ +∞
0u(x)dx ∼ ~ B¡C¢C£ ~
:C;ø C (C¤
24 ¥C¦ 3.5.1§). q¨f© , ª« ö÷ α = 1/10 ¦ β = 1/5
¬. ® γ = 1 < 2 < 3 < 4 ¯ , °±²
10 ³´µI ∫ +∞
0u(x)dx ∼ ~ X¡¶· 10.1 f . [\]^ _ , γ k 4 ¸¹º
2 ¯ , ±C² ~ IC»C¼CC¾½C¿ , À γ = 1 ¯Cª bCc CÁ »C¼CC .SCT C ~
· 130 · ÂÄÃÄÅÇÆ¡ , ÈÉJù , ® α = 1/10 ¦ β = 1/5 ¯ ,
92 6 γ 6 4,
~c ±²»¼Ê ,Ë ∫ +∞
0u(x)dx I Ì :; . ©¶ , ® γ = 3 ¯ , (10.31), »
λ0 = 1.274 92
° ² ∫ +∞
0u(x)dx I Ìc ~ = −1/2 ¯:; , ¶Í 10.1 f . ÎÏ , Ð o
∫ +∞
0 u(x)dx I CÌ :C; , ±C² λ I CÌ (10.22) ÑC:C; , ¶CÍ 10.1 Cf . ÒCÓ , ÔÕ- Ö×´µ (
¤41 ¥¦ 3.5.2
§) ØùÙÚÛÜ λ ¦ ∫ +∞
0u(x)dx
Ì I:; , ¶Í 10.2 f . ÎÏ , [m, m] ³Ô Õ - Ö×´µªÝÞß ~. l , Ð o∫ +∞
0u(x)dx I Ì :; , ±² Ì (10.21) Ñ c g 0 6 t < +∞ ÊX:;à Ì
[CáCâ [100∼102], ¶C· 10.2 Cf (α = 1/10 ¦ β = 1/5).C ^ CC α ¦ β [ ,ã ØäµÚ\º Ì J .
å10.1 α = 1/10, β = 1/5, æ γ çèéêë , 10 ìíîïð ∫ +∞
0u(x)dx ∼ ~ ñóòôóõ÷ö
γ = 1 ; øóù õ÷ö γ = 2; ú õ÷ö γ = 3; ûóøóù õ÷ö γ = 4
ü10.1 γ = 3, λ0 = 1.274 92, --h= −1/2, α = 1/10, β = 1/5 ý , þÿ (10.23)
(10.24) ∫
∞
0u(x)dx λ
∫+∞
0u(x)dx λ
10 1.194 1.014
20 1.196 0.988
30 1.196 0.983
40 1.197 0.983
50 1.197 0.983
60 1.197 0.984
70 1.197 0.985
80 1.197 0.985
ý10 þ Volterra Aÿ · 131 ·
ü10.2 γ = 3, λ0 = 1.274 92, α = 1/10, β = 1/5 ý ,
∫
∞
0u(x)dx λ [m, m]
- [m,m]
∫+∞
0 u(x)dx λ
[5, 5] 1.196 0.982
[10, 10] 1.197 0.987
[15, 15] 1.197 0.986
[20, 20] 1.197 0.986
[25, 25] 1.197 0.986
[30, 30] 1.197 0.986
[35, 35] 1.197 0.986
[40, 40] 1.197 0.986
å10.2 γ =3, ~=−1/2, α=1/10 β=1/5 ë , ê [100∼102] u(t) ïíîïð ! ø ö "#$ ;
ôóõ÷ö10 %&%
; øóù õ÷ö 20 %&% ; ú õ÷ö 50 %&%10.2.2 uv('()yz|~[\]^ _ , (*(+J w0(τ) ÝÞß Ì γ. ,(- ,
A(. γ [ ,/ (0(1(2Ì, kl 1 »¼Ú(3´J .
ß(4³´µ , »λ µ ≈
∫ +∞
0
w0(x)dx (10.52)
ß³´µ , »λ µ ≈
∫ +∞
0
w0(x)dx +
∫ +∞
0
w1(x)dx (10.53)
,(- , Ø (5 4³´µ (10.52) 6(7(8(9 γ [ , :àß³´µ (10.53) ª /
· 132 · ÂÄÃÄÅÇÆ0
λ µ1 . áâ , ;
∫ +∞
0
w1(x)dx = 0 (10.54)
t (10.54)(0 ¶ Ì `a=<
24β γ λ20 + 2(6α2 + 6γ + 4α γ + γ2)λ0 − 3(4α2 + 8αγ + 3γ2) = 0 (10.55)
> J Ì `a (10.30) ¦ (10.55),
α ¦ β [ , Ø\ λ0 ¦ γ@?BA(2=CED [ .
©¶ , α = 1/10 ¦ β = 1/5 ¯ , λ0 ¦ γ I A(2(D [¨λ0 = 1.026 82, γ = 2.275 38 (10.56)
F ?GAH2ICγ [CJHJHKC¨HLHM γ = 2 ¯ ~ IC»C¼CCÄ@ γ = 3 ¦ γ = 4 ¯ »
¼CC 1HN ( ¶C· 10.1 Cf ), ±C² CÌ JC:C;C\ 1 Ü ( ¶CÍ 10.3 Cf ). ÒCÓ , ÔÕ- ÖC×C´CµCØH,CÙCÚHOHP CÌ J :C;HQ ,
¤ Í 10.4. »C^HR _ , α = 1/10 ¦β = 1/5 ¯ , SHT 5 HKHUCCªCÔ λ0 [ , ¯HVHWCHXHY λ I CÌ :C;CàC±CÔC[0.986, ¶Í 10.1∼ Í 10.4 f . ¶Ô ∫ +∞
0 u(x)dx, ¯(V(W(X(Y λ ÝÞß α ¦ β, ® ∫ +∞
0u(x)dx ½¿¯¸(Z . XÒ , ¯(V(W(X(Y λ Ø / m»([h(\U^p .
ü10.3 γ = 2.275 38, λ0 = 1.026 82, --h= −1, α = 1/10, β = 1/5 ý ,
∫
∞
0u(x)dx
λ ∫+∞
0 u(x)dx λ
10 1.195 0.997
20 1.197 0.985
30 1.197 0.985
40 1.197 0.986
50 1.197 0.986
60 1.197 0.986
70 1.197 0.986
80 1.197 0.986
ü10.4 γ = 2.275 38, λ0 = 1.026 82, α = 1/10, β = 1/5 ý ,
∫
∞
0u(x)dx λ
[m, m]
- [m, m]
∫+∞
0u(x)dx λ
[5, 5] 1.197 0.986
[10, 10] 1.197 0.986
[15, 15] 1.197 0.986
[20, 20] 1.197 0.986
[25, 25] 1.197 0.986
[30, 30] 1.197 0.986
[35, 35] 1.197 0.986
[40, 40] 1.197 0.986
]10 ^ Volterra _` · 133 ·
a t (10.30) ¦t (10.55)(0
γ ¦ λ0 I ?GA(2=C [ , Ø 1 »¼ÚJ(b >J Volterra DEFGH . ® α = 1/10, β = 1/10 < 1/5 < 1/2 < 1 ¦ 10 ¯ , u(t) I ÌJ¶· 10.3 f . g(chi Ì j ∫ +∞
0 u(x)dx, ¯(V(W(X(Y λ, :(d? α ¦ β ±²γ ¦ λ0 I ?BA(2=C [ , ¶Í 10.5 f .
å10.3 e ~ = −1, λ0 γ fg (10.30) (10.55) hikjmlnpo ê , æ α = 1/10
β = 1/10 q 1/5 q 1/2 q 1 q 10 ë , u(t) ïíîï êrr [100∼102] ð ü
10.5 α = 1/10 ý , st ∫∞
0u(x)dx λ uvwx λ0 γ kjmyzpo|
β λ0 γ∫+∞
0 u(x)dx λ
1/10 1.199 33 2.486 33 1.100 1.000
1/5 1.026 82 2.275 38 1.197 0.986
1/2 0.754 14 1.877 01 1.418 0.836
1 0.552 74 1.516 53 1.627 0.626
10 0.156 21 0.571 01 2.572 0.157
(c©(YÍÉ , Ô Õ( b`(~ ( ¡(Q( -d( `a(_»¼ .
11
Æ,
U(t) = f [U(t), U(t), U(t)] (11.1)
, t , t , f [U(t), U(t), U(t)] Æ U(t)U(t)
U(t) . , (11.1) () .
, ÆÆ.
, Æ . ω a Æ. , ω . , . , , . , a , . ,
U(0) = 0, U(0) = a (11.2)
a Æ.
11.1
11.1.1
, Æ
cos(mωt) | m = 1, 2, 3, · · · (11.3)
. τ = ωt U(t) = u(τ), (11.1)
ω2u′′(τ) = f [u(τ), ωu′(τ), ω2u′′(τ)] (11.4)
u(τ) = a, u′(τ) = 0, τ = 0 (11.5)
, ′ τ . (11.3), u(τ)
cos(mτ) | m = 1, 2, 3, · · · (11.6)
11 Æ · 135 ·
,
u(τ) =+∞∑k=1
ck cos(kτ) (11.7)
, ck . .
ω0 ω . , (11.7), (11.5),
u0(τ) = a cos τ (11.8)
u(τ) , , a . (11.7),
L[Φ(τ ; q)] = ω20
[∂2Φ(τ ; q)∂τ2
+ Φ(τ ; q)]
(11.9)
,
L (C1 sin τ + C2 cos τ) = 0 (11.10)
(11.4),
N [Φ(τ ; q),Ω(q)] =Ω2(q)∂2Φ(τ ; q)∂τ2
−f[Φ(τ ; q),Ω(q)
∂Φ(τ ; q)∂τ
,Ω2(q)∂2Φ(τ ; q)∂τ2
](11.11)
, Φ(τ ; q) τ q , Ω(q) q . ,
H(τ) .
(1 − q) L [Φ(τ ; q) − u0(τ)] = q H(τ) N [Φ(τ ; q),Ω(q)] (11.12)
Φ(0; q) = a,
∂Φ(τ ; q)∂τ
∣∣∣∣τ=0
= 0 (11.13)
q = 0 , (11.12) (11.13)
Φ(τ ; 0) = u0(τ), Ω(0) = ω0 (11.14)
q = 1 , = 0 H(τ) = 0, (11.12) (11.13) (11.4)
(11.5), Φ(τ ; 1) = u(τ), Ω(1) = ω (11.15)
, q 0 1 , Φ(τ ; q) u0(τ) = a cos τ u(τ), , Ω(q) ω0 ω.
· 136 ·
(11.14) , Φ(τ ; q) Ω(q) q
Φ(τ ; q) = u0(τ) ++∞∑m=1
um(τ) qm (11.16)
Ω(q) = ω0 ++∞∑m=1
ωm qm (11.17)
um(τ) =1m!
∂mΦ(τ ; q)∂qm
∣∣∣∣q=0
, ωm =1m!
∂mΩ(q)∂qm
∣∣∣∣q=0
(11.18)
Æ, (11.12) H(τ). ,
Φ(τ ; q) Ω(q) . H(τ) , Æ q = 1 . , (11.15),
u(τ) = u0(τ) ++∞∑m=1
um(τ) (11.19)
ω = ω0 ++∞∑m=1
ωm (11.20)
11.1.2
,
un = u0(τ), u1(τ), · · · , un(τ) , ωn = ω0, ω1, · · · , ωn
(11.12) (11.13) q m , q = 0, m!,
L [um(τ) − χmum−1(τ)]= H(τ) Rm(um−1,ωm−1) (11.21)
um(0) = u′m(0) = 0 (11.22)
, χm (2.42) ,
Rm(um−1,ωm−1) =1
(m− 1)!dm−1N [Φ(τ ; q),Ω(q)]
dqm−1
∣∣∣∣q=0
(11.23)
(11.16) Æ (11.17) (11.12) Æ (11.13), q , !!"# (11.21) Æ (11.22), !" (11.23). —— !
11 Æ · 137 ·
Æ, um(τ) ωm−1. # um(τ) (11.21) (11.22). , $$, "#%, ωm−1. (11.7), Æ, Rm(um−1,ωm−1)
Rm(um−1,ωm−1) =ϕ(m)∑n=0
bm,n(ωm−1) cos[(2n+ 1)τ ] (11.24)
, bm,n(ωm−1) ωm−1 Æ, ϕ(m) m (11.1)
. (11.7), H(τ) $
H(τ) = cos(2κτ), κ = 0, 1, 2, 3, · · ·
, κ = 0, $%
H(τ) = 1 (11.25)
L (11.10), "
Rm(um−1,ωm−1)
cos τ %, (11.21) Æ& % τ cos τ . &(11.7). , &#$ (11.24) bm,0 . ''%
bm,0(ωm−1) = 0 (11.26)
! ωm−1. ω0( m = 1 ) Æ, #%Æ. , !' (11.21)
um(τ)=χmum−1(τ) +
ω20
ϕ(m)∑n=2
bm,n(ωm−1)(1 − n2)
cos(nτ)
+C1 sin τ + C2 cos τ (11.27)
, C1 C2 . (11.22), C1 = 0. a,
um(0) − um(π) = 0, m = 1, 2, 3, · · · (11.28)
C2 (11.28) . %, ωm−1 um(τ). M "
u(τ) ≈M∑
m=0
um(τ) (11.29)
ω ≈M∑
m=0
ωm (11.30)
· 138 ·
, ÆF[U(t), U(t), U (t), signU(t), signU(t), signU(t)
]= 0 (11.31)
(Æ,
signx =
1, x > 0
−1, x < 0(11.32)
τ = ω t U(t) = u(τ), (11.31) F[u(τ), ωu′(τ), ω2u′′(τ), signu, signu′, signu′′
]= 0 (11.33)
a (a > 0) , u0(τ) = a cos τ Æ. Æ,
signu = signu0 = sign(cos τ) (11.34)
)signu′ = −sign(sin τ), signu′′ = −sign(cos τ) (11.35)
, (11.33)
F[u(τ), ωu′(τ), ω2u′′(τ), sign(cos τ),−sign(sin τ),−sign(cos τ)
]= 0
sign(cos τ) =4π
+∞∑k=0
(−1)k
2k + 1cos[(2k + 1)τ ] (11.36)
sign(sin τ) =4π
+∞∑k=0
12k + 1
sin[(2k + 1)τ ] (11.37)
f [u(τ), ωu′(τ), ω2u′′(τ)]
=F[u(τ), ωu′(τ), ω2u′′(τ), sign(cos τ),−sign(sin τ),−sign(cos τ)
], %, Æ (11.31).
Æ|x| = x signx
, (11.31) G[U(t), U(t), U(t), |U(t)|, |U(t)|, |U (t)|
]= 0 (11.38)
, G U(t)U (t)U (t)|U(t)||U (t)| |U(t)| Æ.
11 Æ · 139 ·
11.2 (
11.2.1 1
Æ
U(t) + U(t) = ε U(t) U2(t) (11.39)
τ = ωt U(t) = u(τ), *
ω2u′′(τ) + u(τ) = ε ω2 u(τ)u′2(τ) (11.40)
) 11.1 #''Æ. (11.23) (11.39),
Rm(um−1,ωm−1)
=m−1∑n=0
⎛⎝ n∑
j=0
ωjωn−j
⎞⎠ u′′m−1−n + um−1
−εm−1∑n=0
(n∑
i=0
un−i
i∑r=0
ωrωi−r
)⎛⎝m−1−n∑j=0
u′ju′m−1−n−j
⎞⎠ (11.41)
m = 1 , (11.26), %
a− aω20 − 1
4a3εω2
0 = 0 (11.42)
ω0 =
1√1 +
14εa2
(11.43)
ω Æ*"
ω ≈ ω0 + (εa2) [2 + (εa2 − 2)ω2
0 ]32(4 + εa2)ω0
ω≈ω0 + (εa2) [2 + (εa2 − 2)ω2
0 ]16(4 + εa2)ω0
+
2(εa2)6 144(4 + εa2)2ω3
0
[39ω4
0(εa2)3 + 4ω2
0(43ω20 + 17)(εa2)2
+4(97ω40 + 98ω2
0 − 3)(εa2) − 192 (9ω40 − 10ω2
0 + 1)]
· 140 ·
+" . = −1 , &, 0 εa2 < 5 , ! 11.1 . Æ, Æ -', &,-,
! 11.1 . , $ εa2 , $" εa2 Æ$.
= −ω20 = −(1 + εa2/4)−1 , &, 0 εa2 < +∞ ,
! 11.1 .
" 11.1 %(& ω .! 1 # (11.9) +/,$#%-('$()&)*0.*+),; /12* = −(1 + εa2/4)−1 *3-%, (11.44); +2* = −(1 + εa2/4)−1 *0-%, (11.45); ..2* = −1/2 *"-%,; +12* = −1/5 * 6 -%,; &..2* = −1/10
*"-%,
= −(1 + εa2/4)−1
"
ω ≈ 256 + 128εa2 + 13(εa2)2
8(4 + εa2)5/2(11.44)
*"
ω ≈ 393 216 + 393 216εa2 + 142 848(εa2)2 + 21 248(εa2)3 + 1 181(εa2)4
768(4 + εa2)9/2(11.45)
" /1 &,
0 εa2 < +∞
', ! 11.1 . #($, , 2#&,)Æ*0.
11 Æ · 141 ·
11.2.2 2
U(t) + U(t) + ε U3(t) = 0 (11.46)
ω =π√
1 + εa2/22K(µ)
(11.47)
, K(µ) 3+41 ,
µ = − εa2
2 + εa2
τ = ωt U(t) = u(τ), (11.46) ω2u′′(τ) + u(τ) + ε u3(τ) = 0 (11.48)
)% 11.1 #''. (11.23) (11.46),
Rm =m−1∑n=0
⎛⎝ n∑
j=0
ωjωn−j
⎞⎠ u′′m−1−n + um−1
+εm−1∑n=0
⎛⎝ n∑
j=0
ujun−j
⎞⎠um−1−n (11.49)
m = 1 , (11.26), %
a+34εa3 − aω2
0 = 0 (11.50)
ω0 =
√1 +
34εa2 (11.51)
ω Æ*"
ω ≈ ω0 +(εa2)128ω3
0
[2(1 − ω2
0) + 3 εa2]
(11.52)
ω≈ω0 +(εa2)
32 768ω70
1 024(ω4
0 − ω60) + 1 536ω4
0(εa2)
−[(576ω6
0 − 640ω40 + 64ω2
0) − (940ω40 − 168ω2
0 − 4)(εa2)
+ (84ω20 + 12)(εa2)2 + 9(εa2)3
](11.53)
· 142 ·
Æ, . −1 < 0 , &, 0 εa2 < +∞ . = −1, "
ω ≈ 256 + 384εa2 + 141ε2a4
32(4 + 3εa2)3/2(11.54)
*"
ω ≈ 131 072 + 393 216εa2 + 440 832ε2a4 + 218 880ε3a6 + 40 599ε4a8
1 024(4 + 3εa2)7/2(11.55)
&, 0 εa2 < +∞ , *" Æ,, 0.09%
0.07%, '! , ! 11.2 .
" 11.2 %(& ω .! 2 #4 = −1 ---,$#%-('$()&)*0.*+),; +2*5 (11.9) 56"6'.(7."3-%, (11.54); 12*5 (11.63) 56"6'27.(7."3-%, (11.64); ..2*5 (11.63) 56"6'27.(7."0-%,
(11.65)
11.2.3 3
ÆU(t) + U(t) + εU(t)|U(t)| = 0 (11.56)
τ = ωt U(t) = u(τ), *ω2u′′(τ) + u(τ) + εu2(τ) sign[u(τ)] = 0 (11.57)
ω2u′′(τ) + u(τ) + εu2(τ) sign[cos τ ] = 0 (11.58)
11 Æ · 143 ·
)% 11.1 #''. (11.23) (11.46),
Rm =m−1∑n=0
⎛⎝ n∑
j=0
ωjωn−j
⎞⎠u′′m−1−n(τ) + um−1(τ)
+ε sign(cos τ)m−1∑n=0
un(τ)um−1−n(τ) (11.59)
m = 1 , (11.26), %
a+8εa2
3π− aω2
0 = 0 (11.60)
ω0 =
√1 +
8εa3π
(11.61)
Æ, . Æ.2" ,8/, −2 < 0 , . = −1 , "
ω ≈√
1 +8εa3π
− 20, 1 789, 3 901, 1 695406, 4 428, 1 993, 5 152
( εaπ
)2(
1 +8εa3π
)−3/2
(11.62)
/1 &, 0 ε a < +∞ ', ! 11.3 .
" 11.3 %(& ω .! 3 #4 = −1 ---,$#%-('$()&)*0.*+),; +2*5 (11.9) 56"6'27.(7."3-%, (11.62); 12*5 (11.63) 56"6'27.( L 7."3-%,; ..2*5 (11.63) 56"6'27.(7."0-%,
· 144 ·
11.3 /&08)'*
., = −1 $(Æ/. # 2 # 3 , = −1
, &, 0 < εa2 < +∞ 0 εa < +∞ ÆÆ. # 2 # 3 , (11.9) L, = −1, +Æ" , ! 11.2 11.3 . , # 1 , = −1 , ÆÆ&, 0 εa2 < 5 . , = −(1+ εa2/4)−1, 2#&,, &, 0 εa2 < +∞ , ! 11.1 .
Æ, (11.9) Æ L ω20 %. "
L[Φ(τ ; q)] =∂2Φ(τ ; q)∂τ2
+ Φ(τ ; q) (11.63)
0% (11.9), # 2 Æ"
ω ≈ ω0 +(εa2)128ω0
[2(1 − ω2
0) + 3 εa2]
(11.64)
*"
ω≈ω0 +(εa2)
32 768ω50
1 024(ω4
0 − ω60) + 1 536ω4
0(εa2)
− ω20
[(576ω6
0 − 640ω40 + 64ω2
0) − (940ω40 − 168ω2
0 − 4)(εa2)
+ (84ω20 + 12)(εa2)2 + 9(εa2)3
](11.65)
, ω0 (11.51) . 9Æ, = −1 , (11.9) ''Æ (11.54) (11.55) 0, " !Æ&,Æ,
! 11.2 . , &
= −ω−20 = −
(1 +
34εa2
)−1
2#&,, &, 0 εa2 < +∞ . 1 , (11.54) (11.55) Æ/1.
), = −1 , (11.9) ''Æ0, # 3
!Æ&,, ! 11.3 . %, &
= −(
1 +8εa3π
)−1
&, 0 εa < +∞ ÆÆ" . 1 , (11.62) Æ/1.
11 Æ · 145 ·
, (11.63) Æ L, # 1 ω Æ"
ω ≈ ω0 + ω0 (εa2) [2 + (εa2 − 2)ω2
0 ]32(4 + εa2)
*"
ω≈ω0 + ω0 (εa2) [2 + (εa2 − 2)ω2
0 ]16(4 + εa2)
+
2ω0(εa2)6 144(4 + εa2)2
[39ω4
0(εa2)3 + 4ω2
0(43ω20 + 17)(εa2)2
+4(97ω40 + 98ω2
0 − 3)(εa2) − 192 (9ω40 − 10ω2
0 + 1)]
= −1 , (11.44) (11.45).
+#$, ', , ),&,)Æ*0. ( 11"-#Æ .
12 .
*:Æ,
U(t) = f [U(t), U(t), U(t)] (12.1)
, t , t , f [U(t), U(t), U(t)] Æ U(t)U(t)
U(t) . 3, Æ/ . ω a
Æ. 2%2;
δ =1T
∫ T
0
U(t)dt (12.2)
, T = 2π/ω . *:Æ, δ 3. Æ *:Æ0#, . , δ ω %$Æ. , a ,
2 , ω δ % a. ,
U(0) = 0, U(0) = a+ δ (12.3)
a Æ.
, " (12.1) () .
12.1
12.1.1
, Æ
cos(mωt) | m = 0, 1, 2, 3, · · · (12.4)
,
U(t) = δ ++∞∑m=1
cm cos(mωt) (12.5)
, cm .
τ = ωt, U(t) = δ + u(τ) (12.6)
12 Æ 93< · 147 ·
(12.2) (12.3) ω2u′′(τ) = f [δ + u(τ), ωu′(τ), ω2u′′(τ)] (12.7)
u(0) = a, u′(0) = 0 (12.8)
, ′ τ . , u(τ) :
cos(mτ) | m = 1, 2, 3, · · · (12.9)
,
u(τ) =+∞∑m=1
cm cos(mτ) (12.10)
*:.
Æ, ω 2%2; δ %. ω0δ0 ω δ
Æ. (12.10) (12.8),
u0(τ) = a cos τ (12.11)
u(τ) , , a . 4, (12.10) (12.7),
L[Φ(τ ; q)] = ω20
[∂2Φ(τ ; q)∂τ2
+ Φ(τ ; q)]
(12.12)
,
L (C1 sin τ + C2 cos τ) = 0 (12.13)
, q 34, Φ(τ ; q) τ q , C1 C2 . (12.7),
N [Φ(τ ; q),Ω(q),∆(q)]
=Ω2(q)∂2Φ(τ ; q)∂τ2
−f[∆(q) + Φ(τ ; q),Ω(q)
∂Φ(τ ; q)∂τ
,Ω2(q)∂2Φ(τ ; q)∂τ2
](12.14)
, Ω(q) ∆(q) 34 q ∈ [0, 1] , $ ω 2%2; δ.
( 1Æ*= Φ(τ ; q)Ω(q) ∆(q)434 q
0 1 , Φ(τ ; q) u0(τ) *= u(τ), ), Ω(q)
· 148 ·
ω0 *= ω, ∆(q) δ0 *=2%2;δ. )Æ*=, /;<Æ( (4 3.6 #)
H[Φ(τ ; q),Ω(q),∆(q), H(τ), H2(τ), , 2, q]
= (1 − q) L [Φ(τ ; q) − u0(τ)] − q H(τ) N [Φ(τ ; q),Ω(q),∆(q)]
−2 H2(τ) (1 − q)(f [∆(q), 0, 0]− f [δ0, 0, 0]) +
[Ω2(q) − ω2
0
]u′′0(τ)
, q ∈ [0, 1] 34, 2 , H(τ) H2(τ) .
H[Φ(τ ; q),Ω(q),∆(q), H(τ), H2(τ), , 2, q] = 0
(1 − q) L [Φ(τ ; q) − u0(τ)]
= q H(τ) N [Φ(τ ; q),Ω(q),∆(q)]
+2 H2(τ) (1 − q) (f [∆(q), 0, 0]− f [δ0, 0, 0])
+2 H2(τ) (1 − q)[Ω2(q) − ω2
0
]u′′0(τ) (12.15)
Φ(0; q) = a,
∂Φ(τ ; q)∂τ
∣∣∣∣τ=0
= 0 (12.16)
q = 0 , (12.11) (12.15), '
Φ(τ ; 0) = u0(τ), Ω(0) = ω0, ∆(0) = δ0 (12.17)
q = 1 , = 0 H(τ) = 0, (12.15) (12.16) *(12.7) (12.8),
Φ(τ ; 1) = u(τ), Ω(1) = ω, ∆(1) = δ (12.18)
, q 0 1 , Φ(τ ; q) u0(τ) = a cos τ u(τ), %, Ω(q) ω0 ω, ∆(q) δ0 2%2; δ.
Æ, (12.15) 2, 5 H(τ)
H2(τ). =, 1 (12.15) (12.16) Φ(τ ; q)Ω(q)
∆(q) q ∈ [0, 1] %,
u[m]0 (τ) =
∂mΦ(τ ; q)∂qm
∣∣∣∣q=0
, ω[m]0 =
∂mΩ(q)∂qm
∣∣∣∣q=0
, δ[m]0 =
∂m∆(q)∂qm
∣∣∣∣q=0
12 Æ 93< · 149 ·
m 1 %. , (12.17), Φ(τ ; q)Ω(q) ∆(q) q
Φ(τ ; q) = u0(τ) ++∞∑m=1
um(τ) qm (12.19)
Ω(q) = ω0 ++∞∑m=1
ωm qm (12.20)
∆(q) = δ0 ++∞∑m=1
δm qm (12.21)
um(τ) =u
[m]0 (τ)m!
, ωm =ω
[m]0
m!, δm =
δ[m]0
m!(12.22)
2 H(τ) H2(τ) , 1 q = 1
. (12.18),
u(τ) = u0(τ) ++∞∑m=1
um(τ) (12.23)
ω = ω0 ++∞∑m=1
ωm (12.24)
δ = δ0 ++∞∑m=1
δm (12.25)
12.1.2
,
un = u0(τ), u1(τ), · · · , un(τ) , ωn = ω0, ω1, · · · , ωn
δn = δ0, δ1, · · · , δn
(12.15) (12.16) q m , q = 0, m!,
L [um(τ) − χmum−1(τ)] = H(τ) Rm(um−1,ωm−1, δm−1)
+2 H2(τ) Sm(τ,ωm, δm) (12.26)
(12.19)5(12.20) Æ (12.21) (12.15) Æ (12.16), q , !!"# (12.26) Æ (12.27), !" (12.28)∼(12.30). —— !
· 150 ·
um(0) = u′m(0) = 0 (12.27)
, χm (2.42) ,
Rm(um−1,ωm−1, δm−1)
=1
(m− 1)!dm−1N [Φ(τ ; q),Ω(q),∆(q)]
dqm−1
∣∣∣∣q=0
(12.28)
Sm(τ,ωm, δm)
=−(
m∑i=0
ωiωm−i − χm
m−1∑i=0
ωiωm−1−i
)a cos τ
+ [Qm(δm) − χmQm−1(δm−1)] (12.29)
5
Qm(δm) =1m!
dmf [∆(q), 0, 0]dqm
∣∣∣∣q=0
(12.30)
Æ, 54um(τ)ωm−1 δm−1( 2 = 0 ), 62um(τ)ωm δm( 2 = 0 ). , um(τ) (12.26)
(12.27). , $$, "#%, ωm−1 δm−1 (2 = 0 ), 62 ωm δm( 2 = 0 ).
(12.10) (12.26), $1 H(τ) H2(τ)
H(τ) = cos(2κ1τ), H2(τ) = cos(2κ2τ)
, κ1 κ2 . , κ1 = κ2 = 0, $%
H(τ) = 1, H2(τ) = 1 (12.31)
(12.10), *:, (12.26) Æ>>%
bm,0 +ϕ(m)∑n=1
bm,n cos(nτ) (12.32)
, ϕ(m) m * (12.1) 5, n > ϕ(m) bm,n . L (12.13), " bm,1 = 0, m (12.26) um(τ)
& % τ cos τ . 4, " bm,0 = 0, um(τ) % bm,0/ω20. ,
%=&(12.10). , &#$ bm,0 bm,1 ,
bm,0 = 0, bm,1 = 0, m = 1, 2, 3, · · · (12.33)
12 Æ 93< · 151 ·
ωm−1 δm−1( 2 = 0 ), 62 ωm δm( 2 = 0 )
Æ%3. )+, $$, 664. Æ, 2 = 0 m = 1 , (12.33) 3Æ, #%5Æ. , 2 = 0 m = 1 , ω0 δ0, &3% (12.33).
, 2 = 0 ,? ω0 δ0 .7@67 2 = 0,
3" !?8'':+Æ" (4@#).
, 7' m (12.26)
um(τ) = χmum−1(τ) +ϕ(m)∑n=2
bm,n
ω20(1 − n2)
cos(nτ) + C1 sin τ + C2 cos τ (12.34)
, C1 C2 1 . (12.27), ', C1 = 0. a,
um(0) − um(π) = 0, m = 1, 2, 3, · · · (12.35)
, C2 . %, um(τ) (m = 1, 2, 3, · · ·) ωm−1δm−1(2 =
0 ) 62 ωmδm (2 = 0 ). M "
u(τ) ≈M∑
m=0
um(τ) (12.36)
ω ≈M∑
m=0
ωm (12.37)
δ ≈M∑
m=0
δm (12.38)
12.2 (
12.2.1 1
U(t) + U(t) + γ U2(t) = 0 (12.39)
, γ . τ = ωt U(t) = δ + u(τ), *
ω2u′′(τ) + δ + u(τ) + γ [δ + u(τ)]2 = 0 (12.40)
· 152 ·
)% 12.1 #''. (12.28) (12.30),
Rm =m−1∑n=0
⎛⎝ n∑
j=0
ωj ωn−j
⎞⎠u′′m−1−n(τ) + vm−1(τ)
+γm−1∑n=0
vn(τ) vm−1−n(τ) (12.41)
Qm = δm + γ
m∑n=0
δn δm−n (12.42)
vk(τ) = δk + uk(τ) (12.43)
Æ, 2. 867 2 = 0 Æ%. , (12.33), ω0 δ0 Æ3%3
a+ 2aγδ0 − aω20 = 0 (12.44)
γa2
2+ δ0 + γδ20 = 0 (12.45)
ω0 =(1 − 2a2γ2
)1/4, δ0 =
ω20 − 12γ
(12.46)
, 2 = 0 , "
ω≈ω0 − (aγ)2
12ω30
, δ ≈ δ0 (12.47)
*"
ω ≈ ω0 − (aγ)2
6ω30
(1 +
2
)+
2(aγ)4
288ω70
, δ ≈ δ0 +
2a4γ3
144ω60
(12.48)
5"
ω≈ω0 − (aγ)2
4ω30
(1 + +
2
3
)
+
2(aγ)2
1 728ω70
(18 + 41) +
3(aγ)6
3 456ω110
(12.49)
δ≈ δ0 +
2a4γ3
48ω60
(1 +
2
3
)(12.50)
12 Æ 93< · 153 ·
/1= . 'Æ a γ, 39:, 9 (4 24
A 3.5.1 #) B; CD. 8/, −2 < 0 , ω δ , #&, . , Æ , 2#&,. #, = −4/5 62 = −ω2
0 , ω 5" &, |aγ| 1/√
2 ', 0" , ! 12.1 . = −1/5 62 = −ω2
0 , γ δ 5" &, |aγ| 1/
√2 '! , ! 12.2 .
" 12.1 2 = 0 -, ! 1 &:,%(& ω .'$()&)*0.*+),; +2*3;A-%, ω = 1 − 5a2γ2/12; /12* = −4/5 *3-%, (12.47); +12* = −4/5 *<-%, (12.49); ..2* = −ω2
0 *3-%, (12.47); &..2* = −ω20 *<
-%, (12.49)
" 12.2 2 = 0 -, ! 1 &=-8E δ %(6.'$&&)*0.*+),; 12*3-%, (12.47); ..2* = −1/5 *<-%, (12.50); +2* = −ω2
0 *<-%, (12.50)
· 154 ·
|aγ| > 1/√
2 , (12.46) ''Æ ω0 δ0 ..
39+7<, ', (12.39) &, |aγ| 3/4 . , (12.46) , &, 1/
√2 |aγ| 3/4 ÆÆ/
1. &, 0 |aγ| 3/4 ÆÆ" , & &,Æ. 9Æ, 2 = 0 , )Æ. /, 8 2 = 0 Æ%. , (12.46) Æ ω0 |aγ| < 1/
√2 &''! Æ" . ;-#Æ, /Æ>
=ÆF9. ,
ω0 =(
1 − 169a2γ2
)1/4
(12.51)
&, 0 |aγ| 3/4 Æ. (12.46),
δ0 =ω2
0 − 12γ
(12.52)
&, 0 |aγ| 3/4 Æ. Æ, ω δ /4 2. , 2 = −1 :#. "
ω ≈ ω0, δ ≈ δ0 +
4γω20
(ω4
0 − 1 + 2a2γ2)
(12.53)
*"
ω≈ ω0 − 2
12ω30
(3 ω4
0 − 3 + 5 a2γ2)
δ≈ δ0 +
16γω60
(ω4
0 − 1 + 2a2γ2) [
8ω40 +
(3 ω4
0 + 1 − 2 a2γ2)]
5"
ω≈ ω0 − 2
48ω70
12ω4
0
(3ω4
0 − 3 + 5a2γ2)
+ [(21ω8
0 − 18ω40 − 3) + 4
(7ω4
0 + 3)a2γ2 − 12a4γ4
](12.54)
δ≈ δ0 +
288γω100
216 ω8
0 (ω40 − 1 + 2a2γ2)
+54 ω40[(3 ω8
0 − 2 ω40 − 1) + 4(ω4
0 + 1)a2γ2 − 4a4γ4]
+2[9(5 ω12
0 − 3 ω80 − ω4
0 − 1) + 18(3 ω80 + 2 ω4
0 + 3)a2γ2
−2(19 ω40 + 54)a4γ4 + 72 a6γ6
](12.55)
+/1= . 39:, 9 (4 24 A 3.5.1 #), B; &,CD. 8/, = −ω0 ω Æ5" = −ω0/2
12 Æ 93< · 155 ·
2%2; δ Æ5" , = &, 0 |aγ| 3/4 ', ! 12.3
! 12.4 . , 1; Æ (12.51) (12.52), 2 2, &, 0 |aγ| 3/4 ÆÆ1" .
" 12.3 2 = −1, ω0 = (1 − 16a2γ2/9)1/4 > = −ω0 -,
! 1 &:, ω %(6.'$&&)*0.*+),; 12*;A, ω = 1 − 5a2γ2/12; ..2*3-%, (12.53); +2*<-%, (12.54)
" 12.4 2 = −1, ω0 = (1 − 16a2γ2/9)1/4 > = −ω0/2 -,
! 1 &=-8E δ %(&.'$&&)*0.*+),; ..2*3-%, (12.53); +2*<-%, (12.55)
12.2.2 2
U(t) − U(t) + U4(t) = 0 (12.56)
· 156 ·
U(t) = δ + u(τ) τ = ωt, *
ω2u′′(τ) − [u(τ) + δ] + [δ + u(τ)]4 = 0 (12.57)
)% 12.1 #''. (12.28) (12.30),
Rm =m−1∑n=0
⎛⎝ n∑
j=0
ωjωn−j
⎞⎠u′′m−1−n(τ) − vm−1(τ)
+m−1∑n=0
[n∑
i=0
vi(τ)vn−i(τ)
]⎡⎣m−1−n∑j=0
vj(τ)vm−1−n−j(τ)
⎤⎦ (12.58)
Qm =−δm +m∑
n=0
(n∑
i=0
δiδn−i
)⎛⎝m−n∑
j=0
δjδm−n−j
⎞⎠ (12.59)
, vk(τ) (12.43) .
2 = 0 , (12.33), ω0 δ0 %3
a− 3a3δ0 − 4aδ30 + aω20 = 0 (12.60)
38a4 − δ0 + 3a2δ20 + δ40 = 0 (12.61)
ω0 =√
4δ30 + 3a2δ0 − 1 (12.62)
δ0 =12
(õ1 +
√2√µ1
− µ1 − 6a2
)(12.63)
µ1 =−2a2 +3a4
µ0+µ0
2(12.64)
µ0 =(
4 − 4a6 + 2√
4 − 8a6 − 50a12)1/3
(12.65)
12 Æ 93< · 157 ·
"
ω≈ω0 +a2
(4δ30 + 6a2δ0 − 1)ω30
[27160
a4 +(
116
− 920a6
)δ0
+34a2δ20 − 9
5a4δ30 +
52δ40 − 15
2a2δ50 − 11δ70
+(
116δ0 − 3
8a2δ20 − 1
4δ40
)ω2
0
](12.66)
δ≈ δ0 +a4δ0
(4δ30 + 6a2δ0 − 1)ω20
(38a2 +
94δ20
)(12.67)
/, ω0, δ0 (12.62) (12.63) ''. ), 39:,$Æ 9 (424 A 3.5.1 #), B; &,CD. 8/, −2 < < 0 , ω δ
. ?1 = −1 ω Æ" = −3/4 2%2; δ Æ" , = '0 , ! 12.5 ! 12.6 . , .#@# 2 2 = 0 Æ%.
87, ?#($, $ 3.6#Æ;<Æ; Æ" . ?@# ( 1Æ1@@2.
" 12.5 2 = 0 > = −1 -, ! 2 &:,%(&.GA'$&&)*0.*+),; 12*3;A-%, ω =
√3(1− 7a2/6); ..2*9A:;, (12.62); +2*3-%,
(12.66)
· 158 ·
" 12.6 2 = 0 > = −3/4 -, ! 2 &=-8E δ %(&.GA'$&&)*0.*+),; ..2*9A:;, (12.63); +2*3-%, (12.67)
13
Liao[39] $( 1, B ;2B<C,
u(t) = f(u, u, u) (13.1)
/, t , · t , f(u, u, u) Æ uu u . , Æ () . 87, $( 1;2B<C.
#, *;Æ2, (4 Kahn[103])
x+ x = ε x(1 − x2 w) (13.2)
w = −ε (w2 − µ x4) (13.3)
, · t , µ ε , x w . , 2B<C. T α = max[x(t)] B<CÆ x(t) Æ. , t = 0,
x(0) = α, x(0) = 0 (13.4)
δ =1T
∫ T
0
w(t)dt (13.5)
ω = T/2π
B<C x(t) . τ = ω t, x(t) = α u(τ), w(t) = δ + v(τ) (13.6)
(13.2) (13.3) ω2 u′′ + u = ε ω u′ (1 − α2δ u2 − α2 u2 v) (13.7)
ω v′ = −ε (δ2 + 2δ v + v2 − µ α4 u4) (13.8)
u(0) = 1, u′(0) = 0 (13.9)
· 160 ·
, ′ τ . 34, (13.5) (13.6), ∫ 2π
0
v(τ)dτ = 0 (13.10)
(13.10) v(τ) Æ<,. Æ αδ ω %.
13.1
13.1.1
, 2B<C . , u(τ) v(τ)
u(τ) =+∞∑n=1
[an cos(nτ) + bn sin(nτ)] (13.11)
v(τ) =+∞∑n=1
[cn cos(nτ) + dn sin(nτ)] (13.12)
, anbncn dn . (13.12) u(τ) v(τ) .
(13.11) (13.12), (13.9) (13.10),
u0(τ) = cos τ, v0(τ) = 0 (13.13)
u(τ) v(τ) . /, BC v(τ) F9, : u(τ) v(τ) ÆF9, v0(τ) = 0. α0δ0 ω0 αδ ω . (13.11) (13.12), (13.7) (13.8),
Luf =∂2f
∂τ2+ f (13.14)
Lvf =∂f
∂τ(13.15)
Lu (C1 cos τ + C2 sin τ) = 0, Lv(C3) = 0 (13.16)
, C1C2 C3 , f ,. , (13.7) (13.8),
13 Æ C=4&D>E · 161 ·
Nu [U(τ ; q), V (τ ; q), A(q),∆(q),Ω(q)]
=Ω2(q)∂2U(τ ; q)∂τ2
+ U(τ ; q)
−ε Ω(q)∂U(τ ; q)∂τ
[1 −A2(q)∆(q)U2(τ ; q) −A2(q)U2(τ ; q)V (τ ; q)
](13.17)
Nv [U(τ ; q), V (τ ; q), A(q),∆(q),Ω(q)]
=Ω(q)∂V (τ ; q)∂τ
+ε[∆2(q) + 2∆(q) V (τ ; q) + V 2(τ ; q) − µ A4(q) U4(τ ; q)
](13.18)
, q ∈ [0, 1]34, U(τ ; q) V (τ ; q) τ q ,, A(q)∆(q)
Ω(q) q ,.
u v , Hu(τ) Hv(τ) .
(1 − q)Lu [U(τ ; q) − u0(τ)]
= q u Hu(τ) Nu [U(τ ; q), V (τ ; q), A(q),∆(q),Ω(q)] (13.19)
(1 − q)Lv [V (τ ; q) − v0(τ)]
= q v Hv(τ) Nv [U(τ ; q), V (τ ; q), A(q),∆(q),Ω(q)] (13.20)
U(0; q) = 1,∂U(τ ; q)∂τ
∣∣∣∣τ=0
= 0,∫ 2π
0
V (τ ; q)dτ = 0 (13.21)
, τ 0, q ∈ [0, 1].
q = 0 , (13.13) , U(τ ; 0) = u0(τ), V (τ ; 0) = v0(τ) (13.22)
q = 1 , (13.19) 1 (13.21) (13.7) 1 (13.10), U(τ ; 1) = u(τ), V (τ ; 1) = v(τ) (13.23)
A(1) = α, ∆(1) = δ, Ω(1) = ω (13.24)
· 162 ·
, 34 q 0 1, U(τ ; q) V (τ ; q) u0(τ) v0(τ) u(τ) v(τ), %, A(q)∆(q) Ω(q) α0δ0 ω0 $Æ αδ ω.
(13.19) (13.20) uv, 5Hu(τ)Hv(τ). =, 1
un(τ)=(
1n!
)∂nU(τ ; q)∂qn
∣∣∣∣q=0
(13.25)
vn(τ)=(
1n!
)∂nV (τ ; q)∂qn
∣∣∣∣q=0
(13.26)
αn =(
1n!
)dnA(q)d qn
∣∣∣∣q=0
(13.27)
δn =(
1n!
)dn∆(q)d qn
∣∣∣∣q=0
(13.28)
ωn =(
1n!
)dnΩ(q)d qn
∣∣∣∣q=0
(13.29)
n 1 %. 56, (13.22), Æ q
U(τ ; q) = u0(τ) ++∞∑n=1
un(τ) qn (13.30)
V (τ ; q) = v0(τ) ++∞∑n=1
vn(τ) qn (13.31)
A(q) = α0 ++∞∑n=1
αn qn (13.32)
∆(q) = δ0 ++∞∑n=1
δn qn (13.33)
Ω(q) = ω0 ++∞∑n=1
ωn qn (13.34)
uvHu(τ) Hv(τ) , 1 q = 1 ,
13 Æ C=4&D>E · 163 ·
(13.23) (13.24),
u(τ) = u0(τ) ++∞∑n=1
un(τ) (13.35)
v(τ) = v0(τ) ++∞∑n=1
vn(τ) (13.36)
α = α0 ++∞∑n=1
αn (13.37)
δ = δ0 ++∞∑n=1
δn (13.38)
ω = ω0 ++∞∑n=1
ωn (13.39)
13.1.2
ÆF,
uk = u0(τ), u1(τ), · · · , uk(τ) , vk = v0(τ), v1(τ), · · · , vk(τ) (13.40)
αk = α0, α1, · · · , αk , δk = δ0, δ1, · · · , δk (13.41)
ωk = ω0, ω1, · · · , ωk (13.42)
(13.19)∼(13.21) q n , n!, q = 0,
Lu [un(τ) − χn un−1(τ)]
=u Hu(τ) Run(un−1,vn−1,αn−1, δn−1,ωn−1) (13.43)
Lv [vn(τ) − χn vn−1(τ)]
=v Hv(τ) Rvn(un−1,vn−1,αn−1, δn−1,ωn−1) (13.44)
un(0) = 0, u′n(0) = 0,∫ 2π
0
vn(τ)dτ = 0 (13.45)
(13.30)∼(13.34) (13.19)∼(13.21), q , !!"# (13.43)∼(13.45), !" (13.46)∼(13.49). —— !
· 164 ·
/, χn (2.42) ,
Run(un−1,vn−1,αn−1, δn−1,ωn−1)
=1
(n− 1)!dn−1Nu [U(τ ; q), V (τ ; q), A(q),∆(q),Ω(q)]
d qn−1
∣∣∣q=0
=n−1∑j=0
u′′n−1−j(τ)
(j∑
i=0
ωiωj−i
)+ un−1(τ) − ε Fn−1(τ)
+εn−1∑j=0
Fn−1−j(τ)j∑
i=0
[δi + vi(τ)]Wj−i(τ) (13.46)
Rvn(un−1,vn−1,αn−1, δn−1,ωn−1)
=1
(n− 1)!dn−1Nv [U(τ ; q), V (τ ; q), A(q),∆(q),Ω(q)]
d qn−1
∣∣∣q=0
=n−1∑j=0
ωj v′n−1−j(τ) + ε
n−1∑j=0
[δj δn−1−j + 2δj vn−1−j(τ)]
+εn−1∑j=0
[vj(τ) vn−1−j(τ) − µ Wj(τ) Wn−1−j(τ)] (13.47)
Fk(τ)=k∑
j=0
ωk−j u′j(τ) (13.48)
Wk(τ)=k∑
j=0
(k−j∑m=0
αm αk−j−m
)[j∑
n=0
un(τ) uj−n(τ)
](13.49)
#2Æ, (13.43) (13.44) Æ8Æ, , !'.
?4un(τ)vn(τ)αn−1δn−1 ωn−1, un(τ) vn(τ) (13.43)(13.44) (13.45). , $$, "#5%, αn−1δn−1 ωn−1. (13.11) (13.12), (13.43)
(13.44), Hu(τ) Hv(τ) !@H@. ,
Hu(τ) = Hv(τ) = 1 (13.50)
n = 1 , (13.13) %4 (13.46) (13.47),
Ru1 = a1,0 cos τ + b1,0 sin τ + b1,1 sin(3τ) (13.51)
13 Æ C=4&D>E · 165 ·
Rv
1 = c1,0 + c1,1 cos(2τ) + c1,2 cos(4τ) (13.52)
, a1,0b1,0b1,1c1,0c1,1 c1,2 τ . " a1,0 = 0 b1,0 = 0,
Lu (13.16), (13.43) u1(τ) & % τ sin τ τ cos τ , &(13.11). 4, " c1,0 = 0, Lv (13.16), (13.44) v1(τ)
& % c1,0 τ , (&(13.12). &(13.43) (13.44),
a1,0 = 0, b1,0 = 0, c1,0 = 0
! 5%
ω0 − α20 δ04
= 0, ω20 − 1 = 0, δ20 − 3α4
0µ
8= 0 (13.53)
α0 =
28√
6µ, δ0 = 4
√6µ, ω0 = 1 (13.54)
, Ru
1 = b1,1 sin(3τ)
Rv
1 = c1,1 cos(2τ) + c1,2 cos(4τ)
(13.43)(13.44) (13.45),
u1(τ) = −( ε
8
)u (3 sin τ − sin 3τ) (13.55)
v1(τ) = −(
4ε√µ
6
)v
(sin 2τ +
18
sin 4τ)
(13.56)
%, 67%3ε2(3u + 4
√176µ v
)− 48 ω1 = 0 (13.57)
(246µ3)1/8 α1 + (46)3/4 δ1 = 0 (13.58)
(126µ)3/8α1 − δ1 = 0 (13.59)
α1δ1 ω1, HÆ* (13.43)∼(13.45), u2(τ) v2(τ). %, αn−1δn−1ωn−1un(τ) vn(τ).
u(τ) v(τ) n "
u(τ) =Mu
n∑k=0
[an,k cos(2k + 1)τ + bn,k sin(2k + 1)τ ]
· 166 ·
v(τ) =Mv
n∑k=1
[cn,k cos(2kτ) + dn,k sin(2kτ)]
/, Mun Mv
n " n Æ! . , w(t) x(t) Æ<.
13.1.3 9!
9! 13.1 "#$%(13.35)∼(13.39)&', (, un(τ) ) vn(τ) *+,(13.43)∼ (13.45),-.(13.46)∼(13.49)/0(2.42)12,,345.6+,(13.7)∼(13.10) %.
7 " (13.35) (13.36) ,
limm→+∞um(τ) = 0, lim
m→+∞ vm(τ) = 0
(13.43), (2.42) (13.14),
u Hu(τ)+∞∑n=1
Run(un−1,vn−1,αn−1, δn−1,ωn−1)
=+∞∑n=1
Lu [un(τ) − χnun−1(τ)]
= limm→+∞
m∑n=1
Lu [un(τ) − χnun−1(τ)]
= limm→+∞Lu [um(τ)]
=Lu
[lim
m→+∞um(τ)]
=0
u = 0 Hu(τ) = 0, ''+∞∑n=1
Run(un−1,vn−1,αn−1, δn−1,ωn−1) = 0
%+∞∑n=1
Rvn(un−1,vn−1,αn−1, δn−1,ωn−1) = 0
13 Æ C=4&D>E · 167 ·
(13.46) (13.47) %4, 2, (13.37)∼(13.39) Æ,
(+∞∑i=0
ωi
)2d2
dτ2
⎡⎣+∞∑
j=0
uj(τ)
⎤⎦ +
+∞∑j=0
uj(τ)
= ε
(+∞∑i=0
ωi
)ddτ
⎡⎣+∞∑
j=0
uj(τ)
⎤⎦
×
⎧⎪⎨⎪⎩1 −
(+∞∑i=0
αi
)2⎛⎝+∞∑
j=0
uj
⎞⎠
2+∞∑k=0
[δk + vk(τ)]
⎫⎪⎬⎪⎭
(
+∞∑i=0
ωi
)ddτ
⎡⎣+∞∑
j=0
vj(τ)
⎤⎦
=−ε
⎧⎪⎨⎪⎩[
+∞∑k=0
δk ++∞∑k=0
vk(τ)
]2
− µ
(+∞∑i=0
αi
)4⎛⎝+∞∑
j=0
uj
⎞⎠
4⎫⎪⎬⎪⎭
(13.13) (13.45), +∞∑i=0
ui(0) = 1,+∞∑i=0
u′i(0) = 0,∫ 2π
0
[+∞∑i=0
vi(τ)
]dτ = 0
3 (13.7)∼(13.10) 0G, , (13.35)∼(13.39) *. :=.
13.2 H I
13.1, " (13.35)∼ (13.39) . Æ, + u v. ,
u = v =
, u(τ)v(τ)ωα δ " . 3, 'Æ ε
µ, 6739:, α ∼ δ ∼ ω ∼ 9 (4 24 A 3.5.1 #) B; CD. #, ε = 1/5 µ = 3 , $ 9$;$ αδ ω Æ Æ&,, ! 13.1 . !$, ε = 1/5 µ = 3 ,
" −3/2 < < 0, (13.37)∼(13.39) . #, u = v = −3/4 , ωα
· 168 ·
δ 1 0.969 681.413 99 2.070 15, 13.1 . ( -
7D" (4 41 A 3.5.2 #), <), 13.2 . 8/,
=# αδ ω , ''Æ u(τ) v(τ) , ! 13.2∼ ! 13.4 (ε = 1/5, µ = 3).
" 13.1 Hu(τ ) = Hv(τ ) = 1, ε = 1/5 > µ = 3 -, 10 A'$& ω ∼ >α ∼ > δ ∼ D
12*δ; ..2*α; +2*ω
> 13.1 --hu = --hv = −3/4, Hu(τ ) = Hv(τ ) = 1, ε = 1/5, µ = 3 8,
ω>α Æ δ 9 m :;<=m ω α δ
1 1.000 00 1.393 54 2.059 77
2 0.970 63 1.404 76 2.063 18
3 0.969 66 1.410 20 2.064 58
4 0.969 63 1.412 51 2.066 68
5 0.969 68 1.413 46 2.068 43
6 0.969 69 1.413 82 2.069 44
7 0.969 69 1.413 95 2.069 89
8 0.969 68 1.413 98 2.070 06
9 0.969 68 1.413 99 2.070 11
10 0.969 68 1.413 99 2.070 13
11 0.969 68 1.413 99 2.070 14
12 0.969 68 1.413 99 2.070 15
13 0.969 68 1.413 99 2.070 15
14 0.969 68 1.413 99 2.070 15
13 Æ C=4&D>E · 169 ·
> 13.2 --hu = --hv = −3/4, Hu(τ ) = Hv(τ ) = 1, ε = 1/5,
µ = 3 8, ω>α Æ δ 9 [m, m] :>? - @?;<=[m, m] ω α δ
[1, 1] 0.968 89 1.393 54 2.059 77
[2, 2] 0.969 77 1.414 13 2.083 45
[3, 3] 0.969 68 1.414 14 2.087 35
[4, 4] 0.969 68 1.413 99 2.070 15
[5, 5] 0.969 68 1.413 98 2.070 16
[6, 6] 0.969 68 1.413 99 2.070 15
[7, 7] 0.969 68 1.413 99 2.070 15
" 13.2 ε = 1/5 > µ = 3 -, D>E& x-x′ =8AI+2*u = v = −3/4, Hu(τ) = Hv(τ) = 1 *B-%; 0.*@,
" 13.3 ε = 1/5 > µ = 3 -, D>E& x-w =8AI+2*u = v = −3/4, Hu(τ) = Hv(τ) = 1 *B-%,; 0.*@,
· 170 ·
" 13.4 ε = 1/5 > µ = 3 -, D>E& x′-w =8AI+2*u = v = −3/4, Hu(τ) = Hv(τ) = 1 *B-%,; 0.*@,
), 'Æ ε µ, *;2B<CÆ. #, " ε , #, , "#;" . #, ε = 3/4 µ = 1/6 , αδ ω 9$, = −3/4 , ! 13.5. , :+Æ/1, "#Æ" , ! 13.6∼ ! 13.8 .
( 1AB$;2B<C [39]. 87#$, $( 1(;2B<C.
" 13.5 Hu(τ )=Hv(τ )=1, ε=3/4 > µ=1/6 -, EA'$& ω ∼ >α ∼ > δ ∼ D12*δ; ..2*α; +2*ω
JB?@ µ = 1, CAC. DD?EE. —— !
13 Æ C=4&D>E · 171 ·
" 13.6 ε = 3/4 > µ = 1/6 -, D>E& x-x′ =8AI+2*u = v = −3/4, Hu(τ) = Hv(τ) = 1 * 20 -%,; 0.*@,
" 13.7 ε = 3/4 > µ = 1/6 -, D>E& x-w =8AI+2*u = v = −3/4, Hu(τ) = Hv(τ) = 1 * 20 -%,; 0.*@,
· 172 ·
" 13.8 ε = 3/4 > µ = 1/6 -, D>E& x′-w =8AI+2*u = v = −3/4, Hu(τ) = Hv(τ) = 1 * 20 -%,; 0.*@,
14 FABCDE
BE<&2CÆ*;GD99, f ′′′(η) +
12f(η)f ′′(η) = 0 (14.1)
EJf(0) = f ′(0) = 0, f ′(+∞) = 1 (14.2)
, ′ η = y√U∞/(νx) , EF f(η) 9
f(η) = ψ/√νxU∞, /, U∞ EFKFÆ09), ν G, x y
G (L4FG [20]).
1908 :, G;MH [104](Blasius) ''
f(η) =+∞∑k=0
(−1
2
)kAkσ
k+1
(3k + 2)!η3k+2 (14.3)
, σ = f ′′(0),
A0 = A1 = 1, Ak =k−1∑r=0
(3k − 1
3r
)Ar Ak−r−1 (k 2) (14.4)
Æ σ , G;MH [104] '' η f(η) KL, 2<HGH" , σ = 0.332. 1938 :, Howarth[105] 39;+Æ σ = 0.332 06. , σ = 0.332 06, (14.3) <&, 0 η < ρ0 Æ, /, ρ ≈ 5.690, ! 14.1 . σ = f ′′(0) , , G;MH (14.3) 8B1B.
87, $( 1, 1/, ''G;MHG9H1.
14.1 N=MIIJI
14.1.1
G;MH [104], 3ηαm+β | m 0
(14.5)
· 174 ·
" 14.1 H(η) = 1 -, -- 6JK f ′′(0) &NJ& (14.26)
. Howarth[105] J6&&)*0.*@,; 12* = −1(K>OK)L..2* = −1/2L&..2* = −1/4L
+2* = −1/8
(14.1) (14.2) ,
f(η) =+∞∑k=0
ak ηαk+β (14.6)
, ak , α > 0 β 0 . G;MHG93/
(14.6), (14.2),
f0(η) =12ση2 (14.7)
f(η) , , σ = f ′′(0). (14.6), (14.1) (14.2),
L0[Φ(η; q)] =∂3Φ(η; q)∂η3
(14.8)
L0
(C0 + C1η + C2η
2)
= 0 (14.9)
, C0C1 C2 , Φ(η; q) η q ,, q ∈ [0, 1] 34. O, (14.1),
N [Φ(η; q)] =∂3Φ(η; q)∂η3
+12Φ(η; q)
∂2Φ(η; q)∂η2
(14.10)
14 Æ M?PLH@ · 175 ·
, H(η) .
(1 − q) L0 [Φ(η; q) − f0(η)] = q H(η) N [Φ(η; q)] (14.11)
EJ
Φ(0; q) = 0,∂Φ(η; q)∂η
∣∣∣∣η=0
= 0,∂2Φ(η; q)∂η2
∣∣∣∣η=0
= σ (14.12)
, q ∈ [0, 1] 34.
q = 0 , (14.7)(14.11) (14.12), '
Φ(η; 0) = f0(η) (14.13)
q = 1 , = 0 H(η) = 0, (14.11) (14.12) (14.1) (14.2),
Φ(η; 1) = f(η) (14.14)
, 34 q 0 1 , Φ(η; q) f0(η) f(η).
(14.13), Φ(η; q)
Φ(η; q) = f0(η) ++∞∑k=1
fk(η) qk (14.15)
fk(η) =1k!∂kΦ(η; q)∂qk
∣∣∣∣q=0
(14.16)
Æ, (14.11) H(η). H(η) =, 1 (14.15) q = 1 , , (14.14),
f(η) = f0(η) ++∞∑k=1
fk(η) (14.17)
m "
f(η) ≈ f0(η) +m∑
k=1
fk(η) (14.18)
· 176 ·
14.1.2
O,
fn = f0(η), f1(η), f2(η), · · · , fn(η) (14.19)
(14.11) (14.12) q k , q = 0, k!,
L0 [fk(η) − χk fk−1(η)] = H(η) Rk(fk−1) (14.20)
EJfk(0) = f ′
k(0) = f ′′k (0) = 0 (14.21)
, χk (2.42) ,
Rk(fk−1) = f ′′′k−1(η) +
12
k−1∑n=0
fn(η)f ′′k−1−n(η) (14.22)
(14.8), (14.20)
fk(η)=χkfk−1(η) +
∫∫∫H(η) Rk(fk−1) dη
+C0 + C1η + C2η2 (14.23)
, C0C1 C2 EJ (14.21) .
14.1.3 9!
9! 14.1 "#$(14.17)&', (, fk(η) *+,(14.20))(14.21), -.(14.22))(2.42)12, 35.6+,(14.1))(14.2) %.
7 (2.42) (14.20),
H(η)m∑
k=1
Rk(fk−1) = L[fm(η)]
" (14.17) , lim
m→+∞ fm(η) = 0
(14.8)
H(η)+∞∑k=1
Rk(fk−1) = limm→+∞L[fm(η)] = L
[lim
m→+∞ fm(η)]
= 0
(14.15) (14.11) Æ (14.12), q , !!"# (14.20) Æ (14.21), !" (14.22). —— !
14 Æ M?PLH@ · 177 ·
= 0 H(η) = 0, ''+∞∑k=1
Rk(fk−1) = 0
(14.22) %4, 2,
d3
dη3
[+∞∑k=0
fk(η)
]+
12
[+∞∑k=0
fk(η)
]d2
dη2
[+∞∑k=0
fk(η)
]= 0
(14.7) (14.21)
+∞∑k=0
fk(0) =+∞∑k=0
f ′k(0) = 0,
+∞∑k=0
f ′′k (0) = σ
, "
f0(η) ++∞∑k=1
fk(η)
, (14.1) (14.2) , :=.
14.1.4 FGHI
14.1, ="P2Æ H(η), (14.17)
. (14.6), H(η) $
H(η) = ηκ
, κ . κ < 0 , (14.20) (14.21) η ln η %, &(14.6). ,
κ 0 (14.24)
Æ, κ. , MI.
1. H(η) = 1 J%KL κ = 0, $%, H(η) = 1. /%, Liao[28] ''Æ m " (14.18)
f(η) ≈
m∑k=0
[(−1
2
)kAkσ
k+1
(3k + 2)!η3k+2
]µm,k
0 () (14.25)
, µm,k0 () 3 2 7 (3 21 A) Æ (2.58) . ,
f(η) = limm→+∞
m∑k=0
[(−1
2
)kAkσ
k+1
(3k + 2)!η3k+2
]µm,k
0 () (14.26)
· 178 ·
(14.26) 7IÆ, QL (14.1) (14.2) N. Æ, µm,k
0 () '/. 3 2 7 (3 21 A) , :$
µm,k0 (−1) = 1
, = −1 , (14.26) G;MH (14.3) . , G;MH (14.3) (14.26) :#. ! Liao[28] ;'Æ, (14.26) &,
ρ0 η ρ0
[2|| − 1
]1/3
(−2 < < 0) (14.27)
Æ, /, ρ0 ≈ 5.690 G;MH (14.3)B0. , −1 0, (14.26)&, η ∈ [−ρ0, ρ0] η ∈ [−ρ0,+∞), ! 14.1
. , 39J 2#), (14.26) &,.
A, (14.26) &, η ∈ [0,+∞) Æ. , G;MH [104], "#Æ η ''KL. f ′(+∞) = 1, Æ , B η = 0 :KÆH η = η0 F, 39%
m∑k=0
[(−1
2
)kAkσ
k+1
(3k + 1)!η3k+10
]µm,k
0 () = 1 (14.28)
f ′′(0) . " m :, (−1 < 0) :, η = 8 η = 9 H='')Æ σ = f ′′(0) = 0.332 06, Howarth [105] ', 14.1 .
> 14.1 KML> --h Æ η0 N8, O (14.28) PMN f ′′(0) 9;<=
m = − 1
10Æ η0 = 8 = − 1
12Æ η0 = 9
20 0.328 81 0.327 43
40 0.331 85 0.331 49
60 0.332 05 0.332 01
80 0.332 06 0.332 05
90 0.332 06 0.332 06
100 0.332 06 0.332 06
QO*Liao Shijun. A kind of approximate solution technique which does not depend upon small
parameters (II)*an application in fluid mechanics. International Journal of Non-Linear Mechanics,
1997, 32:815 ∼822 (RS Elsevier R)
2. H(η) = η J%KL κ = 1, $%, H(η) = η. , $Æ m "
f(η) ≈ σ
2η2 +
4m+2∑k=6
bm,k() ηk (14.29)
14 Æ M?PLH@ · 179 ·
, bm,k() Æ. Æ, η5 %, (14.25) . ''SÆ7IÆ. −1 0, (14.29) Æ&,, ! 14.2 . P', (14.25) , (14.29) &, 0 η < +∞ 1.
" 14.2 H(η) = η -, -- 6JK f ′′(0) &NJ& (14.29)
. Howarth[105] J6&&)*0.*@,L12* = −1(K>OK)L..2* = −1/2L&..2* = −1/4L
+2* = −1/8
3. H(η) =√η J%KL
κ = 1/2, $%, H(η) =√η. , $Æ m "
f(η) ≈ σ
2η2 +
7m+4∑k=11
cm,k() (√η )k (14.30)
, cm,k() Æ. Æ, η11/2 %, (14.25) (14.29) . ''37IÆ. −1 0, (14.30) Æ&,, ! 14.3 . P',
(14.25) (14.29) , (14.30) &, 0 η < +∞ 1.
· 180 ·
" 14.3 H(η) =√
η > I--6-, f ′′(0) NJ& (14.30)
. Howarth[105] J6&&)*0.*@,L12* = −1(K>OK)L..2* = −1/2L&..2* = −1/4;
+2* = −1/8
%, 'Æ κ 0, P', $Æ &, 0 η < +∞ 1. 4, 'Æ ( < 0), H(η) = 1 ''Æ&,.
#2Æ, (14.29) (14.30) G;MH (14.3) >, 2. Æ, (14.29) (14.30) = &, 0 η < +∞, , 0 (14.3) ; .
14.2 NQITMTJIJI
14.2.1 QRST
QL''Æ (14.26)(14.29) (14.30) &, η ∈[0,+∞) Æ, #G;MH (14.3), KB1B, σ = f ′′(0) &'. +&, , || !, "#<%N η +Æ" . , (14.1) (14.2) ÆÆ. 0#, (14.5) Æ:EFKFEJ f ′(+∞) = 1.
!Æ η , G;MH [104] (14.1) EJ f ′(+∞) = 1, ''K"
f ′(η) ≈ 1 +A
∫exp(−η2/4)dη
14 Æ M?PLH@ · 181 ·
, A 1 . , η → +∞ , f(η) → η O;P". f(η) -#ÆK". , EJD)O;'"09).
η → +∞ , f(η) → η O;P", 1η, ηn exp(−m λ η) | n 0,m 1, λ > 0 (14.31)
f(η),
f(η) = η ++∞∑m=1
+∞∑n=0
am,n ηn exp(−m λ η) (14.32)
, λ > 0 QC, am,n . (14.32) G;MHG93*/
14.2.2
(14.32), 2 (14.2),
f0(η) = η +1 − exp(−λ η)
λ(14.33)
f(η) . (14.32), (14.1) (14.2),
L[Φ(η; q)] =∂3Φ(η; q)∂η3
+ λ∂2Φ(η; q)∂η2
(14.34)
L [C0 + C1η + C2 exp(−λ η)] = 0 (14.35)
, C0C1 C2 . H(η) . 1 (14.10) Æ N ,
(1 − q) L[Φ(η; q) − f0(η)
]= q H(τ) N [Φ(η; q)] (14.36)
EJ
Φ(0; q) = 0,∂Φ(η; q)∂η
∣∣∣∣∣η=0
= 0,∂Φ(η; q)∂η
∣∣∣∣∣η=+∞
= 1 (14.37)
, q ∈ [0, 1] 34, Φ(η; q) η q ,.
14.1.1 # ,
f(η) = f0(η) ++∞∑k=1
fk(η) (14.38)
fk(η) =1k!∂kΦ(η; q)∂qk
∣∣∣∣∣q=0
(14.39)
· 182 ·
14.2.3
fn =f0(η), f1(η), f2(η), · · · , fn(η)
%, (14.36) (14.37) q k , q = 0, k!,
L[fk(η) − χkfk−1(η)
]= H(η) Rk(fk−1) (14.40)
EJfk(0) = f ′
k(0) = f ′k(+∞) = 0 (14.41)
, χk (2.42) ,
Rk(fk−1) = f ′′′k−1(η) +
12
k−1∑n=0
fn(η) f ′′k−1−n(η) (14.42)
14.2.4 RUV
G;MHG9$<N, , #''Æ. (14.32), H(η) 1
H(η) = ηm exp(−λ n η), m 0, n 0
,
H(η) = 1 (14.43)
39.2 (14.40) (14.41), 8/, fm(η)
fm(η) = bm,00 +
m+1∑n=1
exp(−nλ η)2(m+1−n)∑
k=0
bm,nk ηk (14.44)
, bm,nk . %4 (14.40) (14.41), ÆOO)
bm,00 =χmb
m−1,00 − λ−1
2m−1∑r=0
Γm,1r Π 1,1
r −m+1∑n=2
(n− 1)Γm,n0 Π n,0
0
+m+1∑n=2
2(m−n+1)∑r=1
Γm,nr
(nΠ n,0
r − Π n,0r − λ−1Π n,1
r
)
14 Æ M?PLH@ · 183 ·
bm,10 =χmb
m−1,10 + λ−1
2m−1∑r=0
Γm,1r Π 1,1
r
+m+1∑n=2
⎡⎣nΓm,n
0 Π n,00 +
2(m−n+1)∑r=1
Γm,nr
(nΠ n,0
r − λ−1Π n,1r
)⎤⎦
bm,1k =χm(1 − χk+3−2m)bm−1,1
k +2m−1∑r=k−1
Γm,1r Π 1,k
r (1 k 2m)
bm,nk =χm(1 − χk+1−2m+2n)bm−1,n
k −2(m−n+1)∑
r=k
Γm,nr Π n,k
r
(2 n m, 0 k 2m− 2n+ 2)
bm,m+10 = −Γm,m+1
0 Π m+1,00
Π 1,kr =
r! (r − k + 2)k! λr−k+3
(0 k r + 1)
Π n,kr =
r!k!(n− 1)r−k+1λr−k+3
[1 −
(1 − 1
n
)r−k+1(1 +
r − k + 1n
)]
(n 2, 0 k r)
Γm,nr =
[(1 − χr+1−2m+2n) dm−1,n
r + δm,nr
](1 n m, 0 r 2m− 2n+ 2)
/
δm,nr =
12
m−1∑k=0
minn,k+1∑j=max1,n+k−m
minr,2(k−j+1)∑i=max0,r−2(m−k−n+j)
×ck,ji bm−1−k,n−j
r−i Λm−1−k,n−jr−i
cm,kn =(n+ 1)(n+ 2)(1 − χn+1−2m+2k) bm,k
n+2
−2(kλ)(n+ 1)(1 − χn−2m+2k) bm,kn+1 + (kλ)2 bm,k
n
dm,kn =(n+ 1)(1 − χn−2m+2k) cm,k
n+1 − kλcm,kn
· 184 ·
Λi,jk =
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
0, i = j = 0, k 2
0, i > 0, j = 0, k 1
0, j > i+ 1
0, k > 2(i+ 1 − j)
1, %
(14.45)
(14.33), 1Æ5
b0,00 = −λ−1, b0,0
1 = 1, b0,10 = λ−1 (14.46)
51,ÆOO),OO' bm,nk (LPOL
4 Liao AF [29]).
(14.44) %4 (14.38),
f(η)= η + limM→+∞
⎡⎣ M∑
m=0
bm,00 +
M+1∑n=1
exp(−nλ η)⎛⎝ M∑
m=n−1
2(m−n+1)∑k=0
bm,nk ηk
⎞⎠⎤⎦
(14.47)
Æ, bm,nk Cλ. ,
ÆMI. 4, (14.47) Æ, η → +∞ f ′(η) → 1 O;P".
14.2.5 9!
9! 14.2 "#$%(14.38)&', (, fk(η) *+,(14.40))(14.41), -.(14.42))(2.42)12, 35.6+,(14.1))(14.2)%.
7 (2.42) (14.40),
H(η)m∑
k=1
Rk(fk−1) = L [fm(η)]
" (14.38) , lim
m→+∞ fm(η) = 0
, (14.34),
H(η)+∞∑k=1
Rk(fk−1) = limm→+∞ L[fm(η)] = L
[lim
m→+∞ fm(η)]
= 0
= 0 H(η) = 0, ''+∞∑k=1
Rk(fk−1) = 0
14 Æ M?PLH@ · 185 ·
(14.42) %4, 2,
d3
dη3
[+∞∑k=0
fk(η)
]+
12
[+∞∑k=0
fk(η)
]d2
dη2
[+∞∑k=0
fk(η)
]= 0
(14.33) (14.41), +∞∑k=0
fk(0) =+∞∑k=0
f ′k(0) = 0,
+∞∑k=0
fk(+∞) = 1
, =#
f0(η) ++∞∑k=1
fk(η)
, (14.1) (14.2) , :=.
14.2.6 FGHI
14.2,="P2Æ Cλ, (14.47) . , 67 λ Æ f ′′(0) . 67 = −1, 2Æ λ . :Æ λ, 0 λ 4, f ′′(0) " 1Æ, ! 14.4 . 'Æ λ (λ 4), f ′′(0) ∼ 9 (4 24 A 3.5.1 #) B; (14.47) CD. #, λ = 4 ,
f ′′(0) ∼ 9MR$Æ&, −3/2 −1/2,! 14.5. λ = 4 = −1 , (14.47) ''Æ f ′′(0)1 0.332 057, Howarth[105] f ′′(0) = 0.332 06 ', 14.2 . 8/, -, f ′′(0) -D. = −3/2 λ = 4 , 25 " ''+ f ′′(0) = 0.332 057.
$( - 7D" (4 41 A 3.5.2#) < f ′′(0) ), 14.3 . 8/, f ′′(0) [m,m] ( - 7D" .
4, f ′′(0) [m,m] ( - 7D" Æ)C λ EP, 14.4. ! 14.4,, = −1 λ 2, f ′′(0)8N. #, λ = 2 = −1
, f ′′(0) 30" −3.7×109. , λ 2 , #, λ = 16 λ = 2,
f ′′(0) ( - 7D" K1 0.332 057, 14.4 . 8/, f ′′(0) [m,m] ( - 7D" , C λ (EP.
· 186 ·
" 14.4 = −1, λ I--6-, O (14.47) -( f ′′(0) ∼ λ D12*10 -%,L..2*20 -%,L+2*30 -%,
" 14.5 λ = 4 -, O (14.47) -( f ′′(0) ∼ D12*10 -%,L..2*20 -%,L+2*30 -%,
8/, =# f ′′(0) , $Æ f(η) f ′(η) &, 0 η < +∞ 1 Howarth[105] . #, λ = 4 = −1 , f ′(η) &, 0 η < +∞ 1 Howarth[105] , 14.5 . ), ( - 7D" < (14.47) )S %, f(η) f ′(η) [m,m]
( - 7D" . ., λ = 2 = −1 , f ′′(0) 8
14 Æ M?PLH@ · 187 ·
N. , λ = 2 = −1 , ( - 7D" , &,0 η < +∞ f(η) , 14.6 . λ = 2 = −1 , ?1 f ′(η) [5,5] ( - 7D" +, ! 14.6 .
Æ, (14.47) H1Æ, &, η ∈ [0,+∞)
SÆ. , (14.1) (14.2) ÆG;MHG9Æ.
#$, ( 1, :Æ, N.
> 14.2 --h = −1 Æ λ = 4 8, OW (14.47) PMN f ′′(0) 9=X;<=-% f ′′(0)
10 0.327 756
20 0.331 851
30 0.332 040
40 0.332 055
45 0.332 057
50 0.332 057
55 0.332 057
QO*Liao Shijun. A uniformly valid analytic solution of two-dimensional viscous flow past a
semi-infinite flat plate. Journal of Fluid Mechanics. Cambridge University Press, 1999, 385: 101∼128
(TP c©1999 Cambridge University Press, RTUR)
> 14.3 --h = −1 Æ λ = 4 8, f ′′(0) 9 [m, m] :>? - @?;<[m, m] f ′′(0) F - QQ-%[4, 4] 0.344 675
[8, 8] 0.332 055
[12, 12] 0.332 056
[16, 16] 0.332 057
[20, 20] 0.332 057
[25, 25] 0.332 057
> 14.4 YL> λ N, f ′′(0) 9 [m, m] :>? - @?;<[m, m] λ = 1 λ = 2 λ = 4 λ = 5 λ = 10
[4, 4] 0.326 857 0.331 867 0.344 675 0.362 964 0.519 751
[8, 8] 0.331 808 0.331 753 0.332 055 0.332 269 0.347 726
[12, 12] 0.332 008 0.332 056 0.332 056 0.332 053 0.332 908
[16, 16] 0.332 043 0.332 057 0.332 057 0.332 057 0.332 084
[20, 20] 0.332 054 0.332 057 0.332 057 0.332 057 0.332 057
[25, 25] 0.332 057 0.332 057 0.332 057 0.332 057 0.332 057
· 188 ·
> 14.5 --h = −1 Æ λ = 4 8, OW (14.47) PMN f ′(η) 9=X;<=ZHowarth[105] [N=9U\
η 20 30 40 50 55 @,0.4 0.132 650 0.132 756 0.132 763 0.132 764 0.132 764 0.132 8
0.8 0.264 412 0.264 488 0.264 707 0.264 709 0.264 709 0.264 7
1.2 0.393 075 0.393 755 0.393 772 0.393 776 0.393 776 0.393 8
1.6 0.514 758 0.516 680 0.516 750 0.516 756 0.516 756 0.516 8
2.0 0.626 372 0.629 553 0.629 754 0.629 764 0.629 764 0.629 8
2.4 0.727 156 0.728 494 0.728 950 0.728 980 0.728 980 0.729 0
2.8 0.814 839 0.810 980 0.811 429 0.811 503 0.811 503 0.811 5
3.2 0.885 026 0.876 124 0.875 982 0.876 066 0.876 066 0.876 1
3.6 0.935 172 0.924 321 0.923 315 0.923 312 0.923 312 0.923 3
4.0 0.966 854 0.957 245 0.955 665 0.955 518 0.955 518 0.955 5
4.4 0.984 622 0.977 780 0.976 154 0.975 900 0.975 900 0.975 9
5.0 0.995 914 0.992 920 0.991 856 0.991 599 0.991 599 0.991 6
6.0 0.999 708 0.999 317 0.999 092 0.999 006 0.999 006 0.999 0
7.0 0.999 987 0.999 961 0.999 939 0.999 926 0.999 926 1.000 0
8.0 1.000 000 0.999 999 0.999 998 0.999 997 0.999 997 1.000 0
QO*Liao Shijun. A uniformly valid analytic solution of two-dimensional viscous flow past a
semi-infinite flat plate. Journal of Fluid Mechanics. Cambridge University Press, 1999, 385*101∼128
(TP c©1999 Cambridge University Press, RTUR)
> 14.6 λ = 2 Æ --h = −1 8, f ′(η) 9 [m, m] :>? - @?;<Z Howarth[105] [N=9U\
η [5, 5] [10, 10] [15, 15] [20, 20] [25, 25] @,0.4 0.133 023 0.132 814 0.132 764 0.132 764 0.132 764 0.132 8
0.8 0.264 655 0.264 688 0.264 709 0.264 709 0.264 709 0.264 7
1.2 0.393 380 0.393 774 0.393 775 0.393 776 0.393 776 0.393 8
1.6 0.516 251 0.516 751 0.516 755 0.516 757 0.516 757 0.516 8
2.0 0.629 577 0.629 759 0.629 765 0.629 766 0.629 766 0.629 8
2.4 0.729 388 0.728 968 0.728 981 0.728 982 0.728 982 0.729 0
2.8 0.812 514 0.811 489 0.811 508 0.811 509 0.811 510 0.811 5
3.2 0.877 471 0.876 059 0.876 079 0.876 081 0.876 081 0.876 1
3.6 0.924 790 0.923 298 0.923 325 0.923 329 0.923 330 0.923 3
4.0 0.956 803 0.955 468 0.955 523 0.955 518 0.955 518 0.955 5
4.4 0.976 987 0.975 798 0.975 872 0.975 871 0.975 870 0.975 9
5.0 0.992 848 0.991 460 0.991 542 0.991 542 0.991 542 0.991 6
6.0 1.000 400 0.998 920 0.998 972 0.998 974 0.998 973 0.999 0
7.0 0.999 989 0.999 920 0.999 920 0.999 922 0.999 921 1.000 0
8.0 0.999 998 0.999 995 0.999 996 0.999 997 0.999 996 1.000 0
14 Æ M?PLH@ · 189 ·
" 14.6 λ=2, =−1 -, f ′(η) & [5, 5] A-G - RR'$. Howarth[105] J6&&)*0.*@,; +2*[5, 5] F - QQ-%
15 V]^_`WaXE
BE<&2CÆ*;GD99. VQEJDÆ 67 Falkner Skan[106] 1931 :''. x 1BE<&2CEÆR2HB, y UY2CÆHB, U 09), ν G, u v 95) xy Æ . Falkner Skan[106] ;', " U !0 xκ, /, κ , , EJD
f ′′′(η) + f(η)f ′′(η) + β[1 − f ′2(η)] = 0 (15.1)
EJf(0) = f ′(0) = 0, f ′(+∞) = 1 (15.2)
β =2κκ+ 1
, η = y
√(1 + κ)U
2νx(15.3)
/, ′ η . 95) uv ''
u = Uf ′(η), v = [f(η) − (κ− 1)ηf ′(η)]
√νU
2(κ+ 1)x(15.4)
Æ, f(η) E β. κ 0 , (15.3), '
0 β 2
κ < 0 , 09) U ∝ xκ x = 0 FW, Falkner-Skan f(η) x = 0 FEÆ. β > 0 β < 0 Æ<& . 1937
:, Hartree[107] Falkner-Skan . 4, ηHartree[107] ''K"
1 − f ′(η) ≈ A exp(−η2/2) η−(2β+1) +B η2β (15.5)
, A B . EJ f ′(+∞) = 1, , η → +∞ , f ∼ η, ,
η → +∞ f → +∞.
Hartree[107] ;', β !, , $ (15.5) B = 0 Æ,
EFKF. β > 0 , N f(η), η → +∞ f ′(η) → 1
15 Æ VZJSUWVX@ · 191 ·
O;P". Hartree[107] 0 β 2 @TÆN. , β
S, (15.5) η( f) ' ∞ A B W, , f ′′(0) =''EFKFÆ. , β , EJ (15.2) N. β < 0 N, Hartree[107]
η → +∞, f ′(η) ' 1, f ′′(0) :0%EFKF, 2'' β0 β 2 @TÆ/1, , β0 = −0.198 $ f ′′(0) = 0. β0 β 2 , Hartree f ′′(0) 0, η → +∞ ,
f ′(η) → 1 Obc[R, X.Y9, (.9)9.
Stewartson[108] :$, β 0 , Falkner-Skan EJ (15.2) N. β < 0 N, Stewartson[108] Fα(η) 0% f(η),
F ′′′α (η) + Fα(η)F ′′
α (η) + β[1 − F ′2α (η)] = 0 (15.6)
EJFα(0) = F ′
α(0) = 0, F ′α(α) = 1 (15.7)
f(η) = lim
α→+∞Fα(η)
/S, Stewartson[108] 8/, &, β0 β < 0 SÆ, 4f ′′(0) < 0, I9.
Stewartson[108] :$4" β < −0.198 8,56, EJ f(0) = f ′(0) =
0 Æ Falkner-Skan , = η &,, &, f ′(η) > 1, 9)&. Hartree[107] Stewartson[108], Libby Liu [109] F, 9)&. , β < β0 '' Æ \. Æ<$, β < β0 , 'Æ f ′′(0) , (15.1) (15.2) , ?1EF.
Æ, Æ#6Æ, #6BB1Æ.
87, $( 1, ''Falkner-Skan EJD9ÆH1.
15.1
15.1.1
(15.5), " β 0, 62 β < 0 B = 0, 56, η → +∞ , f ′(η) → 1 O;P". ,
ηm exp(−n λ η) | m 0, n 0, λ > 0 (15.8)
· 192 ·
f(η),
f(η) =+∞∑m=0
+∞∑n=0
am,n ηm exp(−n λ η) (15.9)
, am,n , λ C. (15.9) Falkner-Skan EJD9.
(15.9), (15.2),
f0(η) = η − 1 − exp(−λ η)λ
+γ[1 − (1 + λ η) exp(−λ η)]
λ2(15.10)
f(η) , , γ . Æ, f ′′0 (0) = λ+ γ (15.11)
(15.9), (15.1) (15.2),
L[Φ(η; q)] =∂3Φ(η; q)∂η3
+ λ∂2Φ(η; q)∂η2
(15.12)
L [C0 + C1η + C2 exp(−λ η)] = 0 (15.13)
, C0C1 C2 , Φ(η; q) η q ,. (15.1),
N [Φ(η; q)] =∂3Φ(η; q)∂η3
+ Φ(η; q)∂2Φ(η; q)∂η2
+ β
1 −
[∂Φ(η; q)∂η
]2
(15.14)
, H(η) . (1 − q) L [Φ(η; q) − f0(η)] = q H(η) N [Φ(η; q)] (15.15)
EJ
Φ(0; q) = 0,∂Φ(η; q)∂η
∣∣∣∣η=0
= 0,∂Φ(η; q)∂η
∣∣∣∣η=+∞
= 1 (15.16)
, q ∈ [0, 1] 34.
q = 0 , Φ(η; 0) = f0(η) (15.17)
q = 1 , q = 0 H(η) = 0, (15.15) (15.16) (15.1) (15.2),
Φ(η; 1) = f(η) (15.18)
15 Æ VZJSUWVX@ · 193 ·
, 34 q 0 1 , Φ(η; q) f0(η) (15.1)
(15.2) f(η). (15.17), Φ(η; q)
Φ(η; q) = f0(η) ++∞∑k=1
fk(η) qk (15.19)
fk(η) =
1k!∂kΦ(η; q)∂qk
∣∣∣∣q=0
(15.20)
Æ, (15.15) H(η), f0(η) γ. =, 1 (15.19) q = 1 , , (15.18),
f(η) = f0(η) ++∞∑k=1
fk(η) (15.21)
(15.21) f0(η) f(η) .
15.1.2
,
fn = f0(η), f1(η), f2(η), · · · , fn(η)
(15.15) (15.16) q k , q = 0, k!,
L [fk(η) − χkfk−1(η)] = H(η) Rk(fk−1) (15.22)
EJfk(0) = f ′
k(0) = f ′k(+∞) = 0 (15.23)
, χk (2.42) ,
Rk(fk−1)= f ′′′k−1(η) +
k−1∑n=0
[fn(η)f ′′
k−1−n(η) − βf ′n(η)f ′
k−1−n(η)]
+β (1 − χk) (15.24)
(15.22) (15.23) !', : "1&ZOT.
(15.19) (15.15) Æ (15.16), q , !!"# (15.22) Æ (15.23), (15.24). —— !
· 194 ·
(15.9) (15.22), H(η) $
H(η) = ηκ1 exp(−λ κ2 η)
, κ1 0 κ2 0 . , κ1 = κ2 = 0, $
H(η) = 1 (15.25)
f∗k (η)
L[f∗k (η)] = Rk(fk−1)
Æ:. (15.13), 3
fk(η) = χk fk−1(η) + f∗k (η) + C0 + C1η + C2 exp(−λ η) (15.26)
, C0C1 C2 EJ (15.23) .
15.1.3 RUdV
39.2 (15.22) (15.23), 8/, fm(η)
fm(η) =m+1∑k=0
Ψm,k(η) exp(−kλ η), m 0 (15.27)
Ψ0,0(η) = b0,00 + b0,0
1 η (15.28)
Ψ0,1(η) = b0,10 + b0,1
1 η (15.29)
Ψm,0(η) = bm,00 , m 1 (15.30)
Ψm,k(η)=2(m+1)−k∑
n=0
bm,kn ηn, m 1, 1 k m+ 1 (15.31)
%4 (15.22) (15.23), JUK [40] bm,nk (m
1, 0 n m+ 1 0 k 2(m+ 1) − n) OO
bm,00 =χmb
m−1,00 − λ−1
2m∑r=0
Γm,1r Π 1,1
r −m+1∑n=2
(n− 1)Γm,n0 Π n,0
0
+m+1∑n=2
2(m+1)−n∑r=1
Γm,nr
(nΠ n,0
r − Π n,0r − λ−1Π n,1
r
)(15.32)
bm,01 =0 (15.33)
15 Æ VZJSUWVX@ · 195 ·
bm,10 =χmb
m−1,10 + λ−1
2m∑r=0
Γm,1r Π 1,1
r +m+1∑n=2
nΓm,n0 Π n,0
0
+m+1∑n=2
2(m+1)−n∑r=1
Γm,nr
(nΠ n,0
r − λ−1Π n,1r
)(15.34)
bm,1k =χm(1 − χk+2−2m) bm−1,1
k +2m∑
r=k−1
Γm,1r Π 1,k
r
1 k 2m+ 1 (15.35)
bm,nk =χm(1 − χk+1−2m+n) bm−1,n
k −2(m+1)−n∑
r=k
Γm,nr Π n,k
r
2 n m, 0 k 2(m+ 1) − n (15.36)
bm,m+1k =−
m+1∑r=k
Γm,m+1r Π m+1,k
r , 1 k m+ 1 (15.37)
/
Π 1,kr =
r! (r − k + 2)k! λr−k+3
, 0 k r + 1
Π n,kr =
r!k!(n− 1)r−k+1λr−k+3
[1 −
(1 − 1
n
)r−k+1(1 +
r − k + 1n
)]
n 2, 0 k r
Γm,nr =
[(1 − χr+1−2m+n) dm−1,n
r + δm,nr + ∆m,n
r
]1 n m, 0 r 2(m+ 1) − n
Γm,m+1r = (δm,m+1
r + ∆m,m+1r )
∆m,nr =−β
m−1∑k=0
minn,k+1∑j=max0,n+k−m
minr,2(k+1)−j∑i=max0,r−2(m−k)+n−j
ak,ji am−1−k,n−j
r−i
δm,nr =
m−1∑k=0
minn,k+1∑j=max1,n+k−m
minr,2(k+1)−j∑i=max0,r−2(m−k)+n−j
ck,ji
× bm−1−k,n−jr−i Λm−1−k,n−j
r−i
m 1, 0 n m+ 1, 0 r 2(m+ 1) − n
· 196 ·
am,kn =(n+ 1)bm,k
n+1Λm,kn+1 − (kλ)bm,k
n Λm,kn (15.38)
cm,kn =(n+ 1)(n+ 2) bm,k
n+2Λm,kn+2 − 2(kλ)(n+ 1)bm,k
n+1Λm,kn+1
+(kλ)2 bm,kn Λm,k
n (15.39)
dm,kn =(n+ 1) cm,k
n+1Λm,kn+1 − (kλ)cm,k
n Λm,kn (15.40)
Λi,jk =
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
0, i = j = 0, k 2
0, i > 0, j = 0, k 1
0, j > i+ 1
0, k > 2(i+ 1) − j
1, %
(15.41)
(15.10), Æ
b0,00 =
γ − λ
λ2, b0,0
1 = 1, b0,10 =
λ− γ
λ2, b0,1
1 = −γλ
(15.42)
+OO, <' bm,nk .
(15.1) (15.2) M "
f(η)≈ η +
(M∑
m=0
bm,00
)
+M+1∑n=1
exp(−nλ η)⎛⎝ M∑
m=n−1
2(m+1)−n∑k=0
bm,nk ηk
⎞⎠ (15.43)
, BE<&2C Falkner-Skan GD99Æ
f(η)=η +
(+∞∑m=0
bm,00
)
+ limM→+∞
M+1∑n=1
exp(−nλ η)⎛⎝ M∑
m=n−1
2(m+1)−n∑k=0
bm,nk ηk
⎞⎠ (15.44)
, η → +∞ , f ′ → 1 O;P".
15.1.4 9!
"#$
f0(η) ++∞∑k=1
fk(η)
15 Æ VZJSUWVX@ · 197 ·
&',(, fk(η)*+,(15.22))(15.23),-.(15.10)(15.12)(15.24))(2.42)12, (5.6+,(15.1))(15.2)%.
7 ", lim
m→+∞ fm(η) = 0
(15.22) (2.42),
H(η)m∑
k=1
Rk(fk−1) = L[fm(η)]
(15.12),
H(η)+∞∑k=1
Rk(fk−1) = limm→+∞L[fm(η)] = L
[lim
m→+∞ fm(η)]
= 0
= 0 H(η) = 0, ''+∞∑k=1
Rk(fk−1) = 0
(15.24) %4, 2,
d3
dη3
[+∞∑k=0
fk(η)
]+
[+∞∑k=0
fk(η)
]d2
dη2
[+∞∑k=0
fk(η)
]
+β
⎧⎨⎩1 −
[ddη
+∞∑k=0
fk(η)
]2⎫⎬⎭ = 0
34, (15.10) (15.23), +∞∑k=0
fk(0) =+∞∑k=0
f ′k(0) = 0,
+∞∑k=0
f ′k(+∞) = 1
, "
f0(η) ++∞∑k=1
fk(η)
, (15.1) (15.2) . :=.
15.2 H I
15.1, =" (15.21) . Æ, (15.44) 5 λ γ. , ?I5Æ
· 198 ·
. η → +∞ , C λ CD" f ′(η) → 0 Æ). (15.11), γ CD" f ′′
0 (0) , , B; f(η) f0(η) . β < 0 , >. #2Æ, ' λ γ , KGÆ , ),2# (15.44)&,).
, f ′′(0) 2C95ÆLY2, -#Æ. (15.21),
f ′′(0) = f ′′0 (0) +
+∞∑k=1
f ′′k (0) (15.45)
β 5 λ γ. 67, = −1, γ = 0 Æ%, λ . U" λ :, ' β , f ′′(0) 1Æ, ! 15.1 . λ 5, 0 β 2 , Æ f ′′(0) . , λ 5. , λ = 5 γ = 0 Æ%. , f ′′(0) ÆCD39:, f ′′(0) ∼ 9 (4 24 A 3.5.1 #) +B;, ! 15.2 . ,
−5/4 −3/4, 0 β 2 , f ′′(0) . λ = 5, γ = 0 = −1, β0 β 2 , f ′′(0) , , β0 = −0.1988. f ′′(0) 10 20 30 "
" 15.1 = −1, γ = 0 > β = 0, 1, 2 -, 20 A'$& f ′′(0) ∼ λ D
15 Æ VZJSUWVX@ · 199 ·
" 15.2 λ = 5, γ = 0 > β = 0, 1, 2 -, 20 A'$& f ′′(0) ∼ D
f ′′(0)
≈0.466 892 061 269 575 + 1.270 377 798 259 161 β
−0.936 606 137 251 929 9 β2 + 0.656 544 480 481 005 2 β3
−0.298 966 715 661 174 3 β4 + 8.714 746 301 295 173 × 10−2 β5
−1.646 263 530 984 164 × 10−2 β6 + 2.009 360 736 004 046 × 10−3 β7
−1.532 383 017 041 316 × 10−4 β8 + 6.654 735 449 025 599 × 10−6 β9
−1.259 647 041 638 817 × 10−7 β10 (15.46)
f ′′(0)
≈0.469 470 560 483 573 + 1.295 165 031 248 947 β
−1.379 744 175 063 81 β2 + 2.191 127 183 953 301 β3
−3.010 696 768 394 167 β4 + 3.217 599 178 710 972 β5
−2.637 727 245 237 923 β6 + 1.672 089 788 089 693 β7
−0.828 892 704 239 146 3 β8 + 0.324 432 161 735 041 8 β9
−0.100 961 072 953 423 9 β10 + 2.508 145 750 314 333 × 10−2 β11
· 200 ·
−4.979 128 795 292 073 × 10−3 β12 + 7.879 252 632 693 684 × 10−4 β13
−9.872 764 016 512 919 × 10−5 β14 + 9.675 273 712 687 701 × 10−6 β15
−7.265 429 168 115 138 × 10−7 β16 + 4.042 349 583 833 827 × 10−8 β17
−1.573 031 139 104 484 × 10−9 β18 + 3.831 177 670 499 221 × 10−11 β19
−4.409 794 935 993 077 × 10−13 β20 (15.47)
f ′′(0)
≈ 0.469 590 361 531 217 7 + 1.298 441 994 559 965 β
−1.491 321 283 547 855 β2 + 3.075 663 557 218 445 β3
−6.529 797 437 132 239 β4 + 12.028 309 716 995 64 β5
−18.164 359 143 708 1 β6 + 22.241 322 748 548 39 β7
−22.201 844 530 357 51 β8 + 18.256 799 864 439 15 β9
−12.504 410 952 225 7 β10 + 7.205 799 232 502 405 β11
−3.523 684 854 407 225 β12 + 1.472 301 275 851 469 β13
−0.528 386 076 949 657 2 β14 + 0.163 475 339 090 429 3 β15
−4.369 783 644 567 797 × 10−2 β16 + 1.010 062 677 099 088 × 10−2 β17
−2.018 052 269 391 153 × 10−3 β18 + 3.479 126 853 101 193 × 10−4 β19
−5.159 762 018 077 551 × 10−5 β20 + 6.552 517 694 061 283 × 10−6 β21
−7.079 292 502 238 022 × 10−7 β22 + 6.449 249 702 165 323 × 10−8 β23
−4.894 052 058 789 618 × 10−9 β24 + 3.041 591 067 712 079 × 10−10 β25
−1.510 937 645 005 482 × 10−11 β26 + 5.783 596 085 142 942 × 10−13 β27
−1.606 885 758 239 585 × 10−14 β28 + 2.896 755 640 650 334 × 10−16 β29
−2.560 020 533 395 366 × 10−18 β30 (15.48)
0 β 2 , f ′′(0) >D, ?1 10 " Hartree[107] White[20]
Æ'0 , 15.1 ! 15.3 . , β0 β < 0 , f ′′(0) >M. $/7D" , (D f ′′(0) ). #, $( -
7D" (4 41 A 3.5.2 #), (15.45) )<, : β0 β < 0 , 15.2 ! 15.3 . 8/, f ′′(0) Æ [m,m] ( - 7D" .
15 Æ VZJSUWVX@ · 201 ·
> 15.1 λ = 5, γ = 0 Æ --h = −1 8, OW (15.45) PMN f ′′(0) 9=X=ZWhite[20] Æ Hartree[107] N[N=9U\
β 10 20 25 30 @,-%, -%, -%, -%,
2.0 1.686 47 1.687 19 1.687 21 1.687 22 1.687 2
1.6 1.517 09 1.521 48 1.521 52 1.521 52 1.521 5
1.2 1.331 47 1.335 78 1.335 72 1.335 72 1.335 7
1.0 1.230 79 1.232 66 1.232 58 1.232 59 1.232 6
0.8 1.122 10 1.120 27 1.120 28 1.120 27 1.120 3
0.6 1.001 07 0.995 72 0.995 85 0.995 84 0.995 8
0.5 0.933 79 0.927 55 0.927 67 0.927 68 0.927 7
0.4 0.860 38 0.854 35 0.854 40 0.854 42 0.854 4
0.3 0.779 22 0.774 83 0.774 74 0.774 75 0.774 8
0.2 0.688 30 0.686 91 0.686 74 0.686 71 0.686 7
0.1 0.595 19 0.587 11 0.587 07 0.587 05 0.587 0
0.0 0.466 89 0.469 47 0.469 56 0.469 59 0.469 6
−0.1 0.329 80 0.323 63 0.321 97 0.320 96 0.319
−0.14 0.268 76 0.253 74 0.249 60 0.246 82 0.239
−0.16 0.236 76 0.215 59 0.209 47 0.205 15 0.190
−0.18 0.203 72 0.174 99 0.166 22 0.159 71 0.128
−0.19 0.186 79 0.153 69 0.143 29 0.135 37 0.086
−0.198 0.173 06 0.136 14 0.124 26 0.115 04 0
QO*Liao Shijun. A uniformly valid analytic solution of two-dimensional viscous flow past a
semi-infinite flat plate. Journal of Fluid Mechanics. Cambridge University Press, 1999, 358:101∼128
(Cambridge University Press, RTUR)
" 15.3 λ = 5, γ = 0 > = −1 -, f ′′(0) &V&.J6()&)*12*10 -%, (15.46)L..2*20 -%, (15.47)L&..2*30 -%, (15.48)L+2*[15,15]
F - QQ-%L+X0.*Hartree[107] @,LNX0.*Stewartson[108] @,
· 202 ·
> 15.2 λ = 5, γ = 0 Æ --h = −1 8, f ′′(0) N [m, m] :>? - @?;<Z White[20] Æ Hartree [107] N[N=9U\
β [5,5] [10,10] [15,15] @,2.0 1.686 36 1.687 22 1.687 22 1.687 2
1.6 1.520 26 1.521 51 1.521 51 1.521 5
1.2 1.333 99 1.335 71 1.335 72 1.335 7
1.0 1.230 63 1.232 60 1.232 59 1.232 6
0.8 1.118 16 1.120 27 1.120 27 1.120 3
0.6 0.993 72 0.995 84 0.995 84 0.995 8
0.5 0.925 63 0.927 69 0.927 68 0.927 7
0.4 0.852 46 0.854 43 0.854 42 0.854 4
0.3 0.772 87 0.774 74 0.774 76 0.774 8
0.2 0.684 78 0.686 70 0.686 71 0.686 7
0.1 0.584 84 0.586 97 0.587 04 0.587 0
0.0 0.467 36 0.469 60 0.469 60 0.469 6
−0.1 0.322 91 0.319 35 0.319 27 0.319
−0.14 0.254 76 0.240 74 0.239 84 0.239
−0.16 0.218 13 0.193 80 0.191 28 0.190
−0.18 0.179 80 0.138 70 0.131 60 0.128
−0.19 0.160 04 0.106 77 0.094 55 0.086
−0.198 0.143 98 0.078 33 0.059 12 0
8/, =# f ′′(0) (15.45) , $Æ f(η) f ′(η) &, 0 η < +∞ . 0 β 2 , f ′(η) !D, 20 " Hartree [107] ', ! 15.4. , β0 β < 0 , β -[" β0, f ′(η)
-M. , "#;Æ" N:+Æ, ! 15.4 .
/1=:, Hartree[107] /A40 β 2 NSβ0 β < 2
f ′′(0) 0, η → +∞ f ′(η) → 1 O;P". , ., β0 β < 0 , Stewartson[108] 8/ f ′′(0) < 0 Æ, η → +∞ f ′(η) → 1 O;P", Y9. Y: Stewartson[108] /1, S γ ,
f ′′0 (0) = γ + λ < 0
' λ γ, :,$Æ f ′′(0) ∼ 9, || !ÆS , $. , 8/, =#, =1 Hartree . #,
γ = −5.5, β = −15/100, λ = 2 = −1/10 , 1 Hartree . ,
(15.44) \'' Stewartson[108] Y9Æ. ), (15.44) \'' Libby Liu[109] )9)Æ. Æ, Stewartson[108]Libby Liu[109] , #!O+Y: η → +∞ f ′(η) → 1
15 Æ VZJSUWVX@ · 203 ·
T]ÆO;P", Æ=<,<ÆBN&,, !FE<KFÆ. , ]ZÆ, Stewartson[108]Libby Liu[109] ''Æ η → +∞ T]ÆO;P". UH"#;Æ: \Z. β < 0
, Falkner-Skan9TEFKF;ÆK[PÆ. ", ^)$( 1+, (W_Æ.
" 15.4 λ = 5>γ = 0 > = −1 -, O (15.43) -( f ′(η)
&&V'$. Hartree[107] J6&&)*+2*β = 2 *@,L&..2*β = 1 *@,L..2*β = 0 *@,L12*β = −0.16 *@,L+X0.*β = 2 * 20 -%,L+XQ*β = 1 * 20 -%,LNX0.*β = 0 * 20 -%,LN
XQ*β = −0.16 * 50 -%,
16 V`^_`WaXE
3, EJD9)XV=O;Y$Æ. , Kuiken[110, 111]
'' +EFKFO%Y$ÆEJD9.
#, Kuiken[111] 1\RZ;,9, 1</ , 38Æ
f ′′′(η) + θ(η) − f ′2(η) = 0 (16.1)
θ′′(η) = 3 σ f ′(η) θ(η) (16.2)
3SEJ
f(0) = f ′(0) = 0, θ(0) = 1 (16.3)
f ′(+∞) = θ(+∞) = 0 (16.4)
, ′ η , σ ÆT:, f(η) θ(η) EJD9Æ)VX G. L4 Kuiken[111] AF.
Kuiken[111] Æ. , 8, B1BÆ. U., ^.W''3Æ1. 87, $( 1, ''3Æ.
16.1
16.1.1 QRST
EJ (16.4), η → +∞ , f ′(η) θ(η) ='" 0. , "#VEFKFK". Kuiken[111] ;', f ′(η) θ(η) η → +∞ =O%Y$.
1ξ = 1 + λ η, F (ξ) = f ′(η), S(ξ) = θ(η) (16.5)
(16.1) (16.2)
λ2 F ′′(ξ) + S(ξ) − F 2(ξ) = 0 (16.6)
λ2 S′′(ξ) = 3 σ F (ξ) S(ξ) (16.7)
16 Æ V]JSUWVX@ · 205 ·
EJF (1) = 0, S(1) = 1 (16.8)
F (+∞) = S(+∞) = 0 (16.9)
, λ > 0 C.
F (ξ) S(ξ) ξ → +∞ K"F ∼ ξα1 , S ∼ ξα2 (16.10)
, α1 α2 [. (16.10) %4 (16.6) (16.7), 2_VÆ0%,
α1 = −2, α2 = −4 (16.11)
, F (ξ) S(ξ) EFKF%Y$, Sξ−n | n 2
(16.12)
F (ξ) S(ξ),
F (ξ)=+∞∑n=2
an
ξn(16.13)
S(ξ)=+∞∑n=4
bnξn
(16.14)
, anbn . (16.13) (16.14) F (ξ) S(ξ) .
16.1.2
(16.13) (16.14), 5EJ (16.8) (16.9),
F0(ξ) = γ(ξ−2 − ξ−3
), S0(ξ) = ξ−4 (16.15)
F (ξ) S(ξ), , γ . (16.13) (16.14),
5 (16.6) (16.7),
LF Φ =(ξ
3
)∂2Φ∂ξ2
+∂Φ∂ξ
(16.16)
LSΦ =(ξ
5
)∂2Φ∂ξ2
+∂Φ∂ξ
(16.17)
LF
(C1 + C2 ξ
−2)
= 0 (16.18)
LS
(C3 + C4 ξ
−4)
= 0 (16.19)
· 206 ·
, C1C2C3C4 . (16.6) (16.7),
NF [Φ(ξ; q),Θ(ξ; q)]=λ2 ∂Φ(ξ; q)∂ξ2
+ Θ(ξ; q) − Φ2(ξ; q) (16.20)
NS [Φ(ξ; q),Θ(ξ; q)]=λ2 ∂2Θ(ξ; q)∂ξ2
− 3 σ Φ(ξ; q) Θ(ξ; q) (16.21)
, q ∈ [0, 1] 34, Φ(ξ; q) Θ(ξ; q) ξ q . 8 F S
, HF (ξ) HS(ξ) . 3(1 − q) LF [Φ(ξ; q) − F0(ξ)]
= q F HF (ξ) NF [Φ(ξ; q),Θ(ξ; q)] (16.22)
(1 − q) LS [Θ(ξ; q) − S0(ξ)]
= q S HS(ξ) NS [Φ(ξ; q),Θ(ξ; q)] (16.23)
EJΦ(1; q) = Φ(+∞; q) = Θ(+∞; q) = 0, Θ(1; q) = 1 (16.24)
q = 0 , Φ(ξ; 0) = F0(ξ), Θ(ξ; 0) = S0(ξ) (16.25)
, F0(ξ) S0(ξ) (16.15). q = 1 ,
F = 0, S = 0, HF (ξ) = 0, HS(ξ) = 0
, (16.22)∼(16.24) * (16.6)∼(16.9), Φ(ξ; 1) = F (ξ), Θ(ξ; 1) = S(ξ) (16.26)
, q 0 1 , Φ(ξ; q) Θ(ξ; q) F0(ξ)S0(ξ) (16.6)∼ (16.9) F (ξ)S(ξ).
, (16.25), S
Φ(ξ; q)=F0(ξ) ++∞∑n=1
Fn(ξ) qn (16.27)
Θ(ξ; q)=S0(ξ) ++∞∑n=1
Sn(ξ) qn (16.28)
Fn(ξ) =
1n!∂nΦ(ξ; q)∂qn
∣∣∣∣q=0
, Sn(ξ) =1n!∂nΘ(ξ; q)∂qn
∣∣∣∣q=0
(16.29)
16 Æ V]JSUWVX@ · 207 ·
C λ, (16.15) γ, FS , 5HF (ξ)HS(ξ) =, 1 q = 1 , (16.26),
F (ξ)=F0(ξ) ++∞∑n=1
Fn(ξ) (16.30)
S(ξ)=S0(ξ) ++∞∑n=1
Sn(ξ) (16.31)
m "
F (ξ)≈F0(ξ) +m∑
n=1
Fn(ξ) (16.32)
S(ξ)≈S0(ξ) +m∑
n=1
Sn(ξ) (16.33)
16.1.3
, F m = F0(ξ), F1(ξ), F2(ξ), · · · , Fm(ξ)
Sm = S0(ξ), S1(ξ), S2(ξ), · · · , Sm(ξ)
(16.22) 1 (16.24) 34 q n , n!, q = 0,
LF [Fn(ξ) − χnFn−1(ξ)]=F HF (ξ) RFn (F n−1,Sn−1) (16.34)
LS [Sn(ξ) − χnSn−1(ξ)]=S HS(ξ) RSn(F n−1,Sn−1) (16.35)
EJFn(1) = Sn(1) = Fn(+∞) = Sn(+∞) = 0 (16.36)
, χn (2.42) ,
RFn (F n−1,Sn−1)=λ2 F ′′
n−1(ξ) + Sn−1(ξ)
−n−1∑j=0
Fj(ξ) Fn−1−j(ξ) (16.37)
RSn(F n−1,Sn−1)=λ2 S′′
n−1(ξ) − 3 σn−1∑j=0
Fj(ξ) Sn−1−j(ξ) (16.38)
(16.27) Æ (16.28) (16.22)∼(16.24), q , !!"# (16.34)∼(16.36), !" (16.37) Æ (16.38). —— !
· 208 ·
(16.34)∼(16.36)8Æ3. , "&Z<T,
!7'L[.
(16.13) (16.14), HF (ξ) HS(ξ) $
HF (ξ) = ξκ1 , HS(ξ) = ξκ2 (16.39)
, κ1 κ2 . 8/, κ1 1 (62)κ2 1 , ln ξ %. &(16.13) (16.14). κ1 −1 (62)κ2 −1 , F (ξ) S(ξ) ξ−2 ξ−4 %. &ecafg. ecaf,
&κ1 = κ2 = 0
$%
HF (ξ) = HS(ξ) = 1 (16.40)
16.1.4 RUdV
39 (16.34)∼(16.36),8/, Fn(ξ) Sn(ξ)
Fn(ξ) = ξ−22n+1∑j=0
an,j ξ−j , Sn(ξ) = ξ−4
2n∑j=0
bn,j ξ−j (16.41)
, an,j bn,j . (16.41) %4 (16.34)∼(16.36), OO) (j 1)
an,j =χn χ2n+1−j an−1,j
+3 F
[χ2n+2−j λ
2 (j + 1)(j + 2)an−1,j−1 + χ2n+1−jbn−1,j−1 −An,j−1
]j(j + 2)
(16.42)
bn,j =χn χ2n−j bn−1,j
+5 S
[χ2n+1−j λ
2 (j + 3)(j + 4)bn−1,j−1 − 3σ Bn,j−1
]j(j + 4)
(16.43)
an,0 = −2n+1∑j=1
an,j, bn,0 = −2n∑
j=1
bn,j (16.44)
16 Æ V]JSUWVX@ · 209 ·
An,i =n−1∑j=0
min2j+1,i∑r=max0,i+2j−2n+1
aj,r an−j−1,i−r (16.45)
Bn,i =n−1∑j=0
min2j+1,i∑r=max0,i+2j−2n+2
aj,r bn−j−1,i−r (16.46)
(16.15), 54a0,0 = γ, a0,1 = −γ, b0,0 = 1 (16.47)
$OO), 5, an,j bn,j .
, Æ
F (ξ)=+∞∑n=0
2n+1∑j=0
an,j
ξj+2, S(ξ) =
+∞∑n=0
2n∑j=0
bn,j
ξj+4(16.48)
) (16.5),
f ′(η)=+∞∑n=0
2n+1∑j=0
an,j
(1 + λ η)j+2, θ(η) =
+∞∑n=0
2n∑j=0
bn,j
(1 + λ η)j+4(16.49)
m "
f ′(η) ≈m∑
n=0
2n+1∑j=0
an,j
(1 + λ η)j+2, θ(η) ≈
m∑n=0
2n∑j=0
bn,j
(1 + λ η)j+4(16.50)
f(η) ≈m∑
n=0
2n+1∑j=0
an,j
λ (j + 1)
[1 − 1
(1 + λ η)j+1
](16.51)
f ′′(0) ≈ −λm∑
n=0
2n+1∑j=0
(j + 2)an,j (16.52)
f(+∞) ≈m∑
n=0
2n+1∑j=0
an,j
λ (j + 1)(16.53)
θ′(0) ≈ −λm∑
n=0
2n∑j=0
(j + 4)bn,j (16.54)
16.1.5 9!
9! 16.1 "#$%(16.30))(16.31)&', (, Fn(ξ) ) Sn(ξ) *+,(16.34)∼(16.36), -.(16.37)(16.38) )(2.42)12, 345.6+,(16.6) ∼(16.9)h%.
· 210 ·
7 " (16.30) (16.31) , lim
m→+∞Fm(ξ) = 0, limm→+∞Sm(ξ) = 0
, (16.16) (16.17),
LF
[lim
m→+∞Fm(ξ)]
= 0, LS
[lim
m→+∞Sm(ξ)]
= 0
(16.34) (16.35), (2.42),
F HF (ξ)m∑
n=1
RFn (F n−1,Sn−1) = LF [Fm(ξ)]
S HS(ξ)m∑
n=1
RSn(F n−1,Sn−1) = LS [Sm(ξ)]
,
F HF (ξ)+∞∑n=1
RFn (F n−1,Sn−1) = LF
[lim
m→+∞Fm(ξ)]
= 0
S HS(ξ)+∞∑n=1
RSn(F n−1,Sn−1) = LS
[lim
m→+∞Sm(ξ)]
= 0
F = 0S = 0HF (ξ) = 0 HS(ξ) = 0,
+∞∑n=1
RFn (F n−1,Sn−1) = 0
+∞∑n=1
RSn(F n−1,Sn−1) = 0
(16.37) (16.38) %4, 2,
λ2 ∂2
∂ξ2
[+∞∑n=0
Fn(ξ)
]+
+∞∑n=0
Sn(ξ) −[
+∞∑n=0
Fn(ξ)
]2
= 0 (16.55)
λ2 ∂2
∂ξ2
[+∞∑n=0
Sn(ξ)
]− 3 σ
[+∞∑n=0
Fn(ξ)
] [+∞∑n=0
Sn(ξ)
]= 0 (16.56)
16 Æ V]JSUWVX@ · 211 ·
(16.15) (16.36), +∞∑n=0
Fn(1) =+∞∑n=0
Fn(+∞) =+∞∑n=0
Sn(+∞) = 0,+∞∑n=0
Sn(1) = 1 (16.57)
(16.55)∼(16.57) (16.6)∼(16.9) L[0G, , (16.30) (16.31) 8 . :=.
16.2 H I
87Æ(Æ4FSλ γ. , IÆÆ. 16.1,"JÆ, (16.30)
(16.31) .
(16.15)Æ F0(ξ) γ. (16.15)
%4 (16.6) (16.7), 2_0%, λ γ Æ%3, 3
λ =
√3σ20
(1 − 9σ
10
)−1/4
, γ =(
1 − 9σ10
)−1/2
(16.58)
[ (16.58) σ < 10/9 Æ, λ γ Æ=ÆF9. (16.58), , σ 1 ,
γ ∼ 1, λ ∼ √σ
, 'ÆÆT: σ, =B;Æ λγF
S CD, 3Æ. #,
σ = 1 Æ%.
f ′′(0) = λ F ′(0), θ′(0) = λ S′(0)
LY2X3, , "-#Æ. 14, 67B;Æ f ′′(0) θ′(0) ÆCD. F = S = −1/2,
γ = 1, 2, 3 , θ′(0) f ′′(0) λ Æ<&,=, &, γ = 3 ,
! 16.1 ! 16.2 . , λ = 1/3, γ = 3, F = S = −1/2 , f ′′(0) θ′(0)
. Y:H, L^:,! 16.3 Æ f ′′(0) ∼ θ′(0) ∼
9, B; λ = 1/3, γ = 3 S = F = CD. σ = 1, λ = 1/3, γ = 3, S = F = −1/2 , f ′′(0) θ′(0) , 16.1 . , 'ÆÆT: σ, Æ λγF S , f ′′(0) θ′(0) . #, σ = 1/10, λ = 1/5, γ = 1,
F = S = −1/2 , 5 σ = 10 λ = 1/3, γ = 1, F = −1/4, S = −1/10 ,
· 212 ·
f ′′(0) θ′(0) %. σ > 10, λ = 1, γ = 1, F = −1/4, S = −1/σ ,
5 σ < 1/10, γ = 1, F = S = −1/2, λ = 1/5(6;) , (. , ! 16.4 ! 16.5 , =# f ′′(0) θ′(0) , $Æ f(ξ)
θ(ξ) & 0 ξ < +∞ =.
" 16.1 σ = 1, F = S = −1/2 -, 24 A'$& θ′(0) ∼ λ D+2*γ = 1L12*γ = 2L..2*γ = 3
" 16.2 σ = 1, F = S = −1/2 -, 24 A'$& f ′′(0) ∼ λ D+2*γ = 1L12*γ = 2L..2*γ = 3
16 Æ V]JSUWVX@ · 213 ·
" 16.3 σ = 1, γ = 3, λ = 1/3, F = S = -, 24 A'$& f ′′(0) ∼ > θ′(0) ∼ D+2*θ′(0)L12*f ′′(0)
> 16.1 σ = 1, λ = 1/3, γ = 3, --hF = --hS = −1/2 8, f ′′(0) Æ θ′(0) N m :=X;<=Z Kuiken[111] [N=9U\
m f ′′(0) θ′(0)5 0.713 814 −0.831 716
10 0.706 453 −0.765 271
15 0.702 547 −0.769 478
20 0.697 170 −0.771 491
25 0.694 380 −0.770 640
30 0.693 538 −0.770 001
35 0.693 342 −0.769 872
40 0.693 268 −0.769 879
45 0.693 227 −0.769 876
50 0.693 213 −0.769 866
Kuiken @, 0.693 212 −0.769 861
( - 7D" (4 41 A 3.5.2 #), <) f ′′(0) θ′(0) , 16.2 16.3 . 8/, S = F = [m,m] ( - 7D" .
Æ, OO) (16.42)∼(16.47), (16.1) (16.2) Æ. EFKFO%Y$. 3 15 7, ( 1B EFKFO;Y$Æ Falkner-Skan EJD9. , ( 1 1/:ÆEJD9%Æ.
· 214 ·
" 16.4 f ′(η) &V&.J6&&)*NX0.*σ = 1/10, F = S = −1/2, λ = 1/5, γ = 1 *, 30 -%,L+X0.*σ = 1,
F =S =−1/2, λ=1/3, γ =3 *, 20 -%,LNXQ: σ=10, F =−1/4, S =−1/10,
λ = 1/3, γ = 1 *, 40 -%,L+2*@,
" 16.5 θ(η) &V&.J6&&)*NX0.*σ = 1/10, F = S = −1/2, λ = 1/5, γ = 1 *, 30 -%,L+X0.*σ = 1,
F =S =−1/2, λ=1/3, γ =3 *, 20 -%,; NXQ: σ=10, F =−1/4, S =−1/10,
λ = 1/3, γ = 1 *, 40 -%,; +2: @,
16 Æ V]JSUWVX@ · 215 ·
> 16.2 f ′′(0) 9 [m, m] :>? - @?;<Z Kuiken N[N= [111] 9U\σ = 1/10 σ = 1 σ = 10
[m, m]λ = 1/5, γ = 1 λ = 1/3, γ = 3 λ = 1/3, γ = 1
[5, 5] 0.952 170 0.705 940 0.433 555
[10, 10] 0.921 936 0.693 438 0.452 229
[15, 15] 0.924 108 0.693 214 0.447 038
[20, 20] 0.924 087 0.693 212 0.447 107
[25, 25] 0.924 088 0.693 212 0.447 117
[30, 30] 0.924 086 0.693 212 0.447 117
[35, 35] 0.924 084 0.693 212 0.447 117
[40, 40] 0.924 083 0.693 212 0.447 117
[45, 45] 0.924 083 0.693 212 0.447 117
[50, 50] 0.924 083 0.693 212 0.447 117
Kuiken "@, 0.924 083 0.693 212 0.447 117
> 16.3 θ′(0) 9 [m, m] :>? - @?;<Z Kuiken N[N= [111] 9U\σ = 1/10 σ = 1 σ = 10
[m, m]λ = 1/5, γ = 1 λ = 1/3, γ = 3 λ = 1/3, γ = 1
[5, 5] −0.347 058 −0.774 151 −1.615 83
[10, 10] −0.350 119 −0.770 018 −1.492 63
[15, 15] −0.350 027 −0.769 866 −1.497 33
[20, 20] −0.350 058 −0.769 861 −1.497 08
[25, 25] −0.350 058 −0.769 861 −1.497 10
[30, 30] −0.350 059 −0.769 861 −1.497 10
[35, 35] −0.350 059 −0.769 861 −1.497 10
[40, 40] −0.350 059 −0.769 861 −1.497 10
[45, 45] −0.350 059 −0.769 861 −1.497 10
[50, 50] −0.350 059 −0.769 861 −1.497 10
Kuiken "@, −0.350 059 −0.769 861 −1.497 10
17 i · jkDlE
87, bc_EF4Y\UY4Y z d) Ω
]e`ZÆVQG95dQÆÆD99. 9*=
1r
∂(rVr)∂r
+1r
∂Vθ
∂θ+∂Vz
∂z=0 (17.1)
W; - H\XH (Navier-Stokes)
Vr∂Vr
∂r+ Vz
∂Vr
∂z− V 2
θ
r= ν
[∂2Vr
∂r2+
1r
∂Vr
∂r+∂2Vr
∂z2− Vr
r2
]− 1ρ
∂p
∂r(17.2)
Vr∂Vθ
∂r+ Vz
∂Vθ
∂z+VrVθ
r= ν
[∂2Vθ
∂r2+
1r
∂Vθ
∂r+∂2Vθ
∂z2− Vθ
r2
](17.3)
Vr∂Vz
∂r+ Vz
∂Vz
∂z= ν
[∂2Vz
∂r2+
1r
∂Vz
∂r+∂2Vz
∂z2
]− 1ρ
∂p
∂z(17.4)
5SEJVθ = rΩ , Vr = Vz = 0, z = 0 F (17.5)
Vr = Vθ = 0, z = +∞ F (17.6)
, ρ 95, ν G, p V2, VrVθVz 0dÆ) .
η = z
√Ων
(17.7)
2 Vr = (rΩ) f(η) (17.8)
Vθ = (rΩ) g(η) (17.9)
Vz =√νΩ w(η) (17.10)
p = −ρνΩ P (η) (17.11)
17 Æ W · YZH]@ · 217 ·
(17.1)∼(17.6) 3 3
f ′′ = f2 − g2 + f ′ w (17.12)
g′′ = g′ w + 2f g (17.13)
w w′ = P ′ + w′′ (17.14)
2f + w′ = 0 (17.15)
EJ
f(0) = f(+∞) = 0, g(0) = 1, g(+∞) = 0, w(0) = 0 (17.16)
, ′ η . (17.15)
f = −w′
2(17.17)
(17.17) %4 (17.12) (17.13) ,
w′′′ − w′′ w +12w′ w′ − 2g2 = 0 (17.18)
g′′ − w g′ + w′ g = 0 (17.19)
EJ
w(0) = w′(0) = w′(+∞) = 0, g(0) = 1, g(+∞) = 0 (17.20)
LPO, L4X · [\ [112](Von Karman) 5 Zandbergen Dijkstra[113] F7.
38Æ#Æ.!B;2, X ·[\ [112]Cochran[114]Hettis[115] Rogers Lance[116] Benton[117] McLeod[118] Zandbergen Dijkstra[119]Ackroyd[120] 5 Hulzen[121], =L[9B;. ^Æ/16Æ, 6BB1Æ. 87, ( 1, ''G9H1Æ.
17.1
X · [\G_9EFKFÆd) Æ, #[-#Æ. Benton[117] ,
γ = −w(+∞) (17.21)
· 218 ·
1w(η) = −γ [1 − s(η)] (17.22)
#, Benton[117], ^434ξ = λ η (17.23)
, λ C. (17.22) (17.23), (17.18) (17.19) γλ3s′′′ + γ2λ2(1 − s)s′′ +
12γ2λ2s′s′ − 2g2 = 0 (17.24)
λg′′ + γ(1 − s)g′ + γs′g = 0 (17.25)
EJs(0) = g(0) = 1, s(+∞) = g(+∞) = 0, s′(0) = s′(+∞) = 0 (17.26)
, ′ ξ . Æ, !ÆC λ, , (17.21) Æ γ Æ.
17.1.1
Rogers Lance[116] ÆB;, X · [\G_9EFKFO;Y$.
1978 :, Dijkstra O'=;ÆK". Hulzen[121] ''%;Æ. ", _aJUK [43] ( 1, 6'' %;Æ. 87, ''
exp(−n ξ) | n 1 (17.27)
Æ s(ξ) g(ξ) ,
s(ξ) =+∞∑n=1
an exp(−nξ), g(ξ) =+∞∑n=1
bn exp(−nξ) (17.28)
, an bn . (17.28) s(ξ) g(ξ) .
ε . (17.28) EJ (17.26),
s0(ξ)=2 exp(−ξ) − exp(−2ξ) (17.29)
g0(ξ)=exp(−ξ) + ε [exp(−2ξ) − exp(−ξ)] (17.30)
s(ξ) g(ξ) . (17.28), 5 (17.24) (17.25),
Lsf =∂3f
∂ξ3+ 2
∂2f
∂ξ2− ∂f
∂ξ− 2f (17.31)
Lgf =∂2f
∂ξ2− f (17.32)
17 Æ W · YZH]@ · 219 ·
\
Ls [C1 exp(ξ) + C2 exp(−ξ) + C3 exp(−2ξ)] = 0 (17.33)
Lg [C1 exp(ξ) + C2 exp(−ξ)] = 0 (17.34)
, C1C2 C3 . O, (17.24) (17.25),
Ns [S(ξ; q), G(ξ; q),Λ(q),Γ (q)]
=Γ (q)Λ3(q)∂3S(ξ; q)∂ξ3
+ Γ 2(q)Λ2(q) [1 − S(ξ; q)]∂2S(ξ; q)∂ξ2
+(
12
)Γ 2(q)Λ2(q)
[∂S(ξ; q)∂ξ
]2
− 2G(ξ; q)2 (17.35)
Ng [S(ξ; q), G(ξ; q),Λ(q),Γ (q)]
=Λ(q)∂2G(ξ; q)∂ξ2
+ Γ (q) [1 − S(ξ; q)]∂G(ξ; q)∂ξ
+Γ (q)G(ξ; q)∂S(ξ; q)∂ξ
(17.36)
, q ∈ [0, 1] 34, S(ξ; q) G(ξ; q) ξ q , Λ(q) Γ (q) q . g s , Hs(ξ) Hg(ξ) .
(1 − q) Ls [S(ξ; q) − s0(ξ)]
= q s Hs(ξ) Ns[S(ξ; q), G(ξ; q),Λ(q),Γ (q)] (17.37)
(1 − q) Lg [G(ξ; q) − g0(ξ)]
= q g Hg(ξ) Ng[S(ξ; q), G(ξ; q),Λ(q),Γ (q)] (17.38)
EJ
S(0; q) = 1, S(+∞; q) = 0,∂S(ξ; q)∂ξ
∣∣∣∣ξ=0
=∂S(ξ; q)∂ξ
∣∣∣∣ξ=+∞
= 0 (17.39)
G(0; q) = 1, G(+∞; q) = 0 (17.40)
q = 0 , (17.29) (17.30), 5 (17.37)∼(17.40),
S(ξ; 0) = s0(ξ), G(ξ; 0) = g0(ξ) (17.41)
· 220 ·
q = 1 , s = 0g = 0Hs(ξ) = 0 Hg(ξ) = 0, (17.37)∼(17.40) = (17.24)∼(17.26),
S(ξ; 1) = s(ξ), G(ξ; 1) = g(ξ), Λ(1) = λ, Γ (1) = γ (17.42)
(17.41), q
S(ξ; q) = s0(ξ) ++∞∑n=1
sn(ξ) qn (17.43)
G(ξ; q) = g0(ξ) ++∞∑n=1
gn(ξ) qn (17.44)
Λ(q) = λ0 ++∞∑n=1
λn qn (17.45)
Γ (q) = γ0 ++∞∑n=1
γn qn (17.46)
, λ0 γ0 λ γ ,
sn(ξ) =1n!
∂nS(ξ; q)∂qn
∣∣∣∣q=0
(17.47)
gn(ξ) =1n!
∂nG(ξ; q)∂qn
∣∣∣∣q=0
(17.48)
λn =1n!
∂nΛ(q)∂qn
∣∣∣∣q=0
(17.49)
γn =1n!
∂nΓ (q)∂qn
∣∣∣∣q=0
(17.50)
(17.37) (17.38) sg Hs(ξ)Hg(ξ). , 1 q = 1 , (17.42)
s(ξ) = s0(ξ) ++∞∑n=1
sn(ξ) (17.51)
g(ξ) = g0(ξ) ++∞∑n=1
gn(ξ) (17.52)
λ = λ0 ++∞∑n=1
λn (17.53)
γ = γ0 ++∞∑n=1
γn (17.54)
17 Æ W · YZH]@ · 221 ·
17.1.2
O, ]
sk = s0(ξ), s1(ξ), s2(ξ), · · · , sk(ξ)
gk = g0(ξ), g1(ξ), g2(ξ), · · · , gk(ξ)
λk = λ0, λ1, λ2, · · · , λk , γk = γ0, γ1, γ2, · · · , γk
(17.37)∼(17.40) q n , n!, 2 q = 0,
Ls [sn(ξ) − χn sn−1(ξ)] = s Hs(ξ) Rsn(sn−1, gn−1,λn−1,γn−1) (17.55)
Lg [gn(ξ) − χn gn−1(ξ)] = g Hg(ξ) Rgn(sn−1, gn−1,λn−1,γn−1) (17.56)
EJ
sn(0) = gn(0) = sn(+∞) = gn(+∞) = 0, s′n(0) = s′n(+∞) = 0 (17.57)
, χn (2.42) ,
Rsn(sn−1, gn−1,λn−1,γn−1)
=1
(n− 1)!∂n−1Ns [S(ξ; q), G(ξ; q),Λ(q),Γ (q)]
∂qn−1
∣∣∣∣q=0
=n−1∑k=0
[αn−1−k s′′′k (ξ) + βn−1−k s
′′k(ξ)]
−n−1∑k=0
βn−1−k
⎡⎣ k∑
j=0
sj(ξ) s′′k−j(ξ)
⎤⎦
+12
n−1∑k=0
βn−1−k
⎡⎣ k∑
j=0
s′j(ξ) s′k−j(ξ)
⎤⎦
−2n−1∑k=0
gn−1−k(ξ) gk(ξ) (17.58)
(17.43)∼(17.46) (17.37)∼(17.40), q , !!"# (17.55)∼(17.57), !" (17.58)∼(17.62). —— !
· 222 ·
Rgn(sn−1, gn−1,λn−1,γn−1)
=1
(n− 1)!∂n−1Ng [S(ξ; q), G(ξ; q),Λ(q),Γ (q)]
∂qn−1
∣∣∣∣q=0
=n−1∑k=0
[λn−1−k g′′k (ξ) + γn−1−k g
′k(ξ)]
+n−1∑k=0
γn−1−k
k∑j=0
[s′j(ξ) gk−j(ξ) − sj(ξ) g′k−j(ξ)
](17.59)
αn =n∑
k=0
λn−k δk (17.60)
βn =n∑
k=0
γn−k δk (17.61)
δn =n∑
k=0
γn−k
k∑j=0
λj λk−j (17.62)
(17.55) (17.56) Æ8Æ, , !', : "&ZOT.
Æ4sn(ξ)gn(ξ)λn−1 γn−1. # sn(ξ) gn(ξ)
(17.55) (17.56). , $$, &34%, λn−1 γn−1. (17.28), 5 (17.55) (17.56), $
Hs(ξ) = exp(κs ξ), Hg(ξ) = exp(κg ξ) (17.63)
, κs κg . (17.29) (17.30),
Rs1(s0, g0,λ0,γ0) =
4∑k=1
c1,k(λ0, γ0) exp(−kξ) (17.64)
Rg1(s0, g0,λ0,γ0) =
3∑k=1
d1,k(λ0, γ0) exp(−kξ) (17.65)
, c1,k(λ0, γ0) d1,k(λ0, γ0) ξ EÆ. /. 3/#$ c1,1(λ0, γ0) d1,1(λ0, γ0) ,
2γ0(γ0 − λ0)λ20 = 0, γ0 − λ0 = 0 (17.66)
17 Æ W · YZH]@ · 223 ·
γ0 = λ0 (17.67)
`", =# c1,1(λ0, γ0) = 0, d1,1(λ0, γ0) = 0 . , /[3.
3*/ c1,1(λ0, γ0) c1,2(λ0, γ0) , 2γ0(γ0 − λ0)λ2
0 = 0, (1 − ε)2 + 3γ20λ
20 − 4γ0λ
30 = 0 (17.68)
Nγ0 =
√|1 − ε|, λ0 =
√|1 − ε| (17.69)
! $$, 4S. , " κs > 0, (17.55) >> exp(−2ξ) (6) exp(−ξ) %. , (17.33), s1(ξ) ξ exp(−2ξ)
(6)ξ exp(−ξ) %, (17.28). , κs 0
3, κs −1 , s(ξ) exp(−3ξ) %5Y^. 66ecaf . , (17.28) ecaf, &
κs = 0 (17.70)
Hs(ξ) = 1. %, κg = 0 (17.71)
Hg(ξ) = 1.
,(17.28)ecaf, 2, &J
Hs(ξ) = Hg(ξ) = 1 (17.72)
2%cn,1(λn−1,γn−1) = 0, cn,2(λn−1,γn−1) = 0 (17.73)
λn−1 γn−1, , cn,1(λn−1,γn−1)cn,2(λn−1,γn−1) S
Rsn(sn−1, gn−1,λn−1,γn−1) =
2n+2∑k=1
cn,k(λn−1,γn−1) exp(−kξ) (17.74)
Rgn(sn−1, gn−1,λn−1,γn−1) =
2n+2∑k=1
dn,k(λn−1,γn−1) exp(−kξ) (17.75)
· 224 ·
% (17.73) Æ3 (%3 n 2 Æ), (17.55) (17.56) 5EJ (17.57). 8/
sn(ξ)=2n+2∑k=1
an,k exp(−kξ) (17.76)
gn(ξ)=2n+2∑k=1
bn,k exp(−kξ) (17.77)
, an,k bn,k . "%4 (17.55)∼(17.57), an,k
bn,k OO).
17.1.3 9!
9! 17.1 "#$(17.51)∼(17.54)&', (, sn(ξ) ) gn(ξ) *+,(17.55)∼(17.57), -.(17.31)(17.32)(17.38))(2.42)12, 345.6+,(17.24)∼(17.26)%.
7 " (17.51) (17.52) ,
limm→+∞ sm(ξ) = 0, lim
m→+∞ gm(ξ) = 0 (17.78)
(17.31)(17.32)(2.42) 5 (17.55) (17.56),
s Hs(ξ)+∞∑n=1
Rsn(sn−1, gn−1,λn−1,γn−1)
= limm→+∞Ls [sm(ξ)] = Ls
[lim
m→+∞ sm(ξ)]
= 0 (17.79)
g Hg(ξ)+∞∑n=1
Rgn(sn−1, gn−1,λn−1,γn−1)
= limm→+∞Lg [gm(ξ)] = Lg
[lim
m→+∞ gm(ξ)]
= 0 (17.80)
s = 0g = 0Hs(ξ) = 0 Hg(ξ) = 0, ''+∞∑n=1
Rsn(sn−1, gn−1,λn−1,γn−1) = 0 (17.81)
+∞∑n=1
Rgn(sn−1, gn−1,λn−1,γn−1) = 0 (17.82)
17 Æ W · YZH]@ · 225 ·
(17.58) (17.59) %4 (17.82), 2, (17.51)∼(17.54),
(+∞∑i=0
γi
)⎛⎝+∞∑j=0
λj
⎞⎠
3
d3
dξ3
[+∞∑k=0
sk(ξ)
]
+
(+∞∑i=0
γi
)2⎛⎝+∞∑
j=0
λj
⎞⎠
2 [1 −
+∞∑k=0
sk(ξ)
]d2
dξ2
[+∞∑k=0
sk(ξ)
]
+12
(+∞∑i=0
γi
)2⎛⎝+∞∑
j=0
λj
⎞⎠
2
ddξ
[+∞∑k=0
sk(ξ)
]ddξ
[+∞∑k=0
sk(ξ)
]
−2
[+∞∑k=0
gk(ξ)
]2
= 0 (17.83)
⎛⎝+∞∑
j=0
λj
⎞⎠ d2
dξ2
[+∞∑k=0
gk(ξ)
]+
(+∞∑i=0
γi
)⎛⎝1 −
+∞∑j=0
sj(ξ)
⎞⎠ d
dξ
[+∞∑k=0
gk(ξ)
]
+
(+∞∑i=0
γi
)⎡⎣+∞∑
j=0
gj(ξ)
⎤⎦ d
dξ
[+∞∑k=0
sk(ξ)
]= 0 (17.84)
3, (17.29)(17.30) (17.57), +∞∑n=0
sn(0) =+∞∑n=0
gn(0) = 1,+∞∑n=0
sn(+∞) =+∞∑n=0
gn(+∞) = 0 (17.85)
+∞∑n=0
s′n(0) =+∞∑n=0
s′n(+∞) = 0 (17.86)
Æ (17.24)∼(17.26) 0G, ', " (17.51)∼(17.54) ,
X · [\G_9. :=.
17.2 H I
17.1, " (17.51)∼(17.54) Æ. 54εsg. , 5ÆI. ,
s = g =
· 226 ·
267b ε γ = −w(+∞) ÆCD.
, 'Æ ε , γ S, 39:,! 17.1 Æ γ ∼ 9 (4 24 A 3.5.1 #) b γ CD. γ ∼ 9, , ε = 0, −1/5 < 0 62 ε = 1/4, −3/5 < 0 , γ . #, ε = 0, s = g = −1/5 62 ε = 1/4, s = g = −1/2 , $Æ γ Benton[117] ', 17.1 . ε = 1/4, −3/5 < 0 0ε = 0, −1/5 < 0 D. $ ε = s = g ),Æ). ( - 7D" (4 41 A, 3.5.2 #) (<)Æ, 17.2 .
! 17.1, ε 0 0.5, Æ&,7f^, #c9<Z$. , $ ε Æ&. ε , ε γ1 = λ1 = 0, $
√1 − ε
(119 − 328ε+ 193ε2
)− 5(3 + 19ε) = 0
ε ≈ 0.261 67 (17.87)
! 17.1 , ε = 0.261 67 $Æ Æ&. , ε = 0.261 67,
s = g = −1/2 , γ ;D, 17.1 17.2 .
" 17.1 19 A'$& γ ∼ D&..2*ε = 0L12*ε = 1/4L..2*ε = 0.261 67L+2*ε = 1/2
17 Æ W · YZH]@ · 227 ·
> 17.1 γ = −w(+∞) 9 m :>?;<=ε = 0 ε = 1/4 ε = 0.261 67
ms = g = −1/5 s = g = −1/2 s = g = −1/2
10 0.879 446 0.882 352 0.882 977
20 0.881 898 0.884 437 0.884 454
30 0.883 607 0.884 477 0.884 477
40 0.884 173 0.884 474 0.884 474
50 0.884 337 0.884 474 0.884 474
> 17.2 γ = −w(+∞) 9 [m, m] :>? - @?;<ε = 0 ε = 1/4 ε = 0.261 67
[m, m]s = g = −1/5 s = g = −1/2 s = g = −1/2
[5, 5] 0.879 337 0.883 856 0.885 038
[10, 10] 0.884 502 0.884 482 0.884 475
[15, 15] 0.884 436 0.884 474 0.884 474
[20, 20] 0.884 474 0.884 474 0.884 474
[25, 25] 0.884 474 0.884 474 0.884 474
8/, n 0, λn = γn
, λ = γ
& Cochran[114] ξ K". X · [\G_9,]Æ/.
8/, =# γ , $Æ s(ξ) g(ξ) & 0 ξ <
+∞ . #, ε = 0, s = g = −1/5 , w(η) g(η) Benton[117] ', ! 17.2 ! 17.3 . , ! 17.4 ! 17.5 , ε = 1/4,
g = s = −1/2 , ?1 g(η) w(η) [1, 1] ( - 7D"
g(η) ≈ ∆1(η)Π1(η)
, w(η) ≈ −γ[∆2(η)Π2(η)
](17.88)
( Benton /1 [117] '! ,
∆1(η)=(
935 649 + 3 881 640√
3)
exp(−γη)
+(
456 252 + 2 097 200√
3)
exp(−2γη)
+(
785 007 − 311 640√
3)
exp(−3γη)
+(
220 464 − 317 520√
3)
exp(−4γη)
· 228 ·
−(
22 212 + 25 200√
3)
exp(−5γη) − 4 608 exp(−6γη)
Π1(η)=(
1 247 532 + 3 022 880√
3)
+(
192 492 + 1 858 080√
3)
exp(−γη)
+(
982 512 + 544 320√
3)
exp(−2γη)
−(
33 552 + 100 800√
3)
exp(−3γη) − 18 432 exp(−4γη)
∆2(η)=(
1 364 904 − 477 008√
3)
−(
2 855 992 − 962 752√
3)
exp(−γη)
+(
1 612 135 − 497 280√
3)
exp(−2γη)
−(
115 206 − 14 336√
3)
exp(−3γη)
−(
6 545 + 2 800√
3)
exp(−4γη) + 704 exp(−5γη)
Π2(η)=(
1 364 904 − 477 008√
3)
+(
28 446 + 8 736√
3)
exp(−γη)
−(
57 777 + 2 800√
3)
exp(−2γη) + 5 184 exp(−3γη)
γ = 0.884 474. 7ÆX · [\91.
" 17.2 w(η) &&V&. Benton[117] J6&&)*0.*@,L+2*ε = 0, s = g = −1/5 * 20 -%,
17 Æ W · YZH]@ · 229 ·
" 17.3 g(η) &&V&. Benton[117] J6&&)*0.*@,L+2*ε = 0, s = g = −1/5 * 20 -%,
" 17.4 ω(η) & [1, 1] A-G - RR'$. Benton[117] J6&&)*0.*@,L+2*[1, 1] F - QQ-% (17.88)
· 230 ·
" 17.5 γ(η) & [1, 1] A-G - RR'$. Benton[117] J6&&)*0.*@,L+2*[1, 1] F - QQ-% (17.88)
87, ?#($, ( 1W; · H\XH5;GD9(ÆÆ. [2L^` Liao[41] %a+94_`9AF.
18 mnopqg
E<Ra) C .[Æ*;-2d. 7d^`hÆc_ (x, y), , x dÆdÆ[L, y dUY. 95EGVQ, a_i2. φ(x, y) )b, ζ(x) d. 95Æ;Æ;H
∇2φ(x, y) = 0, (x, y) ∈ Ω (18.1)
Ω = (x, y) | −∞ < x < +∞,−∞ < y < ζ(x)
)b φ(x, y) EJ
C2φxx + gφy +12∇φ∇(∇φ∇φ) − 2C∇φ∇φx = 0, y = ζ(x) (18.2)
ζ(x) =1g
(Cφx − 1
2∇φ∇φ
), y = ζ(x) (18.3)
5\elim
y→−∞∂φ
∂y= 0 (18.4)
, g -2), _ x y Æc. QL),(18.1) Æ, #EJ (18.2) (18.3) [Æ, Æd^. b]ÆRd7Æ, 19 Uc+a^ !B;2. H\XH [122](Stokes) 7$, d?" [123, 124]. , !B;2 [125∼127] jH\XH$, O';Æ" . <d, 5^2bd, Schwartz[128] ` H\XHÆ, 581" . , &><, "/Æ7D" (Pade approximation) B<d (H/L)max = 0.141 18, /,
H d, L d&.
1 Schwartz[128] , Longuet-Higgins[129] H\XH:Æfd, d_ H/L = 0.1411b. B;$, 'Æd&, )2d_Æ72. 4, Longuet-Higgins[130, 131] ^-2dÆbL[ B;, 28/4d<, cdb, de=, bbZ^. Chen Saffman[132] 39<8/,
· 232 ·
QÆdÆ-2d H/L ≈ 0.13 g. H\XHÆ" [133∼136] L^f -2dÆ:.
87, $( 1, EJÆE.
18.1
18.1.1
)b φ ;Æ;H (18.1) \eEJ (18.4). , φ
exp(mky) sin(nkx) | m 1, n 1 (18.5)
,
φ(x, y) =+∞∑m=1
+∞∑n=1
αm,n exp(mky) sin(nkx) (18.6)
, k = 2π/L d, αm,n . (18.6) )b φ(x, y) . $%, dζ(x)
cos(m k x) | m 0 (18.7)
,
ζ(x) =+∞∑m=0
βm cos(mkx) (18.8)
(18.8) d ζ(x) .
(18.6), 5 Airy dA,
φ0(x, y) = A C0 exp(ky) sin(kx) (18.9)
C0 =√g
k(18.10)
)b φ(x, y) ) C , , A [.
ζ0(x) = 0 (18.11)
ζ(x) , O<. (18.2) Æ%,
L [Φ(x, y; q),Λ(q)] = Λ2(q)∂2Φ(x, y; q)
∂x2+ g
∂Φ(x, y; q)∂y
(18.12)
18 Æ ec#fgh · 233 ·
, q ∈ [0, 1] 34, Λ(q) q , Φ(x, y; q) xy q .
(18.2) (18.3),
N [Φ(x, y; q),Λ(q)]
=Λ2(q)Φxx(x, y; q) + gΦy(x, y; q)
+12∇Φ(x, y; q)∇ [∇Φ(x, y; q)∇Φ(x, y; q)]
−2Λ(q)∇Φ(x, y; q)∇Φx(x, y; q) (18.13)
Z [Φ(x, y; q),Λ(q)]
=1g
[Λ(q) Φx(x, y; q) − 1
2∇Φ(x, y; q)∇Φ(x, y; q)
](18.14)
( 1/*=. Rd, bd4φ(x, y) → Φ(x, y; q)ζ(x) → η(x; q) C → Λ(q), , q 0
1 , Φ(x, y; q)η(x; q) Λ(q) φ(x, y)ζ(x) C
. ,//*=, (18.1)∼(18.4),∇2Φ(x, y; q) = 0, (x, y) ∈ Ω(q) (18.15)
5 y = η(x; q) ÆEJ(1 − q) L [Φ(x, y; q) − φ0(x, y),Λ(q)]
= q 1 H1(x) N [Φ(x, y; q),Λ(q)] (18.16)
(1 − q) [η(x; q) − ζ0(x)]
= q 2 H2(x) η(x; q) −Z[Φ(x, y; q),Λ(q)] (18.17)
\eEJlim
y→−∞∂Φ(x, y; q)
∂y= 0 (18.18)
, q ∈ [0, 1] 34, 12 , H1(x)H2(x) , &
Ω(q) = (x, y) | −∞ < x < +∞,−∞ < y < η(x; q)
q ∈ [0, 1] Z*3.
', q = 0 , ), (18.15) EJ (18.16)∼(18.18)
Φ(x, y; 0) = φ0(x, y), η(x; 0) = ζ0(x), Λ(0) = C0 (18.19)
· 234 ·
, C0 ). q = 1 ,
1 = 0, 2 = 0, H1(x) = 0, H2(x) = 0
(18.15)∼(18.18) = (18.1)∼(18.4),
Φ(x, y; 1) = φ(x, y), η(x; 1) = ζ(x), Λ(1) = C (18.20)
, q 0 1 , (18.15)∼(18.18) ÆE *=.
(18.19), Φ(x, y; q)η(x; q) Λ(q) q
Φ(x, y; q) = φ0(x, y) ++∞∑m=1
φ[m]0 (x, y)m!
qm (18.21)
η(x; q) = ζ0(x) ++∞∑m=1
ζ[m]0 (x)m!
qm (18.22)
Λ(q) = C0 ++∞∑m=1
C[m]0
m!qm (18.23)
φ[m]0 (x, y) =
∂mΦ(x, y; q)∂qm
∣∣∣∣q=0
(18.24)
ζ[m]0 (x) =
∂mη(x; q)∂qm
∣∣∣∣q=0
(18.25)
C[m]0 =
dmΛ(q)dqm
∣∣∣∣q=0
(18.26)
(18.16) (18.17) 12 H1(x)H2(x).
, 1 q = 1 F, , (18.20),
φ(x, y) = φ0(x, y) ++∞∑m=1
φ[m]0 (x, y)m!
(18.27)
ζ(x) = ζ0(x) ++∞∑m=1
ζ[m]0 (x)m!
(18.28)
C = C0 ++∞∑m=1
C[m]0
m!(18.29)
18 Æ ec#fgh · 235 ·
18.1.2
O, ]
φn =φ0(x, y), φ
[1]0 (x, y), φ[2]
0 (x, y), · · · , φ[n]0 (x, y)
ζn =ζ0(x), ζ
[1]0 (x), ζ[2]
0 (x), · · · , ζ[n]0 (x)
Cn =C0, C
[1]0 , C
[2]0 , · · · , C[n]
0
4,
Φ[m](x, y; q) =∂mΦ(x, y; q)
∂qm(18.30)
η[m](x; q) =∂mη(x; q)∂qm
(18.31)
Λ[m] =dmΛ(q)
dqm(18.32)
(18.15) (18.18) q m , 2 q = 0, ∇2φ
[m]0 (x, y) = 0, (x, y) ∈ Ω0 (18.33)
5\eEJlim
y→−∞∂φ
[m]0 (x, y)∂y
= 0 (18.34)
Ω0 = (x, y) | −∞ < x < +∞,−∞ < y ζ0(x)
Æ, EJ (18.16) (18.17) y = η(x; q) , y =
η(x; q) q. , y = η(x; q) , DmΦ(x, y; q)
Dqm=[∂
∂q+ η[1](x; q)
∂
∂y
]m
Φ(x, y; q) (18.35)
, η[1](x; q) (18.31) . & Dm/Dqm % ∂m/∂qm, ''. /,
DmΦ(x, y; q)Dqm
=Φ[m](x, y; q) + Rm[Φ(x, y; q),Λ(q)] (18.36)
JB?!"DmΦ(x, y; q)
Dqm=
[∂
∂p+ η[1](x; q)
∂
∂y
]m
Φ(x, y; q)
gk, ∂∂pd ∂
∂q, CAC, DD?EE. —— !
· 236 ·
, Rm , Φ[m](x, y; q) (18.30) . +, Λ(q) η(x; q), y = η(x; q),
Dmη(x; q)Dqm
=∂mη(x; q)∂qm
= η[m](x; q) (18.37)
DmΛ(q)Dqm
=dmΛ(q)
dqm= Λ[m](q) (18.38)
(18.31) (18.32) &.
(18.16) (18.17) q m , q = 0, y = ζ0(x) FÆEJ
m∑i=0
(m
i
)Di[Λ2(q)
]Dqi
∣∣∣∣∣q=0
Dm−iΦxx(x, y; q)Dqm−i
∣∣∣∣q=0
+gDmΦy(x, y; q)
Dqm
∣∣∣∣q=0
=m χmDm−1L [Φ(x, y; q),Λ(q)]
Dqm−1
∣∣∣∣q=0
+m 1 H1(x)Dm−1N [Φ(x, y; q),Λ(q)]
Dqm−1
∣∣∣∣q=0
(18.39)
ζ[m]0 (x) = m Wm(x, ζm−1,Cm−1) (18.40)
, χm (2.42) ,
Wm(x, ζm−1,Cm−1)=χm ζ[m−1]0 (x)
+2 H2(x)
[ζ[m−1]0 (x) − Dm−1Z [Φ(x, y; q),Λ(q)]
Dqm−1
∣∣∣∣q=0
]
(18.41)
(18.36) %4 (18.39) , y = ζ0(x) F,
C20
∂2φ[m]0 (x, y)∂x2
+ g∂φ
[m]0 (x, y)∂y
= Sm(x,φm−1, ζm,Cm) (18.42)
18 Æ ec#fgh · 237 ·
Sm(x,φm−1, ζm,Cm)
=
m χm
Dm−1L [Φ(x, y; q),Λ(q)]Dqm−1
+m 1 H1(x)Dm−1N [Φ(x, y; q),Λ(q)]
Dqm−1
−C20 Rm [Φxx(x, y; q),Λ(q)] − g Rm [Φy(x, y; q),Λ(q)]
−m∑
i=1
(m
i
)Di[Λ2(q)
]Dqi
Dm−i [Φxx(x, y; q)]Dqm−i
∣∣∣∣∣q=0
(18.43)
EJ (18.40) (18.42) d ζ0(x) . , .J ζ0(x) = 0 *8U .
E m" (18.33)EJ (18.40)(18.42). , Wm(x, ζm−1,Cm−1) % (m− 1) " . , ζ [m]
0 (x) Y[ (18.40) <'. 1, %4φ[m]
0 (x, y) C[m]0 , #
φ[m]0 (x, y) ), (18.33) EJ (18.34) (18.42). ,
$$, "#% C[m]0 .
(18.6) (18.8), (18.40) (18.42), $H1(x) H2(x)
H1(x) = cos(n1kx), H2(x) = cos(n2kx)
, n1n2 . ,
n1 = n2 = 0
$%H1(x) = H2(x) = 1 (18.44)
, (18.6) (18.8), Sm(x,φm−1, ζm,Cm)
Sm(x,φm−1, ζm,Cm) =m∑
n=1
bm,n(Cm) sin(nkx), m 1 (18.45)
, bm,n(Cm) ] Cm . , (18.42), bm,1(Cm) = 0 ,
φ[m]0 (x, y) & %, 66 (18.6). ia
H, &#$bm,1(Cm) = 0, m 1 (18.46)
%
αm(Cm−1) C[m]0 + βm(Cm−1) = 0
· 238 ·
, αm(Cm−1) βm(Cm−1) . , C[m]0 . ),
$$, (18.6) . , !'
φ[m]0 (x, y) =
m∑n=1
am,n exp(nky) sin(nkx) (18.47)
am,n =
bm,n(Cm)(kn)g − C2
0 (kn)2, 2 n m (18.48)
Æ, am,1 K. d H , 1
ζ[m]0 (0) − ζ
[m]0 (L/2) =
H, m = 1
0, m 2(18.49)
(18.49) %γm am,1 + δm = 0
, γm δm %. , am,1. (18.9) Æ φ0(x, y) A , (18.40) (18.49) ,
A = − gH
2 2 k C20
(18.50)
1,Rd C[m]0 am,1 Æ%." C
[m]0
am,1 , !' m " φ[m]0 (x, y). <>&Z. fl
, ζ[m]0 (x)C [m]
0 φ[m]0 (x, y).
m 1 , Dm/Dqm S9. y = η(x; q) Æ)b Φ(x, y; q) q = 0 FL[
Φ(x, y; q) =+∞∑m=0
DmΦ(x, y; q)Dqm
∣∣∣∣q=0
(qm
m!
)(18.51)
%, Φ(x, y; q) ( y = η(x; 0) FL[
Φ(x, y; q) =+∞∑n=0
+∞∑r=0
∂nΦ[r](x, y; q)∂yn
∣∣∣∣q=0
(qr
n! r!
)[η(x; q) − η(x, 0)]n (18.52)
0G Φ(x, y; q) , 2 (18.19) (18.22) +∞∑m=0
DmΦ(x, y; q)Dqm
∣∣∣∣q=0
(qm
m!
)
=+∞∑n=0
+∞∑r=0
∂nΦ[r](x, y; q)∂yn
∣∣∣∣q=0
(qr
n! r!
)[+∞∑s=1
(qs
s!
)ζ[s]0 (x)
]n
(18.53)
18 Æ ec#fgh · 239 ·
>>%, 20G q , m 1 Dm/Dqm . O&Z<. [24JUK Cheung[50] AF.
18.2 H I
41 2. , IÆ. O,
1 = 2 =
, ) C)b φ(x, y)d ζ(x) d_. , (18.27)(18.28) (18.29) &) . #,h, " <%. (18.27)(18.28) (18.29) M "
φ(x, y) ≈ φ0(x, y) +M∑
m=1
φ[m]0 (x, y)m!
(18.54)
ζ(x) ≈ ζ0(x) +M∑
m=1
ζ[m]0 (x)m!
(18.55)
C ≈ C0 +M∑
m=1
C[m]0
m!(18.56)
B;22P) C d H hN. Schwartz[128] '') M " (
C
C0
)2
≈M∑
j=0
aj (kH)2j (18.57)
, aj . $/7D" , Schwartz YL Æ, Æd_ (H/L)max = 0.141 18. 87'') M "
C
C0≈
M∑j=0
bj (kH)2j (18.58)
, bj . , 'Æ kH , CD39:, C/C0 ∼ 9 (4 24 A 3.5.1 #) . 8/, =#), )b φ(x, y) d ζ(x) .
( - 7D" (4 41 A 3.5.2 #), <YL))
· 240 ·
. 8/, ) [κ, κ] ( - 7D"
C
C0≈
1 +κ(κ+1)/2∑
n=1Γ2κ,n (kH)2n
1 +κ(κ+1)/2∑
n=1∆2κ,n (kH)2n
(18.59)
, Γ2κ,j∆2κ,j . Æ, [κ, κ] ( - 7D" (18.59) O(H2κ2+2κ), $0 Schwartz[128] ''Æ [κ, κ]/7D" Æ O(H2κ) #.
18.1'') C2/C20 `( –7D" (18.59),5 Schwartz[128]
O(H116)0G. d_ H/L 0.10 , [6, 6]( -7D" Schwartz
2=. , EF) O(H82), 0 Schwartz ?. C2 [10, 10] ( - 7D" O(H220), d_ H/L > 0.12 ,
Schwartz +,W. ( - 7D" !D, Æd_@T,
20 22 " 6!". ($ 20 " , 5( - 7D" Æ.
> 18.1 C2/C20 9 [κ, κ] :>? - @?;<Z Schwartz[128] rs9U\
H/L Schwartz , κ = 6 κ = 8 κ = 10 κ = 11
0.040 1.015 92 1.015 92 1.015 92 1.015 92 1.015 92
0.070 1.049 55 1.049 55 1.049 55 1.049 55 1.049 55
0.100 1.103 67 1.103 67 1.103 67 1.103 67 1.103 67
0.120 1.151 82 1.151 90 1.151 84 1.151 82 1.151 81
0.130 1.178 20 1.178 65 1.178 34 1.178 21 1.178 21
0.135 1.189 96 1.191 48 1.190 61 1.190 03 1.190 03
0.140 1.193 0 1.201 50 1.198 33 1.193 69 1.193 85
QO*Liao, Cheung. Homotopy analysis of nonlinear progressive waves in deep water. Journal
of Engineering Mathematics. Kluwer Academic Publishers, 2003, 45(2): 105∼116 (Kluwer Academic
Publishers TP c©(2003) Kluwer Academic Publishers, R Kluwer Academic Publishers TUR)
18.2 '' C/C0 [10, 10] [11, 11] ( - 7D" Longuet-Higgins 0G. H/L 0.121 921 , ( - 7D" Longuet-
Higgins . H/L 0.131 249 , [10, 10] ( - 7D" '. '"B<d, Æ)0G, ! 18.1 . ( 1 Longuet-Higgins =d_ H/L = 0.138 712 F''),
=$)d_Æ72. )( -7D" d_ H/L 0.14
Longuet-Higgins'! , #;Æd_ ($'"B<dd JB?@! 1, CAC, DD?EE. —— !
18 Æ ec#fgh · 241 ·
^), $;D. H/L = 0.14 , Schwartz[128] ''Æ) A0c?.
!9i/B;fÆ,-2dÆ6i!^e, ÆB;B<d^*0_Æ. B<d^bÆ, (. $( 1B; Chen Saffman[132] 8/ÆH/L ≈ 0.13
-2dÆ g/f>Æ.
> 18.2 C/C0 9 [κ, κ] :>? - @?;<Z Longuet-Higgins[129] rs9U\H/L Longuet-Higgins , κ = 10 κ = 11
0 1.000 00 1.000 00 1.000 00
0.045 266 1.010 16 1.010 16 1.010 16
0.064 351 1.020 65 1.020 65 1.020 65
0.079 187 1.031 43 1.031 43 1.031 43
0.091 809 1.042 47 1.042 47 1.042 47
0.102 959 1.053 66 1.053 66 1.053 66
0.108 093 1.059 26 1.059 26 1.059 26
0.112 962 1.064 82 1.064 82 1.064 82
0.117 572 1.070 29 1.070 29 1.070 29
0.121 921 1.075 58 1.075 58 1.075 58
0.125 993 1.080 59 1.080 60 1.080 60
0.129 760 1.085 16 1.085 17 1.085 17
0.133 178 1.089 04 1.089 06 1.089 06
0.136 178 1.091 84 1.091 88 1.091 88
0.136 723 1.092 22 1.092 28 1.092 28
0.137 249 1.092 55 1.092 60 1.092 60
0.137 755 1.092 75 1.092 84 1.092 85
0.138 242 1.092 90 1.093 00 1.093 01
0.138 712 1.092 95 1.093 06 1.093 08
0.139 170 1.092 91 1.093 02 1.093 05
0.139 610 1.092 79 1.092 85 1.092 90
0.140 060 1.092 58 1.092 50 1.092 58
0.140 530 1.092 40 1.091 89 1.092 02
0.141 100 1.092 30 1.090 66 1.090 89
QO*Liao, Cheung. Homotopy analysis of nonlinear progressive waves in deep water. Journal
of Engineering Mathematics. Kluwer Academic Publishers, 2003, 45(2):105∼116 (Kluwer Academic
Publishers TP c©(2003) Kluwer Academic Publishers, R Kluwer Academic Publishers TUR)
· 242 ·
" 18.1 ec#fghgda C/C0 >hb H/L &c"12*[10,10] F - QQ-%,L+2: [11,11] F - QQ-%; NX0*Schwartz[128] ,;
+X0*Longuet-Higgins[129] ,QO*Liao, Cheung. Homotopy analysis of nonlinear progressive waves in deep water. Journal of
Engineering Mathematics. Kluwer Academic Publishers, 2003, 45(2):105∼116 (Kluwer Academic
Publishers TP c©2003 Kluwer Academic Publishers, R Kluwer Academic Publishers TUR)
m t u v
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wxy 2 Mathematica nz
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coV ′(t) + V 2(t) = 1,
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<<Graphics‘Graphics‘;
(************************************************************************
orospq?————————————————————————————————————–
BASE = 1 ndpoq (2.50)
BASE = 2 ndppq (2.62)
BASE = 3 ndpqq (2.70)
BASE = 4 ndpoqrqqr (2.82)
************************************************************************)
BASE = 1;
(************************************************************************
56kQ chi[ m ]
dkQ7. chi[ m ] 56 (2.42)
*************************************************************************)
chi[ m ] := If[ m <= 1, 0, 1 ];
(************************************************************************
56kQ GetR[ m ]
dkQ7. R[ m ] 56 (2.43)
*************************************************************************)
GetR[ m ] := Module[ temp, n,temp[ 1 ] = D[ v[ m-1 ], x ];
temp[ 2 ] = Sum[ v[ n ] * v[ m-1-n ], n, 0, m-1 ];
qlG 2 Æ Mathematica ps · 251 ·
temp[ 3 ] = temp[ 1 ] + temp[ 2 ] + chi[ m ] -1;
R[ m ] = temp[ 3 ]//Expand;
];
(************************************************************************
56kQ GetRHS[ m ]
dkQ7. (2.39) trq*************************************************************************)
GetRHS[ m ] := Module[ ,GetR[ m ];
RHS[ m ] = Expand[ hbar * H[ x ] * R[ m ] ];
];
(************************************************************************
566'q H[ x ]
*************************************************************************)
H[ x ]:=Module[ temp,If[ BASE == 1 || BASE == 4, temp = 1 ];
If[ BASE == 2, temp = 1/x ];
If[ BASE == 3, temp = Exp[ -x ] ];
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];
(************************************************************************
56kQ GetInitial
dkQ7. V(t) 9A:;,*************************************************************************)
GetInitial = Module[ ,If[ BASE == 1, v[ 0 ] = x ];
If[ BASE == 2, v[ 0 ] = 1 - 1/x ];
If[ BASE == 3 || BASE == 4, v[ 0 ] = 1 - Exp[ -x ] ];
Print[ ‘‘ initial guess = ’’, v[ 0 ] ];
];
(************************************************************************
566'27.( L
dkQ7.6'27.(56 (2.23)
*************************************************************************)
L[ f ] := Module[ ,If[ BASE == 1, gamma[ 1 ] = 1; gamma[ 2 ] = 0 ];
If[ BASE == 2, gamma[ 1 ] = x; gamma[ 2 ] = 1 ];
If[ BASE == 3 || BASE == 4, gamma[ 1 ] = 1; gamma[ 2 ] = 1 ];
Expand[ gamma[ 1 ] * D[ f,x ] + gamma[ 2 ] * f ]
];
· 252 ·
(************************************************************************
56kQ Linv[ f ]
dkQ566'27.( (2.23) m.( L−1
s: n75q f, t!on, u:
L[u] = f
*************************************************************************)
Linv[ f ] := Module[ temp, EQ, u, solution,EQ = L[ u[ x ] ] - f;
temp = DSolve[ EQ == 0, u[ x ], x ];
solution = temp[ [ 1, 1, 2 ] ] /. C[ ] -> 0;
Expand[ solution ]
];
(************************************************************************
m.( L−1 7uLinv[ a * f + b * g ] := a * Linv[ f ] + b * Linv[ g ]
*************************************************************************)
Linv[ p Plus ] := Map[ Linv, p ];
Linv[ c * f ] := c * Linv[ f ] /; FreeQ[ c, x ];
(************************************************************************
56kQ GetSPECIAL[ m ]
dkQ7.# (2.39) D,*************************************************************************)
GetSPECIAL[ m ] := Module[ temp,temp = Expand[ RHS[ m ] ];
SPECIAL = Linv[ temp ];
];
(************************************************************************
56kQ GetCOMMON[ m ]
dkQ7.# (2.39) p,*************************************************************************)
GetCOMMON[ m ] := Module[ temp, u, solution,temp = DSolve[ L[ u[ x ] ] == 0, u[ x ], x ];
solution = temp[ [ 1, 1, 2 ] ];
COMMON = SPECIAL + chi[ m ] * v[ m-1 ] + solution;
];
(************************************************************************
56kQ Getv[ m ]
dkQ7.# (2.39) Æ (2.40) ,
qlG 2 Æ Mathematica ps · 253 ·
*************************************************************************)
Getv[ m ] := Module[ temp, EQ, x0, res,If[ BASE == 1 || BASE == 3 || BASE == 4, x0 = 0 ];
If[ BASE == 2, x0 = 1 ];
EQ = Expand[ COMMON/.x->x0 ];
temp = Solve[ EQ == 0, C[ 1 ] ];
res = COMMON/.temp[ [ 1, 1 ] ];
v[ m ] = res//Expand;
];
(************************************************************************
56kQ HP[ F,m,n ]
dkQ7.
F ≈+∞∑k=0
fk
[ m,n ] F - QQ-%, ut §2.3.7 Æ §3.5.2************************************************************************)
hp[ F , m , n ] := Block[ i, k, dF, temp, q,dF[ 0 ] = F[ 0 ];
For[ k = 1, k< = m + n, k = k + 1, dF[ k ] = Expand[ F[ k ] - F[ k-1 ] ] ];
temp = dF[ 0 ] + Sum[ dF[ i ] * q∧i, i, 1, m + n ];
Pade[ temp, q, 0, m, n ]/.q->1
];
(************************************************************************
56q (2.58)
*************************************************************************)
mu0[ m , n , h ] := (-h)∧n * Sum[ Binomial[ n - 1 + j, j ]*(1+h)∧j, j, 0, m-n ];
(************************************************************************
56q (2.85)
*************************************************************************)
sigma0[ m , n , k , h ] := ( mu0[ m, n+k, h ] + mu0[ m, n+k-1, h ] )/2;
(************************************************************************
56q (2.91)
*************************************************************************)
mu[ m , n , alpha , h ] := If[ n>m, 0, (-h)∧(n - alpha)
*(1+Sum[ (-1)∧j * Binomial[ alpha - n, j ] * (1 + h)∧j, j, 1, m - n ]) ];
(************************************************************************
56q (2.96)
· 254 ·
*************************************************************************)
sigma[m , n , k , alpha , h ] := (mu[m, n+k, alpha, h]+mu[m, n+k-1, alpha, h])/2;
(************************************************************************
vkQ*************************************************************************)
ham[ m0 , m1 ] := Module[ temp, k, variable,For[ k = Max[ 1, m0 ], k <= m1, k = k + 1,
Print[ ‘‘k = ’’, k ];
GetRHS[ k ];
GetSPECIAL[ k ];
GetCOMMON[ k ];
Getv[ k ];
If[ BASE == 1 || BASE == 3 || BASE == 4, variable = t ];
If[ BASE == 2, variable = 1 + t ];
If[ k == 1, V[ 0 ] = v[ 0 ] /. x -> variable ];
V[ k ] = Simplify[ V[ k-1 ] + v[ k ] /. x -> variable ];
Vtt0[ k ] = D[ V[ k ], t, 2 ] /. t ->0 //Expand;
Vttt0[ k ] = D[ V[ k ], t, 3 ] /. t ->0 //Expand;
If[PRN == 1,
Print[‘‘V’’(0)=’’, N[Vtt0[k],20],‘‘delta=’’,N[Vtt0[k]-chi[ k ]*Vtt0[k-1],20]];
Print[‘‘V’’(0) = ’’, N[Vttt0[k],20],‘‘ delta = ’’,N[Vttt0[k] - chi[k] * Vttt0[k-1],20]];
];
];
Print[ ‘‘ Successful !’’ ];
];
(************************************************************************
osqUtovq erpn7?
1 ndp ‘‘o’’
0 ndp ‘‘s’’
*************************************************************************)
PRN = 1;
(************************************************************************
ku@*************************************************************************)
hbar = h;
(************************************************************************
qUvvq*************************************************************************)
Print[ ‘‘The main code is ham[ N start,N end ]’’ ];
Print[ ‘‘BASE = ’’, BASE ];
qlG 2 Æ Mathematica ps · 255 ·
Print[ ‘‘Auxiliary function = ’’, H[ x ] ];
Print[ ‘‘Initial guess = ’’, v[ 0 ] ];
Print[ ‘‘PRN = ’’, PRN ];
Print[ ‘‘hbar = ’’, hbar ];
(* R 5 -% *)
ham[ 1, 5 ];
(* R V (t) [ 1,1 ] F - QQ-% *)
hp[ V, 1, 1 ]//Simplify;
wx 6s7 Mathematica nz
(************************************************************************
cov′′ + ε(v − v3) = 0
tugjv(0) = v(π) = 0
ruj"pq sin[(2m + 1)κξ] | m 0, κ > 1
gkkappa = 1 ndpw 6 wkappa > 1 ndpw 7 w
———————————————————————————————————
!wo56kappa => Ex, gp5pq (7.4) 56a[0] => (7.5) k A 9A:;@u[0] => (7.6) , u(x) "9A:;,u[k] => k (7.22) Æ (7.23) ,U[k] => u(x) k -%A[k] => A k -%
ux[k] => u′k(x)
uxx[k] => u′′k(x)
uu[k] =>k∑
n=0
un(x) uk−n(x)
aa[n] =>n∑
k=0
ak an−k
R[k] => !" (7.24)
RHS[k] => k (7.22) trqSPECIAL => k (7.22) D,
dhde !. jfh E-mail ! [email protected],
kgijp* http://numericaltank.sjtu.edu.cn/code.htm
*************************************************************************)
<<Calculus‘Pade‘;
<<Graphics‘Graphics‘;
(************************************************************************
56 u(x) 9A:;,Æ A 9A:;@
ql9 6>7 Æ Mathematica ps · 257 ·
************************************************************************)
GetInitial := Module[ ,
u[ 0 ] = Sin[ kappa * x ];
U[ 0 ] = u[ 0 ];
a[ 0 ] = 2/Sqrt[ 3 ] * Sqrt[ 1 - kappa∧2/epsilon ];
A[ 0 ] = a[ 0 ];
];
(************************************************************************
566'q************************************************************************)
H[ x ] := 1;
(************************************************************************
56q chi[ k ]
************************************************************************)
chi[ k ] := If[ k<=1 , 0 , 1 ];
(************************************************************************
566'27.( L
************************************************************************)
L[ f ] := Module[ temp ,
Expand[ D[ f , x , 2 ] + kappa∧2 * f ]
];
(************************************************************************
566'27.(m.( L−1
************************************************************************)
Linv[ Sin[ m * x ] ] := Sin[ m * x ]/(1 - m∧2);Linv[ Cos[ m * x ] ] := Cos[ m * x ]/(1 - m∧2);Linv[ c ] := c /; FreeQ[ c , x ];
(************************************************************************
m.( L−1 7uLinv[ a * f + b * g ] := a * Linv[ f ] + b * Linv[ g ] L
************************************************************************)
Linv[ p Plus ] := Map[ Linv , p ];
Linv[ c * f ] := c * Linv[ f ] /; FreeQ[ c , x ];
(************************************************************************
56kQ GetR[ k ]
dkQ7. R[ k ] 56 (7.24)
************************************************************************)
GetR[ k ] := Module[ temp ,
· 258 ·
temp = Sum[ aa[ n ] * uuu[ k - 1 - n ] , n , 0 , k - 1 ];
R[ k ] = TrigReduce[ uxx[ k - 1 ] + epsilon * (u[ k - 1 ] - temp) ]//Expand;
];
(************************************************************************
56kQ GetRHS[ k ]
dkQ7. (7.22) trq************************************************************************)
GetRHS[ k ] := Module[ temp ,
GetR[ k ];
RHS[ k ] = Expand[ TrigReduce[ hbar * H[ x ] * R[ k ] ] ];
];
(************************************************************************
56kQ CheckRHS[ k ]
dkQvm RHS[ k ] oswx Sin[ kappa * x ] q************************************************************************)
CheckRHS[ k ] := Module[ temp , C1 ,
temp[ 0 ] = TrigReduce[ RHS[ k ] ]//Expand;
C1 = Coefficient[ temp[ 0 ] , Sin[ kappa * x ] ];
temp[ 1 ] = temp[ 0 ] - C1 * Sin[ kappa * x ];
RHS[ k ] = Expand[ temp[ 1 ] ];
temp[ 2 ] = RHS[ k ] /. Sin[ kappa * x ] - >0;
RHS[ k ] = Expand[ temp[ 2 ] ];
];
(************************************************************************
56kQ GetuAll[ k ]
dkQ7.rxr u(x) v"q************************************************************************)
GetuAll[ k ] := Module[ ,
uu[ k ] = Sum[ u[ j ] * u[ k - j ] , j , 0 , k ]//Expand;
uuu[ k ] = Sum[ u[ j ] * uu[ k - j ] , j , 0 , k ]//Expand;
uxx[ k ] = Expand[ D[ u[ k ] , x , 2 ] ];
];
(************************************************************************
56kQ Geta[ k ]
dkQq, (7.26), R a[ k ]
************************************************************************)
Geta[ k ] := Mpdule[ temp , eq ,
If[ k ==1 , Print[ ‘‘ a[ 0 ] is given by (7.34)’’ ] ];
If[ k > 1 ,
temp[ 0 ] = Expand[ RHS[ k ] ];
ql9 6>7 Æ Mathematica ps · 259 ·
temp[ 1 ] = TrigReduce[ temp[ 0 ] ]//Expand;
eq = Coefficient[ RHS[ k ] , Sin[ kappa * x ] ];
temp[ 0 ] = Solve[ eq == 0 , a[ k - 1 ] ];
a[ k - 1 ] = temp[ 0 ][ [ 1 , 1 , 2 ] ]//Expand;
];
];
(************************************************************************
56kQ Getaa[ k ]
************************************************************************)
Getaa[ k ] : =Module[ ,
aa[ k ] = Expand[ Sum[ a[ j ] * a[ k - j ] , j , 0 , k ] ];
];
(************************************************************************
56kQ GetSPECIAL[ k ]
dkQ7. (7.22) D,************************************************************************)
GetSPECIAL[ k ] := Module[ temp ,
temp = TrigReduce[ RHS[ k ] ]//Expand;
SPECIAL = Linv[ temp ];
];
(************************************************************************
56kQ Getu[ k ]
dkQ7. (7.22) Æ (7.23) ,************************************************************************)
Getu[ k ] := Module[ temp , C1 ,
temp = SPECIAL + chi[ k ] * u[ k - 1 ];
C1 = - temp /. x - >Pi/2/kappa;
u[ k ] = Expand[ temp + C1 * Sin[ kappa * x ] ];
];
(************************************************************************
56kQ HP[ F,m,n ]
dkQ7.
F ≈+∞∑k=0
fk
[ m,n ] F - QQ-%, ut 2.3.7 xÆ 3.5.2 x***********************************************************************)
hp[ F , m , n ] := Block[ i , k , dF , temp , q ,
dF[ 0 ] = F[ 0 ];
For[ k = 1 , k <= m + n , k = k + 1 , dF[ k ] = Expand[ F[ k ] - F[ k - 1 ] ] ];
· 260 ·
temp = dF[ 0 ] + Sum[ dF[ i ] * q∧i , i , 1 , m + n ];
Pade[ temp , q , 0 , m , n ]/.q - >1
];
(************************************************************************
vkQ************************************************************************)
ham[ m0 , m1 ] := Module[ temp , k , n ,
For[ k=Max[ 1 , m0 ] , k <= m1 , k = k + 1 ,
Print[ ‘‘k = ’’ , k ];
Getaa[ k - 1 ];
GetuAll[ k - 1 ];
GetRHS[ k ];
Geta[ k ];
A[ k - 1 ] = Expand[ Sum[ a[ n ] , n , 0 , k - 1 ] ];
Print[‘‘ A = ’’, A[ k - 1 ], ‘‘ increment= ’’, a[ k - 1 ] - chi[ k - 1 ]*a[ k - 2 ]];
CheckRHS[ k ];
GetSPECIAL[ k ];
Getu[ k ];
U[ k ] = U[ k - 1 ] + u[ k ];
];
Print[ ‘‘Successful !’’ ];
];
(************************************************************************
R u(x) 9A:;,Æ A 9A:;@************************************************************************)
GetInitial;
(************************************************************************
ku@************************************************************************)
kappa = 2;
epsilon = N[ 10 , 50 ];
hbar = - 1;
(************************************************************************
qUvvq************************************************************************)
Print[ ‘‘ kappa = ’’ , kappa ];
Print[ ‘‘ epsilon = ’’ , epsilon ];
Print[ ‘‘ hbar = ’’ , hbar ];
Print[ ‘‘ a[ 0 ] = ’’ , a[ 0 ] ];
Print[ ‘‘ u[ 0 ] = ’’ , u[ 0 ] ];
ql9 6>7 Æ Mathematica ps · 261 ·
Print[ ‘‘ H(x) = ’’ , H[ x ] ];
(* R 10 -% *)
ham[ 1, 10 ];
(* R u(x) [ 3,3 ] F - QQ-% *)
hp[ U, 3, 3 ]//Simplify;
wx 8 Mathematica nz
(************************************************************************
cou′′(x) + λ u(x) + ε u3(x) = 0
tugju(0) = u(1) = 0
n75 ε, sqytDy@ λn ÆDyq un(x), rzu′′
n(x) + λn un(x) + ε u3n(x) = 0
Ætugjun(0) = un(1) = 0.
dhde !. jfh E-mail ! [email protected]
kgijp* http://numericaltank.sjtu.edu.cn/code.htm
************************************************************************)
<<Calculus‘Pade‘;
<<Graphics‘Graphics‘;
(************************************************************************
orx3yy———————————————————————————————————
NORMALIZATION = 1 ndpx3yy (8.38)
NORMALIZATION = 2 ndpx3yy (8.39)
************************************************************************)
NORMALIZATION = 1;
(************************************************************************
56 un(x) 9A:;,(************************************************************************)
u[ 0 ] = Sqrt[ 2 ] * Sin[ n * Pi * x ];
U[ 0 ] = u[ 0 ];
(************************************************************************
566'q H(x)
************************************************************************)
H[ x ] := 1;
(************************************************************************
56q chi[ k ]
************************************************************************)
chi[ k ] := If[ k <= 1 , 0 , 1 ];
qlr 8 Æ Mathematica ps · 263 ·
(************************************************************************
566'27.( L
************************************************************************)
L[ f ] := Module[ temp ,
Expand[ D[ f , x , 2 ] + (n * Pi)∧2 * f ] ];
(************************************************************************
566'27.(m.( L−1
************************************************************************)
Linv[ Sin[ m * x ] ] := Sin[ m * x ]/(n∧2 * Pi∧2 - m∧2);Linv[ Cos[ m * x ] ] := Cos[ m * x ]/(n∧2 * Pi∧2 - m∧2);
(************************************************************************
m.( L−1 7uLinv[ a * f + b * g ] := a * Linv[ f ] + b * Linv[ g ] L
************************************************************************)
Linv[ p Plus ] := Map[ Linv , p ];
Linv[ c * f ] := c * Linv[ f ] /; FreeQ[ c , x ];
Linv[ c ] := c /; FreeQ[ c , x ];
(************************************************************************
56kQ GetR[ k ]
dkQ7. R[ k ] 56 (8.25)
************************************************************************)
GetR[ k ] := Module[ temp , n ,
temp[ 1 ] = uxx[ k - 1 ];
temp[ 2 ] = lambdau[ k - 1 ];
temp[ 3 ] = uuu[ k - 1 ];
temp[ 4 ] = TrigReduce[ temp[ 1 ] + temp[ 2 ] + epsilon * temp[ 3 ] ];
R[ k ] = Expand[ temp[ 4 ] ];
];
(************************************************************************
56kQ GetRHS[ k ]
dkQ7. (8.23) trq************************************************************************)
GetRHS[ k ] := Module[ temp ,
GetR[ k ];
RHS[ k ]= Expand[ TrigReduce[ hbar * H[ x ] * R[ k ] ] ];
];
(************************************************************************
56kQ GetuAll[ k ]
dkQ7.rxr un(x) v"q
· 264 ·
************************************************************************)
GetuAll[ k ] := Module[ temp ,
uxx[ k ] = Expand[ D[ u[ k ] , x , 2 ] ];
temp = Sum[ u[ j ] * u[ k - j ] , j , 0 , k ];
uu[ k ] = TrigReduce[ Expand[ temp ] ];
temp = Sum[ u[ j ] * uu[ k - j ] , j , 0 , k ]//Expand;
uuu[ k ] = TrigReduce[ temp ];
lambdau[ k ] = Sum[ lambda[ j ] * u[ k - j ] , j , 0 , k ]//Expand;
];
(************************************************************************
56kQ Getlambda[ k ]
dkQq, λn,k−1
************************************************************************)
Getlambda[ k ] := Module[ temp , eq ,
temp[ 1 ] = TrigReduce[ RHS[ k ] ];
temp[ 2 ] = Expand[ temp[ 1 ] ];
eq = Coefficient[ temp[ 2 ] , Sin[ n * Pi * x ] ];
temp[ 3 ] = Solve[ eq == 0 , lambda[ k - 1 ] ];
lambda[ k - 1 ] = temp[ 3 ][ [ 1 , 1 , 2 ] ];
];
(************************************************************************
56kQ CheckRHS[ k ]
dkQvm RHS[ k ] oswx Sin[ n * Pi * x ] q************************************************************************)
CheckRHS[ k ] := Module[ temp , C1 ,
temp[ 0 ] = Expand[ RHS[ k ] ];
C1 = Coefficient[ temp[ 0 ] , Sin[ n * Pi * x ] ];
temp[ 1 ] = temp[ 0 ] - C1 * Sin[ n * Pi * x ];
RHS[ k ] = Expand[ temp[ 1 ] ];
temp[ 0 ] = RHS[ k ] /. Sin[ n * Pi * x ] ->0;
RHS[ k ] = Expand[ temp[ 0 ] ];
];
(************************************************************************
56kQ GetuSpecial
dkQ7. (8.23) D,************************************************************************)
GetuSpecial[ k ] := Module[ temp ,
temp[ 0 ] = Expand[ RHS[ k ] ];
temp[ 1 ] = Coefficient[ temp[ 0 ] , Sin[ n * Pi * x ] ];
temp[ 2 ] = Expand[ temp[ 1 ] * temp[ 1 ] ];
If[ temp[ 2 ] == 0 , , Print[ ‘‘ GetuSpecial: something is wrong ! ’’ ] ];
qlr 8 Æ Mathematica ps · 265 ·
uSpecial = Linv[ temp[ 0 ] ];
];
(************************************************************************
56kQ Getu[ k ]
dkQ7. (8.23) Æ (8.24) ,************************************************************************)
Getu[ k ] := Module[ eq , temp , C1 , w , j , alpha , beta ,
temp[ 1 ] = uSpecial + chi[ k ] * u[ k - 1 ];
u[ k ] = Expand[ temp[ 1 ] ];
w = Expand[ TrigReduce[ Sum[ u[ j ] , j , 0 , k ] ] ];
alpha = Coefficient[ w , Sin[ n * Pi * x ] ]/2;
beta = TrigReduce[ w∧2 - 1 ]/.Cos[ j ] ->0;
temp[ 2 ] = Simplify[ 4 * alpha∧2 - 2 * beta ];
If[ NORMALIZATION == 1 , C1 = - 2 * alpha + Sqrt[ temp[ 2 ] ] ];
If[ NORMALIZATION == 2 , C1 = - 2 * alpha - Sqrt[ temp[ 2 ] ] ];
u[ k ] = temp[ 1 ] + C1 * Sin[ n * Pi * x ];
];
(************************************************************************
56kQ HP[ F,m,n ]
dkQ7.
F ≈+∞∑k=0
fk
[ m,n ] F - QQ-%, ut 2.3.7 xÆ 3.5.2 x************************************************************************)
hp[ F , m , n ] := Block[ i , k , dF , temp , q ,
dF[ 0 ] = F[ 0 ];
For[ k = 1 , k< = m + n , k = k + 1 , dF[ k ] = Expand[ F[ k ] - F[ k - 1 ] ] ];
temp = dF[ 0 ] + Sum[ dF[ i ] * q∧i , i , 1 , m + n ];
Pade[ temp , q , 0 , m , n ]/.q - >1
];
(************************************************************************
vkQ************************************************************************)
ham[ m0 , m1 ] := Module[ temp , k , j ,
For[ k = Max[ 1 , m0 ] , k <= m1 , k = k + 1 ,
Print[ ‘‘ k = ’’ , k ];
GetuAll[ k - 1 ];
GetRHS[ k ];
Getlambda[ k ];
lambda[ k - 1 ] = Expand[ lambda[ k - 1 ] ];
· 266 ·
LAMBDA[ k - 1 ] = Expand[ Sum[ lambda[ j ] , j , 0 , k - 1 ] ];
Print[k-1, ‘‘th approx. of lambda/(n*Pi)∧2 = ’’,N[LAMBDA[ k - 1 ]/n∧2/Pi∧2,30] ];
CheckRHS[ k ];
GetuSpecial[ k ];
Getu[ k ];
U[ k ] = U[ k - 1 ] + u[ k ];
];
Print[ ‘‘Successful !’’ ];
];
(************************************************************************
ku@************************************************************************)
n = 1;
epsilon = N[ - 50 , 100 ];
hbar = - 1/2;
(************************************************************************
qUvvq************************************************************************)
Print[ ‘‘ n = ’’ , n ];
Print[ ‘‘ epsilon = ’’ , epsilon ];
Print[ ‘‘ H(x) = ’’ , H[ x ] ];
Print[ ‘‘ u0 = ’’ , u[ 0 ] ];
Print[ ‘‘ hbar = ’’ , hbar ];
Print[ ‘‘ NORMALIZATION = ’’ , NORMALIZATION ];
(* R 10 -% *)
ham[ 1 , 10 ];
(* R λn/(nπ)2 [ 3,3 ] F - QQ-% *)
hp[ LAMBDA , 3 , 3 ]/n∧2/Pi∧2
wx| 9 Mathematica nz
(************************************************************************q, Thomas-Feimi J(kz (9.6)
λ3 (τ − 1)[u′′(τ)]2 − u3(τ) = 0
rztugj (9.7)
u(1) = 1, u(+∞) = 0
gkτ = 1 + λ x
snQOJA (9.1) Æ (9.2).
dhde !. jfh E-mail ! [email protected]
kgijp* http://numericaltank.sjtu.edu.cn/code.htm************************************************************************)
<<Calculus‘Pade‘;
<<Graphics‘Graphics‘;
(************************************************************************
56 u(τ) 9A:;,************************************************************************)
u[ 0 ] = 1/t∧3;U[ 0 ] = u[ 0 ] /. t-> 1 + lambda * x;
Ux[ 0 ] = D[ U[ 0 ] , x ] /. x-> 0;
(************************************************************************
566'q H[ t ]
This module defines the auxiliary function H[ t ]
************************************************************************)
H[ t ] := t∧4;
(************************************************************************
566'27.( L
************************************************************************)
L[ f ] := t * D[ f , t , 2 ]/4 + D[ f , t ];
(************************************************************************
566'27.(m.( L−1
************************************************************************
Linv[ f ] := Block[ temp , G , g , solution ,
temp = DSolve[ L[ g[ t ] ] == f , g[ t ] , t ];
G = Expand[ temp[ [ 1 , 1 , 2 ] ] /. C[ 2 ] -> 0 ];
temp = Solve[ BC[ G ] == 0 , C[ 1 ] ];
· 268 ·
solution = G /. temp[ [ 1 ] ];
Expand[ solution ]
];
(************************************************************************
m.( L−1 7uLinv[ a * f + b * g ] := a * Linv[ f ] + b * Linv[ g ] L
************************************************************************)
Linv[ p Plus ] := Map[ Linv , p ];
Linv[ c * f ] := c * Linv[ f ] /; FreeQ[ c , t ];
(************************************************************************
56q chi[ m ]
************************************************************************)
chi[ m ] := If[ m <= 1 , 0 , 1 ];
(************************************************************************
56kQ GetuAll[ k ]
dkQ7.rxr u(τ) v"q************************************************************************)
GetuAll[ k ] := Module[ ,
utt[ k ] = Expand[ D[ u[ k ] , t , 2 ] ];
uu[ k ] = Expand[ Sum[ u[ i ] * u[ k - i ] , i , 0 , k ] ];
uuu[ k ] = Expand[ Sum[ uu[ i ] * u[ k - i ] , i , 0 , k ] ];
uttutt[ k ] = Expand[ Sum[ utt[ i ] * utt[ k - i ] , i , 0 , k ] ];
];
(************************************************************************
56kQ R[ k ]
dkQ7. R[ k ] 56 (9.24)
************************************************************************)
R[ k ] := Expand[ lambda∧3 * (t - 1) * uttutt[ k - 1 ] - uuu[ k - 1 ] ];
(************************************************************************
56tugj (9.23)
************************************************************************)
BC[ f ] := f /. t -> 1;
(************************************************************************
56kQ HP[ F,m,n ]
dkQ7.
F ≈+∞∑k=0
fk
qlt 9 Æ Mathematica ps · 269 ·
[ m,n ] F - QQ-%, ut 2.3.7 xÆ 3.5.2 x************************************************************************)
hp[ F , m , n ] := Block[ i , k , dF , temp , q ,
dF[ 0 ] = F[ 0 ];
For[ k = 1 , k < = m + n , k = k + 1 , dF[ k ] = Expand[ F[ k ] - F[ k - 1 ] ] ];
temp = dF[ 0 ] + Sum[ dF[ i ] * q∧i , i , 1 , m + n ];
Pade[ temp , q , 0 , m , n ] /. q - >1
];
(************************************************************************
vkQ(************************************************************************
ham[ begin , end ] := Block[ uSpecial ,
For[ k = begin , k <= end , k = k + 1 ,
Print[ ‘‘k = ’’ , k ];
GetuAll[ k - 1 ];
RHS = Expand[ hbar * H[ t ] * R[ k ] ];
uSpecial = Linv[ RHS ];
u[ k ] = Expand[ uSpecial + chi[ k ] * u[ k - 1 ] ];
U[ k ] = Expand[ U[ k - 1 ] + u[ k ] ] /. t -> 1 + lambda * x;
Ux[ k ] = D[ U[ k ] , x ] /. x -> 0;
Print[ ‘‘u’(0) = ’’ , N[ Ux[ k ] , 24 ] ];
];
Print[ ‘‘successful ’’ ];
];
(************************************************************************
ku@************************************************************************)
hbar = -1;
lambda = 1/4;
(************************************************************************
qUvvq************************************************************************)
Print[ ‘‘ hbar = ’’ , hbar ];
Print[ ‘‘ lambda = ’’ , lambda ];
(* R 10 -% *)
ham[ 1 , 10 ];
(* R u′(0) [ 3 , 3 ] F - QQ-% *)
hp[ Ux , 3 , 3 ]
270
B
134,146
38
211
71
204
231
! 231
" 231
232
232
#$%&'() 190
*+,-./ 173
C
01 54
23456 160
23457 13,46,76,87,98,135,147,174,181
192,205,232
D
89:; 231
<=> 39
<?> 38
@A 144
BCDE 109
B6 201
B@ 159
E
=FGHI 146
F
HJ@KL 4
MNL 12,64
OPQRSTU V6LWW11W
OPQRSTU V6KLWW59
XYTZ[UQ\ ]^_`LWW62W
Va 75,81,91,96
Vab 94
cd` 14,46,77,87,98,112,123,135,148,161,
175, 192,206,219
cd 14,46,77,87,98,112,123,135,148,161
174,192,202,219
cdIef 14,46,76,87,98,111,123,135,147,
160,174,192,205,218,232
G
"ghiK 15,48,78,88,100,113,125,136,
149,163,175,182,193,207,221,235
J
jkl 159
mno 111
mp 111,121
6n 53,76,98,111,135,147,160,174,181,192
6nqr 19
6stqr 19,54
uvV6L 71
K
wxyzf 181,184,185,192,198,218
L
+|+-K 231
~ 271
K 216
KL 6
ghiK 13,46,60,77,87,98,112,123,175
N
---K 216
P
41,55,239
Q
HI 134
i 6,13,98,112,123
49
R
211
]^_` 4
S
239
J@KL 3
17,50,79,90,102,115,128,166,176,
196,209,224
231
z 204
z 231
T
MN 15,47,60,77,88,112,124,136,149,
162,175,193,206,220,234
DE 97
DE7 97
5
- 42,55,69,94,104,131,185,213,
226,239
KL 6
--K 110
--qfG 110
W
V-¡VK 122
¢zV* 204
X
£¤qr 19,54,80,91
¥i¦ 204,216
¥i 173,216
¥6 190
hi 6,47
hi§ 15,47,148,235
Z
¨© '(:; 231
¨©ª@ 134,146
.)/ 173,190
AdomianB1o 60,64
AdomianV6L 5,11
AdomianV6KL 59
Duffingªf 75,94
Duffing oscillator 86
Falkner-Skan 203
Falkner-Skan '()/ 191
high-order deformation equation 66
homotopy 46
Lyapunov]^_`L 4,62
VolterraG 122
MNL 12,64
h«¬®x 24
h¯I 24,28,55,69,81,91,104,117,129
h°¬®± 28,55,69,81,104,129
