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. 74
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ff ?
f g ? g(g + ? g) ? ? ? = 2(1 + ?)g + ?(2g + ??),
? ? ? ? g
f h ? g(h + ? h) = 2(1 + ?)h + ?(2h + ? h), f? ? (g + ? g) = 0.
? ? ? ? ?
12? . ? 1 (f 2 + g 2 ) + (f ? ?g)f? ? ? + ? = 2(?f ? f? + ?g + (?2 + 1)f ),
?
? ?
2
?(f ? ?g)g + ?? = 2[g + g + ?f? ? (?2 + 1)?],
? ? ? g
?f h + 2(f ? ?g)h = h ? 4h + 4(?2 + 1)h, f? ? ?g = 0.
? ? ? ?
13? . ?f 2 ? g 2 + ?f f? ? ? 2 hf? ? 2? + ? ? = ?(?f + ? f ) + ? 3 (2f? + ? f ),
? ?
?
f g ? ? 2 hg = ?(?g + ??) + ? 3 (2g + ??),
? ? g ? g
? ? ? ? ? ?
f (?h + ? h) ? ? 2 hh ? ? 2 ? = h ? ? h + ? 2 h + ? 3 (2h + ? h),
?
f? ? ? h = 0.
? (2.3)
14? . f 2 ? g 2 + 2?f f? + 2? = 4(2f? + ? f ),
?
?
g + ? g ? 2f (g + ? g) = ?(2g + ??),
? ? ? g
? ? ? ? f + ? f? = 0.
? 1
2h + ? h + 2?f h = 4(h + ? h),
15? . f 2 ? g 2 + 2?f f? + 2? = 4(2f? + ? f ),
?
?
g + ? g ? 2f (g + ? g) = ?4(2g + ??),
? ? ? g
? ? ? ? f + ? f? +
? 1 1
2h + ? h + 2?f h + h = 4(h + ? h), = 0.
2

16? . ? 1 (f + ? f?) + hf? = f , ? 1 (g + ? g) + hg = g ,
? ? ??
2 2
? ? ?? ?
? 1 (h + ? h) + hh + ? = h, h = 0.
2
?
? 1 (f + ? f?) + hf? + ?g = f , ? 1 (g + ? g) + hg
?
17 . ? ? + g = g,
?
2 2
? ? ?? ?
? 1 (h + ? h) + hh + ? = h, h + 1 = 0.
2
18? . ? ? ? 1 (g ? ? g) + hg = g ,
2 (f ? ? f ) + hf = f ,
1
? ??
2
? ? ? ?
? 1 (h + ? h) + hh = h, h + 2 = 0.
2
19? . ? 1 (f + ? f?) + hf? = f , 1 (g ? ? g) + hg = g ,
? ? ??
2 2
? ? ?? ?
? 1 (h + ? h) + hh + ? = h, h + 1 = 0.
2

Equations 1? –19? in (2.3) correspond to that of ansatze in table 1; dot means diffe-
rentiation with respect to corresponding ?.
Equations 1? –10? (2.3) can easily be solved and their general solutions are as
follows:
1? . f = c1 , g = c2 , h = c3 , ? = ?(?)
(here and in what follows, c with a subscript denotes an arbitrary constant; ? = ?(?)
means that ? is an arbitrary differentiable function of ?).
310 W.I. Fushchych, W.M. Shtelen, S.L. Slavutsky
?
? c1 ec3 ? + c2 , c3 = 0,
2? . f = c3
?c ? + c , c3 = 0,
1 2
?
c4 c3 ?
?e + c5 , c3 = 0,
g = c3
?c ? + c , c3 = 0,
4 5

h = c3 , ? = c6 .
?
?
?c1 + c2 ec3 ? + c4 ? ? 1 ec3 ? ? c5 ?,
? c3 = 0,
c2 c3 c3
?
3. f= 3
?
?c + c ? + 1 c ? 3 + 1 c ? 2 ,
?1 c3 = 0,
2 4 5
6 2
?
? c4 ec3 ? + c5 , c3 = 0,
g = c3
?c ? + c , c3 = 0,
4 5

h = c3 , ? = c6 .
?
? c1 exp ? 1 c ? + c , c = 0,
3 2 3
4? . f = c3 2
?
c1 + c2 ?, c3 = 0,
?
? c4 exp ? 1 c ? + c , c = 0,
3 5 3
g = c3 2
?
c4 ? + c5 , c3 = 0, (2.4)
1
h = c3 , ? = ? + c6 .
2
?1
?? ? + c1 ec3 ? + c2 , c3 = 0,
?
c2
c3
? 3
5. f=
?1 2
? ? +c ?+c , c3 = 0,
1 2
?2
? c4 ec3 ? + c5 , c3 = 0,
g = c3
?c ? + c , c3 = 0,
4 5

h = c3 , ? = c6 .
?
?
?c1 + c2 exp ? 1 c3 ? + c5 ? ?
?
?
? 2 2c3
?
?
? c4 ? 1 1
6? . f = ?2 exp ? c3 ? ,
+ c3 = 0,
? c3 2 c3 2
?
?
?1
?
? 1 1
? c1 + c2 ? + c5 ? 2 + c4 ? 3 ,
? c3 = 0,
4 2 6
?
? c4 exp ? 1 c ? + c , c = 0,
3 5 3
g = c3 2
?
c4 ? + c5 , c3 = 0,
1
h = c3 , ? = ? + c6 .
2
Reduction and exact solutions of the Navier–Stokes equations 311
?
c? ??
?c exp
?1 + c2 ?
? , c = 0,
2(?2 + 1) 2(?2 + 1)c
7? . f=
? ?? 2
?
? + c1 ? + c2 , c = 0,
2[2(?2 + 1)]2
?
c?
?c exp + c4 , c = 0,
3
2(?2 + 1)
g=
?
c3 ? + c4 , c = 0,
?
h = ?f ? c, ? = + c5 .
2 + 1)
2(?
?
?? 2 ?2 + 1
? ?? c3
? ? c4 + ?? + c1 ?
?22 +
? 2c (? + 1)
? c c2 c c
?
?
? c?
8? . ? exp
f= + c2 , c = 0,
? ?2 + 1
?
?
?
?2 ?? 4
? c3 c4
?(? + 1)?1
? + ? 3 + ? 2 + c1 ? + c2 , c = 0,
24(?2 + 1)2 6 2
?
? ??? c?
?
? + c3 exp + c4 , c = 0,
c(?2 + 1) ?2 + 1
g=
? ?
? ? 2 + c3 ? + c4 ,
? c = 0,
2 + 1)
2(?
?
h = ?g ? c, ? = 2 + c6 .
? +1
c c2
9? . f = 2 , = c1 ? c + 2 , h = c3 ? c + c4 ,
? ?
?
c2 2c1 c2 c c2 + c2
?
? ?? + c5 , c = ?1, 0,
1 2
2(c+1)
?
? 2(c + 1) ? +
? 2
c 2?
?
? 2
? = c2 ln ? ? 2c1 c2 ? c2 + 1 + c5 , c = ?1,
?1
? 2? 2
?
?
?
?1 2 2
? 2
? c ? + 2c1 c2 ln ? ? c2 + c5 , c = 0.
21 2? 2
10? . f , g and ? are the same as in the previous case 9? ,
? 2
??
? c
? 2(2 ? c) + c3 ? + c4 , c = 2, 0,
?
?
?
?
? c3 ln ? + c4 ,
h= c = 0,
?4
?
?2

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