<<

. 62
( 122 .)



>>

By the term antireduction for a partial differential equation (PDE) we mean the
construction of an ansatz which transforms the PDE to a system of differential equati-
ons for several unknown differentiable functions. As a rule, such procedure reduces
the PDE under consideration to a system of PDE with fewer numbers of independent
variables and greater number of dependent variables [1–4].
Antireduction of the nonlinear acoustics equation

ux0 x1 ? (ux1 u)x1 ? ux2 x2 ? ux3 x3 = 0 (1)

is carried out in the paper [2] with the use of the ansatz
1 1
x1 ?1 (x0 , x2 , x3 ) ? x2 ?2 (x0 , x2 , x3 ) + ?3 (x0 , x2 , x3 ). (2)
u=
61
3
In [3] antireduction of the equation for short waves in gas dynamics

2ux0 x1 ? 2(2x1 + ux1 )ux1 x1 + ux2 x2 + 2?ux1 = 0 (3)

is carried out via the following ansatz:
3/2
u = x1 ?1 + x2 ?2 + x1 ?3 + ?4 , (4)
?i = ?i (x0 , x2 ).
1

Ansatzes (2), (4) reduce equations (1), (3) to system of PDE for three and four
functions, respectively.
In the present paper we adduce some new results on antireduction for the nonlinear
heat equations of the form

(5)
ut = a(u)ux + F (u).
x

The antireduction of equation (5) is performed by means of the ansatz

(6)
h t, x, u, ?1 (?), ?2 (?), . . . , ?N (?) = 0

where ? = ?(t, x, u) is a new independent variable. Ansatz (6) reduces equation (5) to
a system of ordinary differential equations (ODE) for the unknown functions ?i (?),
i = 1, N .
J. Nonlinear Math. Phys., 1994, 1, 1, P. 60–64.
248 W.I. Fushchych, R.Z. Zhdanov

Below we list, without derivation, explicit forms of a(u) and F (u), such that
equation (5) admits an antireduction of the form (6). For each case the reduced ODE
are given.
?1
? ? ?
1. a(u) = ?(u)?(u), F (u) = ?1 + ?2 ?(u) ?(u) ,
?
?(u) = ?1 (t) + ?2 (t)x, ?1 = (?2 + ?2 )?1 + ?1 , ?2 = (?2 + ?2 )?2 ;
? ?
2 2
?1
? ?
2. a(u) = u?(u), F (u) = ?1 + ?2 ?(u) ?(u) ,
?(u) = ?1 (t) + ?2 (t)x, ?1 = ?2 ?1 + ?2 + ?1 , ?2 = ?2 ?2 ;
? ?
2
?1
? ?
3. a(u) = ?(u), F (u) = ?1 + ?2 ?(u) ?(u) ,
?(u) = ?1 (t) + ?2 (t)x, ?1 = ?2 ?1 + ?1 , ?2 = ?2 ?2 ;
? ?
a(u) = ?uk , F (u) = ?1 u + ?2 u1?k , uk = ?1 (t) + ?2 (t)x + ?3 (t)x2 ,
4.
?1 = 2??1 ?3 + ?k ?1 ?2 + k?2 , ?2 = 2?(1 + 2k ?1 )?2 ?3 + k?1 ?2 ,
? ?
2
?3 = 2?(1 + 2k ?1 )?2 + k?1 ?3 ;
? 3
a(u) = ?e , F (u) = ?1 + ?2 e?u , eu = ?1 (t) + ?2 (t)x + ?3 (t)x2 ,
u
5.
?1 = 2??1 ?3 + ?1 ?1 + ?2 , ?2 = 2??2 ?3 + ?1 ?2 , ?3 = 2??2 + ?1 ?3 ;
? ? ? 3
a(u) = ?u?3/2 , F (u) = ?1 u + ?2 u5/2 ,
6.
u?3/2 = ?1 (t) + ?2 (t)x + ?3 (t)x2 + ?4 (t)x3 ,
2 3 3
?1 = 2??1 ?3 ? ??2 ? ?1 ?1 ? ?2 ,
? 2
3 2 2
2 3
?2 = ? ??2 ?3 + 6??1 ?4 ? ?1 ?2 ,
?
3 2
22 3 3
?3 = ? ??3 + 2??2 ?4 ? ?1 ?3 , ?4 = ? ?1 ?4 ;
? ?
3 2 2
7. a(u) = 1, F (u) = (? + ? ln u)u, ln u = ?1 (t) + ?2 (t)x,
?1 = ??1 + ?2 + ?, ?2 = ??2 ;
? ?
2
a(u) = 1, F (u) = ? + ? ln u ? ? 2 (ln u)2 u, ln u = ?1 (t) + ?2 (t)e?x ,
8.
?1 = ? + ??1 ? ? 2 ?2 , ?2 = (? + ? 2 ? 2? 2 ?1 )?2 ;
? ?
1
F (u) = ?u(1 + ln u2 ) ? + ?(ln u)?1/2 ,
9. a(u) = 1,
ln u ?1/2
2?? + 4?? 1/2 + ?2 (t) d? = x + ?1 (t),
?1 = 0, ?2 = 4? 2 ? 2??2 ;
? ?
a(u) = 1, F (u) = ?2(u3 + ?u2 + ?u),
10.
(a) ? = ? = 0
?1
u = ?1 (t) + 2?2 (t)x 1 + ?1 (t)x + ?2 (t)x2 ,
?1 = ?6?1 ?2 , ?2 = ?6?2 ;
? ? 2
2
(b) ? = 4? = 0
?1
? ?
u = ? ?1 (t) + 1 ? x ?2 (t) e?x/2 + ?1 (t) + ?2 (t)x ,
2 2
2
?2
?
?1 = ? ?1 ? ??2 , ?2 = ? ?2 ;
? ?
4 4
(c) ?2 > 4?
u = (A + B)?1 (t)eBx + (A ? B)?2 (t)e?Bx ?
Antireduction and exact solutions of nonlinear heat equations 249

?1
? e?Ax + ?1 (t)eBx + ?2 (t)e?Bx ,
? 1
A = ? , B = (?2 ? 4?)1/2 ,
2 2
2
? ?
? 3? ? (?2 ? 4?)1/2 ?1 ,
?1 =
?
2 2
?2 ?
? 3? + (?2 ? 4?)1/2 ?2 ;
?2 =
?
2 2
2
(d) ? < 4?
u = ?1 (t)(A cos Bx ? B sin Bx) + ?2 (t)(A sin Bx +
?1
+ B cos Bx) e?Ax + ?1 (t) cos Bx + ?2 (t) sin Bx ,
?2 ?
? 3? ?1 ? (4? ? ?2 )1/2 ?2 ,
?1 =
?
2 2
2
? ?
? 3? ?2 + (4? ? ?2 )1/2 ?1 .
?2 =
?
2 2
In the above formulae ? = ?(u) ? C 2 (R1 , R1 ) is an arbitrary function; ?, ?1 , ?2 ,
?, ?, ? are arbitrary real constants; overdot means differentiation with respect to the
corresponding argument.
Most of above adduced system of ODE can be integrated. As a result, one obtains
a number of new exact solutions of the nonlinear heat equation (5). Detailed study of
reduced systems of ODE and construction of exact solutions of equation (5) will be a
topic of our future paper. Here we present some exact solutions of the nonlinear heat
equation
ut = uxx + F (u)
obtained with the help of ansatzes 7–10 which are listed above.
F (u) = ? + ? ln u ? ? 2 (ln u)2 u,
1)
(a) ? = ? 2 + 4?? 2 > 0
?2
?1/2 t ?1/2 t
1
2
???
e?x+? t + 2 1/2
u = C cos tg ;
2 2? 2
(b) ? = ?? 2 ? 4?? 2 > 0
?2
?1/2 t ?1/2 t
1
2
e?x+? t
? + ?1/2 th
u = C ch + ;
2? 2
2 2
? = ? + 4?? 2 = 0
2
(c)
1
2
u = Ct?2 e?x+? t
+
(?t + 2);
2? 2 t
F (u) = ?u(1 + ln u2 ) ? + ?(ln u)?1/2 ,
2)
(a) ? = 0
ln u
?1/2
2?? + 4?? 1/2 + Ce?2?t + 2? 2 ??1 d? = x;
(b) ? = 0
ln u
?1/2
4?? 1/2 + 4? 2 t d? = x;
250 W.I. Fushchych, R.Z. Zhdanov

F (u) = ?2u(u2 + ?u + ?),
3)
(a) ?2 = 4?
?1
? ? ?t
u = 1 ? (x ? ?t) x ? ?t + C exp x+ ;
2 2 2
2
(b) ? > 4?
u = (A + B)C1 exp (A + B)(x ? ?t) + (A ? B)C2 ?

? exp (A ? B)(x ? ?t) exp(3?t) + C1 exp (A + B)(x ? ?t) +
?1
+ C2 exp (A ? B)(x ? ?t) ,
? 1
A = ? , B = (?2 ? 4?)1/2 ;
2 2
2
(c) ? < 4?
u = (?AC1 ? BC2 ) cos B(x ? ?t) + (AC2 + BC1 ) ?
? sin B(x ? ?t) exp 3?t ? A(x ? ?t) +
?1
+ C1 cos B(x ? ?t) + C2 sin B(x ? ?t) ,
? 1
A=? , (4? ? ?2 )1/2 .
B=
2 2
In the above formulae C, C1 , C2 are arbitrary constants.
It is worth noting that the above solutions can not be obtained with the use of
the classical Lie symmetry reduction technique [6]. That is why they are essentially
new. Another impotant feature is that solutions 3(a) and 3(c) are soliton-like soluti-
ons. Consequently, nonlinear heat equation with cubic nonlinearity admits soliton-like
solutions.

1. Fushchych W.I., Conditional symmetry of equations of nonlinear mathematical physics, Ukr. Math. J.,
1991, 43, 11, 1456–1470.
2. Fushchych W.I., Mironyuk P.I., Conditional symmetry and exact solutions of nonlinear acoustics
equation, Dopovidi of the Ukrainian Academy of Sciences, 1991, 6, 23–29.
3. Fushchych W.I., Repeta V., Exact solutions of equations of gas dynamics and nonlinear acoustics,
Dopovidi of the Ukrainian Academy of Sciences, 1991, 8, 35–38.
4. Fushchych W.I., Conditional symmetry of equations of mathematical physics, in Proceedings of the
International Workshop “Modern Group Analysis”, Editors N. Ibragimov, M. Torrisi and A. Valenti,
Kluwer Academic Publishers, 1993, 231–240.
5. Fushchych W.I., Zhdanov R.Z., Antireduction of the nonlinear wave equations, Dopovidi of the Ukrai-
nian Academy of Sciences, 1993, 11, 37–41.
6. Fushchych W., Shtelen W., Serov N., Symmetry analysis and exact solutions of equations of nonlinear
mathematical physics, Kluwer Academic Publishers, 1993.
W.I. Fushchych, Scientific Works 2003, Vol. 5, 251–254.

Conditional symmetry and anti-reduction
of nonlinear heat equation
W.I. FUSHCHYCH, R.Z. ZHDANOV
Q- iii i i i -
i i–. i ii i-
ii i i (,

<<

. 62
( 122 .)



>>