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б)
2x0 x0 2x0
v = 2 ln(x1 + c1 ) ? ln(?2?x0 ).
Итак, приведенные результаты говорят о том, что нелинейные уравнения обла-
дают скрытыми нелокальными симметриями, которые к настоящему времени со-
вершенно не изучены и не использованы для их интегрирования.

1. Овсянников Л.В., Групповые свойства уравнения нелинейной теплопроводности, Доклады АН
СССР, Сер. А, 1959, 125, № 3, 492–495.
2. Фущич В.И., Серов Н.И., Амеров Т.К., Условная инвариантность нелинейного уравнения тепло-
проводности, Докл. АН УССР, Сер. А, 1990, № 11, 15–18.
3. Фущич В.И., Штелень В.М., Серов Н.И., Симметрнйный анализ и точные решения нелинейных
уравнений математической физики, Киев, Наук. думка, 1989, 336 с.
W.I. Fushchych, Scientific Works 2003, Vol. 5, 89–92.

The conditional invariance and exact
solutions of the nonlinear diffusion equation
W.I. FUSHCYCH, N.I. SEROV, L.A. TULUPOVA
Исследована условная инвариантность нелинейного уравнения диффузии. Операто-
ры условной инвариантности использованы для построения анзацев, редуцирующих
данное уравнение к обыкновенным дифференциальным уравнениям. Найдены неко-
торые точные решения исходного уравнения.

Let us consider the nonlinear diffusion equation
(1)
H(u)u0 + u11 = F (u),
2
where u = u(x) ? R1 , x = (x0 , x1 ) ? R2 , u0 = ?x0 , u11 = ? u , H(u) and F (u) are
?u
?x21
arbitrary smooth functions.
Usually the equation (1) is investigated in the equivalent form
(2)
u0 + ?1 (f (u)u1 ) = g(u).
In this way, for example, in papers [1, 2] Lie invariance of this equation was investi-
gated.
The present paper is a continuation of the works [3, 4], where the Q-conditional
invariance of the equation (1) was studied when H(u) ? 1 and H(u) = u?1 , F (u) = 0.
In this paper Q-conditional invariance of the equation (1) is studied when H(u) and
F (u) are arbitrary functions. Using obtained operators of Q-conditional invariance
exact solutions of the given equation are found.
Let
(3)
Q = A(x, u)?0 + B(x, u)?1 + C(x, u)?u ,
where A, B, C are smooth functions, be a differential operator of the first order,
acting on the manifold (x, u) ? R3 .
The following theorem is proved analogously, as in [5].
Theorem 1. The equation (1) is Q-conditionally invariant under the operator (3), if
the functions A, B, C satisfy the following conditions:
Case 1. A = 1.
Buu = 0, Cuu = 2(B1u + HBBu ),
3Bu F = 2(C1u + HBu C) ? (HB0 + B11 + 2HBB1 + Hu BC), (4)
CFu ? (Cu ? 2B1 )F = HC0 + C11 + 2HCB1 + Hu C 2 ;
Case 2. A = 0, B = 1.
Hu
CFu ? Cu F = HC0 + C11 + 2CC1u + C 2 Cuu + C(F ? C1 ? CCu ). (5)
H
Доповiдi АН України, 1993, № 4, С. 37–40.
90 W.I. Fushchych, N.I. Serov, L.A. Tulupova

In formulae (4), (5) and everywhere below a subscript means differentiation with
respect to corresponding argument.
Theorem 2. The equation (1) is Q-conditionally invariant under the operator

(6)
Q = ?0 + u?1 + C(u)?u ,

if it has form

?2 ?2
(7)
3?1 + u0 + u11 = 2?1 + P3 (u),
u u

where P3 (u) = ?1 u3 +?2 u2 +?3 u+?4 is arbitrary third-order polynomial of u, ?k are
arbitrary constants, k = 1, 4. In this case C(u) = P3 (u).
Proof. Substituting B = u, C = C(u) into (4), we have

1
(2H ? uHu )C,
Cuu = 2uH, F= uHuu + 2Hu = 0.
3
Whence it appears that

?2 ?2
H = 3?1 + , C = P3 (u), F= 2?1 + P3 (u),
u u

The theorem is proved.
We use the operator (6) for finding solutions of the equation (7). The ansatz
obtained with the help of the operator (6) has the form

udu du
x1 ? ? = x0 ? (8)
= ?(?), .
P3 (u) P3 (u)

The ansatz (8) reduces the equation (7) to the ordinary differential equation (ODE)

(9)
? + P3 (?) = 0.
? ?

Integration of the equation (9) depends on a form of the roots of the polynomial P3 .
There are seven essentially different cases. We give one example of each case.

P3 (u) = (u ? 1)3 , (? ? ?)2 = 2?,
1)
x1 ? x0
u=1+ ;
x0 ? 1 (x1 ? x0 )2
2
1
P3 (u) = (u + 1)(u ? 1)2 , th (? ? ?) ? 1 =
2) ,
?+?
u?1
1
x0 + x1 + u?1
= th (x0 ? x1 );
?u
1
x0 + x1 + u?1
P3 (u) = (u ? 2)(u2 ? 1), exp 3(? ? ?) ? 3 exp(? + ?) + 2 = 0,
3)
exp 3(x1 ? x0 ) + 3 exp(x1 + x0 ) ? 4
u=? ;
exp 3(x1 ? x0 ) ? 3 exp(x1 + x0 ) + 2
Conditional invariance and exact solutions of the diffusion equation 91

P3 (u) = (u ? 1)(u2 + 2u + 2),
4)
3 cos(2? ? 2?) + 4 sin(2? ? 2?) + 5 = 2 exp(?4? ? 6?),
exp(?3x0 ? 2x1 ) + 3 sin(x0 ? x1 ) ? cos(x0 ? x1 )
u= ;
exp(?3x0 ? 2x1 ) + 2 sin(x0 ? x1 ) ? cos(x0 ? x1 )
P3 (u) = (u ? 1)2 , ? = ? + ln ?, u = 1 + exp(x1 ? x0 );
5)
ch x0 ? exp x1
P3 (u) = u2 ? 1, ? = ln ch ?, u =
6) ;
sh x0
? cos x0 + exp x1
P3 (u) = u2 + 1, ? = ln cos ?, u =
7) .
sin x0

Theorem 3. The equation

(10)
u0 + uu11 = ?1 u + ?2 , (?1 , ?2 = const)

is Q-conditionally invariant under the operator
u
(11)
Q = ?0 + ?1 + (?1 u + ?2 )?u .
x1

Proof. If we find a prolongation of the operator (11) and act on the equation (10),
then we have
2u 3u1 ?2
?
Q(u0 + uu11 ? ?1 u ? ?2 ) = 2 ? x + 2?1 + u ?
x1 1
2u 2u1 ?2
? (u0 + uu11 ? ?1 u ? ?2 ) ? ? ?
+ ?1 +
x2 x1 u
1
uu1
? u0 + ? ?1 u ? ?2 ,
x1

i.e.

?
QS = ?S + ?Qu,

The theorem is proved.
The ansatz

x2
? = u ? ?2 x0 ? ?1 1 ,
?1 x0
(12)
?1 v + ?2 = e ?(?),
2
obtained with the help of the operator (11) reduces the equation (10) to the following
ODE

(13)
? = 0.
?

Solving the equation (13) and using the ansatz (12), we find the solution of the
equation (10):

x2
?1 u + ?2 = e?1 x0 c1 u ? ?2 x0 ? ?1 1
+ c2 . (14)
2
92 W.I. Fushchych, N.I. Serov, L.A. Tulupova

Now we give some more results on the Q-conditional invariance of the equation (1).
The results are written in the following order — an equation (1), a corresponded
operator, an ansatz, a reduced equation, a solution of the equation (1).
(?1 u2 + ?2 )u0 + u11 = ?2 u3 , Q = ?2 x2 ?0 + 3x1 ?1 + 3u?u ,
1) 1
?2
?2
u = x1 ?(?), ? = x0 ? x2 , ? 1 ?2 ? + 2 ? = ? 3 ?3 ,
?
61 9
(?1 u2 + ?2 )u0 + u11 = ?3 u3 + 2u,
2)
Q = ?2 (1 + cos 2x1 )?0 ? 3 sin 2x1 ?1 + 6u?u , u = ctg x1 ?(?),
3 ?1
x0 + ln sin x1 , ? ? 3? + 2? + 3 ?2 ? = ?3 ?3 ,
?= ? ? ?
?2 ?2
3u
(?1 u2 + ?2 )u0 + u11 = ?3 u3 ? 2u, Q = ?2 ?0 + 3 th x1 ?1 ?
3) ?u ,
ch2 x1
3 ?1 2
x0 ? ln sh x1 , ? ? = ? 3 ?3 ,
u = cth x1 ?(?), ?= ? + 3? + 2? + 3
? ? ?
?2 ?2
eu u0 + u11 = eu ,
4)
ex0 + 2
a) Q = x1 ?1 ? 2?u , u = ?(x0 ) ? 2 ln x1 , e? ? + 2 = e? ,
? u = ln ,
x21
x1 x1 1
u = ?(x0 ) ? 2 ln cos e? ? + = e? ,
b) Q = ?1 + tg ?u , , ?
2 2 2
ex0 + 12
u = ln ,
2 x1
cos 2
x1 x1 1
c) Q = ?1 + th ?u , u = ?(x0 ) ? 2 ln ch , e? ? ? = e? ,
?
2 2 2
e ?2 1
x0
u = ln ,
cos2 x1
2
5) ?uu0 + u11 = ?u2 ,
1 1 x0
a) Q = ?0 + u + 2 ?u , u = 2 + e ?(x1 ),
?x1 ?x1
x1 ? ? 6(2? ? 1)? = 0,
2
?
1 1
b) Q = ?0 ? u ? W (x1 ) ?u , u = W + ex0 ?(x1 ), ? = W ?, ?
? ?
1
u = W (x1 ) + ex0 ?(x1 ),
?
where W (x) is the Weierstrass function, ?(x) is the Lame function.

1. Овсянников Л.В., Групповой анализ дифференциальных уравнений, М., Наука, 1978, 400 с.
2. Дородницын В.А., Князева И.В., Свищевский С.Р., Групповые свойства уравнени теплопрово-
дности с источником в двумерном и трехмерном случаях, Дифференц. уравнения, 1983, 19,

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