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53
(? + c3 x0 )(? ? 4c3 x0 ) = 4
c x1 + c2 x0 ,
41
where c1 , c2 , c3 are arbitrary constants of integration.
On non-local symmetries of nonlinear heat equation 463

6. The invariance of equation (1) under the transformations (6)–(8). For
the invariance of equation (1) under non-local transformations (6)–(8) the following
condition must be satisfied

z ?2 F (z ?1 ) = F (z). (41)

The solution of equation (41) can be written down in the form

F (z) = z ?1 f (ln z), (42)
where f is arbitrary differentiable even function. So transformations (6)–(8) are non-
local invariance transformations of equation
f (ln u)
(43)
u 0 = ?1 u1 , (f (??) = f (?)).
u
Using this fact, we construct generating formula for solutions of equation (43):
1
2
u(x0 , x1 ) = (44)
,
1
u(x0 , ? )
where ? = ? (x0 , x1 ) is the functional parameter which is a solution of the equations
1 ?11
(45)
?1 = , ?0 = f (ln ?1 ) .
1 ?1
u(x0 , ? )
Example. Let us consider the solution
x0
1
u(x0 , x1 ) = (46)
1 + cos x1
of the equation
u1
(47)
u 0 = ?1 .
u
By means of formulas (44), (45) we construct new solution
2x0
2
u(x0 , x1 ) = (48)
x2 + x2
0 1

of the equation (47). It should be noted that the solutions (46) and (48) have essen-
tially different properties (boundaryness, periodicity, the behavior at zero and at the
infinity and so on). If we will apply Lie transformations to manifold of the solutions of
equation (47), then the majority of those properties of the solutions will be conserved.

1. Овсянников Л.В., Групповые свойства уравнения нелинейной теплопроводности, Доклады АН
СССР, 1959, 125, № 3, 492–495.
2. Олвер П., Приложения групп Ли к дифференциальным уравнениям, М., Мир, 1989, 639 с.
3. King I.R., Some non-local transformations between nonlinear diffusion equations, J. Phys. A: Math.
Gen., 1990, 23, 5441–5464.
W.I. Fushchych, Scientific Works 2002, Vol. 4, 464–469.

Conditional symmetry and exact solutions
of equations of nonstationary filtration
W.I. FUSHCHYCH, N.I. SEROV, A.I. VOROB’EVA

Дослiджена умовна iварiантiсть, одержанi нелiївськi анзаци та побудованi точнi
розв’язки рiвняння нестацiонарної фiльтрацiї з нелiнiйною правою частиною. Ре-
зультати узагальнено для n-вимiрного нелiнiйного рiвняння теплопровiдностi.

In describing filtration processes of gas the following nonlinear equation is widely
used [1]

? 2 ?(v)
?v N ??(v)
(1)
+ + = ?(v),
2
?x0 ?x1 x1 ?x1

where v = v(x), x = (x0 , x1 ) ? R2 , N = const; ?(v), ?(v) are given smooth functions.
Substitution u = ?(v) reduces equation (1) to equivalent equation

N
(2)
H(u)u0 + u11 + u1 = F (u),
x1
2
where u0 = ?x0 , u1 = ?x1 , u11 = ? u .
?u ?u
?x2
1
Lie symmetry of equation (2) under N = 0 was studied in [2, 3] and its conditional
symmetry was studied in [4–7].
In present paper we study conditional symmetry of equation (2) with N = 0.
Operators of conditional symmetry are used to construct ans?tze which reduce (2)
a
to ordinary differential equations (ODE). By means of this method we obtain exact
solutions of equations (2) and then exact solutions of multidimensional nonlinear heat
equation. Below we will use terms and definitions given in [4–7].
Theorem 1. Equation (2) is Q-conditionally invariant under the operator

(3)
Q = A(x, u)?0 + B(x, u)?1 + C(x, u)?u ,

iff function A, B, C satisfy the following system of equations:
Case I. A = 0 (without lose of generality one can put A = 1)

N
Cuu = 2 B1u + HBBu ? 3Bu F = 2(C1u + HBu C) ?
Buu = 0, Bu ,
x1
N N ?
? HB0 + B11 ? (4)
B1 + 2 B + 2HBB1 + HBC ,
x1 x1
N
? ?
C F ? (Cu ? 2B1 )F = HC0 + C11 + C1 + 2N CB1 + HC 2 ;
x1
Доповiдi АН України, 1992, № 6, C. 20–24
Conditional symmetry and exact solutions 465

Case II. A = 0, B = 1,
?
H N N
?
CF ? C1 ? 2 C + 2CC1u + C 2 Cuu ?
Cu + C F = HC0 + C11 +
H x1 x1
(5)
?
H N
?C CCu + C1 + C.
H x1
In formulas (4), (5) and everywhere below subscripts mean differentiation with
respect to corresponding arguments.
To prove the theorem one should use the method described in [4–7].
To find the general solution of equations (4), (5) is impossible, but we succeeded
in obtaining several partial solutions.
Theorem 2. Equation (2) is Q-conditionally invariant under operator (3) with
H(u) = 1, A = 1, Du = 0 it is leeally equivalent to the equation
3
u1 = ?u3 (6)
u0 + u11 + (? = const),
2x1
and in this case operator (3) takes the form
v
3 1 3 1
?1 + u 2?u2 ? 2 (7)
Q = ?0 + 2?u + ?u .
2 x1 4 x1
To prove the theorem one has to solve equations (4) under H(u) = 1, B(u) = 0.
By means of operator (7) we construct an implicit ansatz
v
v
x2 1 + 2?ux1
5
15 x0 ? 1 ? + 4 2?x1 = ?(?), ? = v
2
(8)
,
3 u x1
which reduces equation (6) to the ODE ? = 0. Having solved this latter one and
?
taking into account (8), we obtain the following solution of equation (6), u(x0 , x1 ) is
a new solution
x3 ? 3x0
5
u = ?v 1
3 ? 15x x + c vx (9)
(c1 = const).
2? x1 01 1 1

All inequivalent ans?tze of Lie type are given by one of formulae
a
?1 ?1
u = x0 2 ?(x0 2 x1 ). (10)
u = ?(x1 ),
It is obvious that (9) does not belong to (10).
The above solutions of equation (6) can be multiplied by means of formulae of
generating solutions using Lie symmetry:
2
(11)
u(x0 , x1 ) = ?1 f (?1 x0 + ?0 , ?1 x1 ),
where ?0 , ?1 are group parameters, f (x0 , x1 ) is a known solution of equation (6),
u(x0 , x1 ) is a new solution.
Theorem 3. Equation
1 N 1
(12)
u0 + u11 + u1 = (?1 u + ?2 ) (?1 , ?2 = const),
u x1 u
466 W.I. Fushchych, N.I. Serov, A.I. Vorob’eva

is Q-conditionally invariant under operator
u
(13)
Q = ?0 + (N + 1) ?1 + (?1 u + ?2 )?u .
x1
Proof. To prove the theorem it is sufficient to show that the following relation holds
true
? ?
? (14)
QS = ?1 S + ?2 Qu,
where
1 N 1
u0 + u11 + u1 ? (?1 u + ?2 ),
S=
u x1 u
u
Qu = u0 + (N + 1) u1 ? (?1 u + ?2 ),
x1
??
?
Q is corresponding prolongation of operator Q: ?1 , ?2 are some functions.
?
On acting operator Q on S we get after rather tedious calculations,
N +1
? (2u + 3x1 u1 ) S ?
QS = ?1 +
x2
1
N +1 N +1 ?1 u + ?2
? u1 ? 2 (2u + 3x1 u1 ) ? Qu.
u2
x1 u x1 u
So, the theorem is proved.
Operator (13) results in the ansatz
x2 udu du
? ? = x0 ?
1
(15)
= ?(?), ,
2(N + 1) ?1 u + ?2 ?1 u + ?2
which reduces equation (12) to the ODE
?? = ?1 ? + ?2 . (16)
? ?
Having integrated equation (16) and using once more (15) one finds solution of
equation (12)
?1 ?2 x0 + 1 ?2 (N + 1)?1 x2
21 1
?1 u + ?2 = , ?1 = 0,
1 + ?3 exp(?x0 )
(17)
?2 x2 + ?3 + (N + 1)?1 x2
0 1
u= , ?1 = 0,
2x0
where ?3 is a constant of integration.
It is not difficult to verify that solutions (17) cannot be obtained by means of Lie
ans?tze (analogously to above solutions of equation (6)).
a
The rest results obtained for equation (2) are collected in table, where ?1 , ?2 , ?3
are arbitrary constants, W = W (u) an arbitrary smooth function.
Let us give some solutions of equation (2) obtained as a result of integration of
reduced equations listed in the table.
N ?1
N ? 3 x0 x1
1. u=? + ,
?2 x1 2
N ?1
2
?3
?= , ?1 = 0, N = 1, 2;
(N ? 1)(N ? 2)
Conditional symmetry and exact solutions 467
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