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operators
1 1
?0 , ?1 , G(1) = x0 ?1 + ?W 1 ? W 1 ?W 2 ? x1 W 3 ?W 3 ,
2 2
D = 2x0 ?0 + x1 ?1 ? W ?W 1 ? 2W ?W 2 , I (1) = W 3 ?W 3 ,
(1) 1 2

5
?(1) = x0 x0 ?0 + x1 ?1 ? W 1 ?W 1 ? 2W 2 ?W 2 ? W 3 ?W 3 + (15)
2
x2
1 1
+ x1 ?W 1 ? W 1 ?W 2 ? ?W 2 ? 1 W 3 ?W 3 ,
2 2 4
X = (f0 + f1 W ? f W )?W 3 .
1 2

where f = f (x0 , x1 ) is an arbitrary solution of (1), that is f0 = f11 .
Theorem 3. The Lie maximal invariance algebra of equation (12) is given by the
operators
D(2) = 2x0 ?0 + x1 ?1 + u?u , D(3) = u?u + v?v ,
?0 , ?1 ,
1 1
G(2) = x0 ?1 ? x1 (u?u + v?v ) ? u?v ,
2 2
(16)
x2
1 3 x1
?(2) = x0 x0 ?0 + x1 ?1 ? u?u ? v?v ? 1 (u?u + v?v ) ? u?v ,
2 2 4 2
R = f ?u + f1 ?v (f0 = f11 ).
482 W.I. Fushchych, W.M. Shtelen, M.I. Serov, R.O. Popovych

One can get the proofs of these two theorems by means of the standard Lie’s
algorithm.
Operators (15), (16) can be used to find exact solutions of equations (10), (12).
In particular, using the formula of generating solutions at the expense of invariance
under ?(2)
?x2 ? x1 u
v II (x0 , x1 , u) = (1 ? ?x0 )?3/2 exp 1
v 1 (x0 , x1 , u ) + ,
4(1 ? ?x0 ) 1 ? ?x0 2
x0 x1
x0 = , x1 =
, (17)
1 ? ?x0 1 ? ?x0
1 ?x2
u = (1 ? ?x0 )1/2 exp ? 1
u (? = const)
4 1 ? ?x0

one can construct new solutions of equations (12) starting from known ones.
Solutions of equations (10), (12) can be obtained by the use of reduction on
subalgebras of the invariance algebras (15), (16). For example, using the subalgebra
(1)
of the algebra (15) we find the following solution of the system (10)
?0 + ai

C1 ? C3
2 2
W1 = ,
?C1 tg(C 1 x1 + C 2 ) + C3 tg(C3 x1 + C4 )
C1 tg(C3 x1 + C4 ) ? C3 tg(C1 x1 + C2 ) (18)
W 2 = ?C1 C3 ,
?C1 tg(C 1 x1 + C 2 ) + C3 tg(C3 x1 + C4 )
W 3 = (?11 ? W 1 ?1 ? W 2 ?)eax0 ,

where C1 , . . . , C4 are arbitrary constants, ? = ?(x1 ), ?11 = a?.
Theorem 4. The system (10) is reduced to the system of disconnected heat equations

(z = z(x0 , x1 ) = {z 1 , z 2 , z 3 }) (19)
z0 = z11

with the help of the nonlocal transformation

z11 z 2 ? z 1 z11 z11 z1 ? z1 z11
1 2 12 12
W =? 1 2 W =? 1 2
1 2
2, 2,
z1 z ? z 1 z1 z1 z ? z 1 z1 (20)
?W z .
3 3
W 1 z1
3 23
W= z11 +

Expressions (20) result in (after using the corresponding operator (7), (9)) the
ansatz
z2
1 3
(21)
u = z ?(?) + z , ?= 1
z
(z 1 , z 2 , z 3 are solutions of (19)), and the reduced equation is ? = 0. This means that

u = C1 z 1 + C2 z 2 + C3 z 3 . (22)

So, we get just the well-known superposition principle for the heat equation.
Letting W 2 = W 3 = 0 we get from (10) the Burger’s equation

W0 + 2W 1 W1 = W11 .
1 1 1
(23)
Q-conditional symmetry of the linear heat equation 483

Using Hopf–Cole transformation one obtains solutions of equation (23) in the form
f1
W 1 = ??1 ln f = ? (24)
(f0 = f11 ).
f
This result in the operator
Q = f ?0 ? f1 ?1 . (25)
Q-conditional symmetry of equation (1) under the operator (25) lead to the following
statement.
Theorem 5. If function f is an arbitrary solution of the heat equation (1) and u is
the general integral of the ODE
(26)
f1 dx0 + f dx1 = 0,
then u satisfies equation (1).
Proof. We note that equation (26) is a perfect differential equation and therefore its
general solution u(x0 , x1 ) = C possesses the following property
(27)
u0 = f1 , u1 = f.
Having used (27) we obtain
u0 ? u11 = f1 ? f1 = 0
and the theorem is proved.
Theorem 5 may be considered as another algorithm of generating solutions of
equation (1). Indeed, even starting from a rather trivial solution of the heat equation
u = 1 we get the chain of quite interesting solutions
x2 x3
1 > x1 > x0 + > x0 x1 + 1 > · · · ,
1
(28)
2! 3!
and among them the solutions
x0 x2m?2 x2 x2m?4 xm?1 x2
x2m xm
+ ··· +
1 0 1
+ 0,
1 1 0
(29)
+ +
1! (2m ? 2)! 2! (2m ? 4)! (m ? 1)! 2!
(2m)! m!

x2m+1 x2 x2m?3
x0 x2m?1
+ ···
+0
1 1
+
1! (2m ? 1)! 2! (2m ? 3)!
(2m + 1)! (30)
xm?1 x3 xm x1
··· + 1
+0
0
.
(m ? 1)! 3! m! 1!

It will be also noted that supposing function v in (12) to be independent on x1 and
denoting
1
(31)
v=
w(x0 , u)
we get instead of (12) the following remarkable nonlinear heat equation
w0 = ?u (w?2 wu ). (32)
484 W.I. Fushchych, W.M. Shtelen, M.I. Serov, R.O. Popovych

One easily sees that the operator

(33)
Q = w(x0 , u)?1 + ?u

sets the connection between equations (32) and (1):
u0 ? u11
1
w0 ? ?u (w?2 wu ) = ?1 ,
u1 u1
(34)
1
[w0 ? ?u (w?2 wu )]du
u0 ? u11 =
w
by means of the change of variables
?x1 (x0 , u) ?u(x0 , x1 ) 1
(35)
w(x0 , u) = , = .
?u ?x1 w(x0 , u)
This result has been obtained differently in [7, 8].
It suppose v from (12) to have the form

(36)
v = ?(x0 , x1 )u

then (12) is reduced to the Burger’s equation for ?

(37)
?0 = 2??1 + ?11

and one may say that operator

(38)
Q = ?1 + ?u?u

sets the connection between equation (37) and (1) via the substitution

(39)
? = f1 /f.

Letting

(40)
v = ?(x0 , x1 )u + h(x0 , x1 )

and substituting it into (12) one finds the Burger’s equation (37) for function ? and
the following equation for h

(41)
h0 = 2h?1 + h11 .

System of equation (37), (41) was also obtained in [6] when considering the sys-
tem (10). Having made the change of variables

h = (f1 /f )g ? g1 (42)
? = f1 /f,

we reduced (37), (41) to two disconnected heat equations

(43)
f0 = f11 , g0 = g11 .

Now we consider how to linearise the equation (12) in general case. Let us
introduce the notations

S 1 (x0 , x1 , u, v) = v0 ? (v11 + 2vv1u + v 2 vuu ). (44)
Q-conditional symmetry of the linear heat equation 485

After changing the variables
z1
v=? (45)
, z = z(x0 , x1 , u)
zu
we get
1
S 1 (x0 , x1 , u, v) = ? (?1 + v?u )S 2 (x0 , x1 , u, z), (46)
zu
where
2
z1 z1
S (x0 , x1 , u, v) = z0 ? z11 + 2 z1u ? 2 zuu .
2
(47)
zu zu
Having applied the hodograph transformation

(48)
y0 = x0 , y1 = x1 , y2 = z, R=u

we get
1
S 2 (x0 , x1 , u, z) = ? (R0 ? R11 ), (49)
R2
where R = R(y0 , y1 , y2 ).

1. Фущич В.И., Штелень В.М., Серов Н.И., Симметрийный анализ и точные решения нелинейных
уравнений математической физики, Киев, Наук. думка, 1989, 336 с.
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