CSMT JOURNAL OF STUDENT RESEARCH V.2014 (5)

College of Science, Mathematics and Technology
CSMT JOURNAL OF STUDENT RESEARCH V.2014 (5)
Student’s participation in research is one of the most effective avenues for attracting students to,
and retaining them in careers in science and engineering, including careers in teaching and
education research. The main goal of this journal is to attract students to a publishable research
and share the results between students and advisors at CSMT.
EDITORAL BOARD
Editor-in-Chief:
Ranis Ibragimov
Associate Editors:
Wei Lin; Yong Zhou; Nazmul Islam; Immanuel Edinbarough
Carlos Cintra Buenrostro; Cristina Torres
Department of Mathematics
University of Texas at Brownsville
Brownsville TX 78520, USA
Tel: (956) 882 – 6672 FAX: (956) 882 – 6637 E-mail: [email protected]
TITLE OF THE PROJECT
Symmetry properties and invariant solutions of linear wave equations
FIELDS OF SCIENCE
Applied Mathematics, Group analysis of differential equations
KEY WORDS
Symmetries of differential equations, group classification, invariant solutions, integration ODE by using symmetries
ABSTRACT:
Investigation of the symmetry properties of nonlinear wave equations is based on the Ovsiannikov's classification of
hyperbolic normal forms. For construction of additional operators and invariant solutions we find the
transformation of considered equations to the canonical forms from Ovsiannikov's classification
NAMES OF THE AUTHORS
o Advisor(s): Nail H. Ibragimov, Rafail K. Gazizov, Aleksei A. Kasatkin
o Graduate student(s): n\a
o Undergraduate student(s): Elena N. Karimova
STATEMENT OF ORIGINALITY:
We considered the linear wave equation
We investigated the symmetry properties of equations of the third- and forth- orders for the unknown functions a(x) ,
when the original equation has additional symmetries. Some invariant solutions have been constructed. We propose a
new method of integrating the forth-orders equations.
We investigated the symmetry properties of equations of the third- and forth- orders for the unknown
functions a(x), b(t), when the equations
has additional symmetries.
.
Was this project previously published or submitted for publication? No
College of Science, Mathematics and Technology
CSMT JOURNAL OF STUDENT RESEARCH V.2014 (5)
Student’s participation in research is one of the most effective avenues for attracting students to,
and retaining them in careers in science and engineering, including careers in teaching and
education research. The main goal of this journal is to attract students to a publishable research
and share the results between students and advisors at CSMT.
EDITORAL BOARD
Editor-in-Chief:
Ranis Ibragimov
Associate Editors:
Wei Lin; Yong Zhou; Nazmul Islam; Immanuel Edinbarough
Carlos Cintra Buenrostro; Cristina Torres
Department of Mathematics
University of Texas at Brownsville
Brownsville TX 78520, USA
Tel: (956) 882 – 6672 FAX: (956) 882 – 6637 E-mail: [email protected]
If yes, please write the citation: N/A
Symmetry properties and invariant solutions
of linear wave equations
E. K. Karimova
Laboratory “Group analysis of mathematical models in natural and
engineering sciences”, Ufa State Aviation Technical University, Ufa, Russia
Abstract
Investigation of the symmetry properties of nonlinear wave equations is based
on the Ovsyannikov’s classification of hyperbolic normal forms. For construction of additional operators and invariant solutions we find the transformation of considered equations to the canonical forms from Ovsyannikov’s classification.
Key words:
Symmetries of differential equations, group classification, in-
variant solutions, integration ODE by using symmetries.
1
Introduction
Nonhomogeneous nonstationary dielectric media has wide opportunities in
the formation and transformation of signals in optical fiber communications.
We consider the equations
∂V
∂I
+ L0
+ RI = 0,
∂x
∂t
∂I
∂V
+ C0 a2 (x)b2 (t)
+ GV = 0,
∂x
∂t
Email addresses: [email protected] (E. N. Karimova)
1
describing propagation of electromagnetic waves in these medium [1]. Here
a(x), b(t) are dimensionless functions, which describe the time-spatial distribution of the dielectric permeability, I a current, V a voltage, L0 , C0 , R, G
are unperturbed values of inductance, capacity, resistance, and the leakage
parameter per unit length.
The dielectric permeability of the media is characterized by spatial and temporal dependence (x, t) = n20 a2 (x), where n0 is the index of refraction at the
boundary x = 0.
∂u
Expressing the current and voltage though the potential I =
, V =
∂t
∂u
−C0 a2 (x)b2 (t) , we reduce the system of first-order equations to a single
∂x
second-order equation:
utt − a2 (x)b2 (t)uxx = 0.
(1.1)
Here we consider a particular case of this equation:
utt − a2 (x)uxx = 0.
(1.2)
The purpose of our work is to study symmetry properties of Eq. (1.2).
For arbitrary a(x) the symmetry Lie algebra is infinite dimensional and is
spanned by X0 = u∂u , X1 = ∂t , X∞ = u0 (x, t)∂u , where u0 (x, t) is any solution of the considered equation.
In Section 2 we consider cases when the algebra expanded by using the
Ovsyannikov’s classification for hyperbolic normal form. As a result, we
obtain equations of the fourth- and third-orders for the unknown function
a(x), when the Eq. (1.2) has additional symmetries.
In Section 3, we integrate the third-order equation, and fourth-order equation to reduce to the first-order equation. Then in Section 4 we obtain the
symmetry of the Eq. (1.2).
In section 5 we give some invariant solutions of the Eq. (1.2).
In section 6 we investigate the symmetry properties of the Eq. (1.1).
2
2
2.1
Group classification of linear wave equation
Ovsiannikov’s classification
To the analysis symmetry properties of Eq. (1.2), we use the results of
Ovsyannikov’s classification [2]. L. V. Ovsiannikov investigated symmetry
properties of the equation of the form
uξη + A(ξ, η)uξ + B(ξ, η)uη + C(ξ, η)u = 0.
(2.1)
The latter equation is a reduced form of Eq. (1.2) in the characteristic
variables.
The first step of our classification is the calculation of the Laplace invariants
h = Aξ + AB − C, k = Bη + AB − C.
(2.2)
The algebra extends in the following cases:
1. h = 0, k = 0.
In this case Eq. (2.1) is equivalent to the wave equation
uξη = 0.
2. h 6= 0 (or k 6= 0). In this case Eq. (2.1) admits more than one additional
operator if the functions
p=
k
1
, q = (ln h)ξη
h
h
(2.3)
are constant.
2.1. q 6= 0.
In this case Eq. (2.1) is equivalent to the Euler-Poisson equation
vµν −
2/q
2p/q
4p/q 2
vµ −
vν +
v=0
µ+ν
µ+ν
(µ + ν)2
3
(2.4)
which admits four additional operators:
X1 = ∂µ − ∂ν ,
X2 = µ∂µ + ν∂ν ,
2
X3 = µ2 ∂µ − ν 2 ∂ν + (pµ − ν)v∂v ,
q
X4 = v∂v .
2.2. q = 0.
In this case Eq. (2.1) reduces to the equation
vµν + µvµ + pνvν + pµνv = 0,
(2.5)
which admits four additional operators:
X1 = µ∂µ − ν∂ν ,
X2 = ∂µ − νv∂ν ,
X3 = ∂ν − pµv∂v ,
X4 = v∂v .
2.2
The results of applying Ovsyannikov’s classification
The linear wave equation
utt − a2 (x)uxx = 0
(1.1)
in the characteristic variables
Z
ξ =t−
1
dx, η = t +
a
Z
1
dx
a
is written as
(2.6)
a0 (x)
(uξ − uη ) = 0
(2.7)
4
where t = η + ξ, x Z= ζ(η − ξ), provided that ζ is the inverse function of the
1
function η − ξ = 2
dx.
a(x)
According to the formulas (2.2), the Laplase invariants are
uξη −
h = k, k = 2aa00 − (a0 )2 .
4
1. h = 0, k = 0.
This case was considered by N. H. Ibragimov in [3] for Eq. (1.2). This equation can be reduced to the wave equation vξη = 0 by a linear transformation
of the dependent variable.
In this case by solving the equation
2aa00 − (a0 )2 = 0,
we obtain that a(x) = (C1 x + C2 )2 , where C1 , C2 are arbitrary constants.
Hence, the general form of Eqs. (1.2) reduced to the wave equation in the
form
utt − (C1 x + C2 )2 uxx = 0,
which is easily solved
2
u = (C1 x + C2 ) ϕ t +
1
C1 (C1 x + C2 )
1
+ψ t−
C1 (C1 x + C2 )
,
where ϕ, ψ are arbitrary function.
2. h 6= 0 (or k 6= 0).
In this case, the additional functions p and q are
00
1
16
p = 1, q =
.
ln
2aa00 − (a0 )2
2aa00 − (a0 )2
2.1. q 6= 0.
In this case, Eq. (2.7) is reduced to Eq. (2.4) with additional symmetries,
and for the unknown function a(x) we obtain the 4th-order equation
00
16
1
ln
= C1 ,
2aa00 − (a0 )2
2aa00 − (a0 )2
a(IV ) =
2C1 (2aa00 − (a0 )2 ) 2aa000
4C1 (a000 )2
−
+
a3
a
2aa00 − (a0 )2
(2.8)
2.2. q = 0.
In this case, Eq. (2.7) is reduced to Eq. (2.5) with additional symmetries,
and for the unknown function a(x), we obtain the 3rd-order equation
0
16
ln
= C1 ,
2aa00 − (a0 )2
5
a000 = 2C3
2.3
aa00
(a0 )2
+
C
.
3
a2
a2
(2.9)
Summarizes results
Let us summarize all above given results.
utt − a2 (x)uxx = 0. (1.1)
2C1 (2aa00 − (a0 )2 )
2a0 a000
4C1 (a000 )2
−
+
, (2.8)
a3
a
2aa00 − (a0 )2
0
2
00
(a )
aa
(2.9)
= 2C1 2 + C1 2 .
a
a
Symmetries?
if q 6= 0 ⇒ a(IV ) =
if q = 0 ⇒ a(III)
Z
ξ =t−
∀ a(x) − L2 + L∞ ,
uξη −
a(x) from (2.8) q 6= 0 − L4 + L∞ ,
a(x) from (2.9) q = 0 − L4 + L∞ .
2/q
2p/q
4p/q 2
vµ −
vν +
v = 0 (2.4)
µ+ν
µ+ν
(µ + ν)2
X1 = ∂µ − ∂ν ,
X3 =
µ
−
ν2∂
ν
1
dx
a
q=0
vµν + µvµ + pνuν + pµνv = 0 (2.5)
X1 = µ∂µ − ν∂ν ,
X2 = ∂µ − νv∂ν ,
X2 = µ∂µ + ν∂ν ,
µ2 ∂
Z
a0 (x)
(uξ − uη ) = 0. (2.7)
4
Symmetries?
q 6= 0 Transformations?
vµν −
1
dx, η = t +
a
+ 2/q(pµ − ν)v∂v ,
X3 = ∂ν − pµ∂v ,
X4 = v∂v .
X4 = v∂v
Pic.1.Statement of the problem
Let us investigate the symmetry properties of Eqs. (2.8) and (2.9). Then
let’s obtain the operators of the linear wave equation (1.1) from the operators
of Eqs. (2.4) and (2.5).
6
3
Integration of ODE’s obtained for a(x).
3.1. Let us consider Eq. (2.8)
a(IV ) =
2C1 (2aa00 − (a0 )2 ) 2a0 a000
4C1 (a000 )2
+
−
a3
a
2aa00 − (a0 )2
obtained in the case q 6= 0 of Ovsyannikov’s classification.
This equation admits three additional operators
X1 = ∂x ,
X2 = x∂x + a∂a ,
X3 =
x2
∂x + xa∂a .
2
This Lie algebra is non-solvable. We integrate Eq. (2.8) by using all three
operators.
For each operator we construct the second prolongation
X1 = ∂x ,
X2 = x∂x + a∂a − a00 ∂a00 − 2a000 ∂a000 ,
X3 = x2 ∂x + 2xa∂a + 2a∂a0 + (2a0 − 2a000 x)∂a00 − 4a000 ∂a000 .
Differential invariants for the three-dimensional algebra are a2 a000 , 2aa00 − a02 .
Let us use the notation
t = a2 a000 , v(t) = 2aa00 − a02 .
Then Eq. (2.8) is reduced to the first-order equation
v0 =
C1 t2 + v 2
.
vt
The solution of this equation is
p
v = ± 8C1 ln(t) + C2 .
Then Eq. (2.8) can be written as the third-order equation
p
(2aa00 − (a0 )2 ) 2C1 ln(2aa00 − (a0 )2 ) + C2
000
a =±
,
a2
7
(3.10)
which admits three additional operators
x2
∂x + xa∂a .
2
Repeating the same procedure for the Eq. (3.10), we obtain the invariants
X1 = ∂x ,
X2 = x∂x + a∂a ,
X3 =
s = a0 , z(s) = aa00 .
Then the third-order Eq. (3.10) can be written as the first-order equation
p
(2z − s2 ) 2C1 ln(2z − s2 ) + C2
0
+ C3 .
z =±
z
To integrate Eq. (2.8), we also use the method of ”differential invariants”,
which is described in [4].
Using this method, one can be reduce Eq. (2.8) to the second-order equation
2C1 (2v − t2 ) v 0 − t
2v
00
0
v =1+
+
−v ,
v2
v
2v − t2
where v(t) = aa00 , t = a0 .
The particular solutions of the Eq. (2.8) are a(x) = AeBx , a(x) = (Ax +
B)C , C 6= 0, 2.
3.2. Now let us consider Eq. (2.9)
a(III) = 2C1
(a0 )2
aa00
+
C
1
a2
a2
from case of Ovsiannikov’s classification when q = 0. This equation admits
three additional operators
x2
X1 = ∂x , X2 = x∂x + a∂a , X3 = ∂x + xa∂a .
2
Integrating Eq. (2.9) as illustrated in [5], we obtain the Riccati equation
vz =
v2
1
−
,
2C3 4C3 z
where v = aa00 , z = a0 .
The solution of this equation can be found in the handbook V.F. Zaitsev
and A.D. Polyanin [6]
r
r
√
z
z
v = x C4 J1
− 2 + C 5 Y1
− 2
,
8C3
8C3
8
where J1 (z), Y1 (z) are Bessel functions.
The particular solution of Eq. (2.9) is a(x) = (Ax + B).
4
Transformation of Eqs. (2.4), (2.5) to Eq.
(1.2)
4.1
Transformation of Eqs. (2.4), (2.5) to Eq. (2.7)
Let’s find a change of variables which transforms Eq. (2.4)
vµν −
2/q
2p/q
4p/q 2
vµ −
vν +
v=0
µ+ν
µ+ν
(µ + ν)2
to Eq. (2.7)
uξη −
a0 (x)
(uξ − uη ) = 0
4
(case when q 6= 0).
We are looking for a change of variables in the form
µ = φ(ξ), ν = ψ(η), v(µ, ν) = u(ξ, η)eθ(ξ,η) .
(4.11)
The change of variables (4.11) leads to the following equations for the total
differentiations:
Dξ = φ0 Dµ , Dη = ψ 0 Dν .
Using the notation (4.11), we obtain
φ0 vµ = (uξ + uθξ )eθ ,
ψ 0 vν = (uη + uθη )eθ ,
φ0 ψ 0 vµν = (uξη + uη θξ + uξ θη + u(θξη + θξ θη ))eθ .
It follows that Eq. (2.4) is written
!
!
2ψ 0
2φ0
uξ + θξ −
uη
uξη + θη −
q(φ + ψ)
q(φ + ψ)
!
2φ0 θη
2ψ 0 θξ
4φ0 ψ 0
+ θξη + θξ θη −
−
+
u = 0.
q(φ + ψ) q(φ + ψ) q 2 (φ + ψ)2
9
Let us use the notation
a0 (x)|x=ζ(η−ξ) = λ(η − ξ).
(4.12)
So, we can rewrite the Eq. (2.7) as
uξη −
λ(η − ξ)
(uξ − uη ) = 0.
4
(4.13)
Then Eq. (4.13) can be reduced to Eq. (2.4) by a change of variables (4.11),
when
λ
2ψ 0
=− ,
q(φ + ψ)
4
0
2φ
λ
θξ −
= ,
q(φ + ψ)
4
0
2φ θη
2ψ 0 θξ
4φ0 ψ 0
θξη + θξ θη −
−
+ 2
= 0.
q(φ + ψ) q(φ + ψ) q (φ + ψ)2
Condition of compatibility for this system is written as
θη −
θξη =
λ0
2φ0 ψ 0
− 2
4
q (φ + ψ)2
Substituting the first two equations of the system in the third equation, and
using condition of compatibility the letter system we can rewritten as
2φ0 ψ 0
λ0 λ2
−
= 2
.
4
16
q (φ + ψ)2
(4.14)
Differentiating (4.14) with respect to ξ, η, adding together the resulting equations, and then dividing by φ0 ψ 0 , and differentiating with respect to ξ, η once
more, we obtain the equation for φ, ψ
φ00 ψ 0 + φ0 ψ 00 = 0.
(4.15)
The general solution is
φ = C2 eC1 ξ + C3 ,
ψ = C4 e−C1 η + C5 ,
C12 C2 C4 6= 0,
(4.16)
where C1 , C2 , C3 , C4 , C5 are arbitrary constant.
Then, using the solution (4.16), we obtain the change of variables:
µ = C1 ξ + C2 , ν = C3 η + C4 ,
Z
λ
2C3
v=−
dζ −
ln(C1 ξ − C3 η + C2 − C4 ),
4
qC1
10
(4.17)
The condition (4.14) is rewritten
λ0 λ2
2C1 C3
−
=
,
4
16
C1 ξ + C3 η + C2 + C4
and according to the condition (4.12), it is necessary that C3 = −C1 . From
this we can obtain the equation for the unknown function a(x):
2aa00 − a02 = −
2C12
.
C1 (ξ − η) + C2 + C4
(4.18)
Now we differentiate Eq. (4.18) eliminating the constants. We obtain Eq.
(2.8) from Ovsyanikov’s classification.
Applying the formula for change of variables (4.17) to the operator of the
canonical equation (2.4), we obtain the operators of the characteristic equation (2.7):
X1 =e−C1 ξ ∂ξ + eC1 η ∂η ,
X2 =(C1 ξ + C2 )e−C1 ξ ∂ξ − (C3 η + C4 )eC1 η ∂η ,
X3 =(C1 ξ + C2 )2 e−C1 ξ ∂ξ + (C3 η + C4 )2 eC1 η ∂η
λ
+ ((C1 ξ + C2 )2 e−C1 ξ − (C3 η + C4 )2 eC1 η ) u∂u ,
4
X4 =u∂u .
Let us find a change of variables which transforms Eq. (2.5)
vµν + µvµ + pνvν + pµνv = 0,
to Eq. (2.7)
uξη −
a0 (x)
(uξ − uη ) = 0
4
(case when q = 0).
Like wise, we seek the change of variables in the form
µ = φ(ξ), ν = ψ(η), v(µ, ν) = u(ξ, η)eθ(ξ,η) .
Repeating for Eq. (2.5) the same procedure as for Eq. (2.4), we find that
µ = C2 eC1 ξ + C3 , ν = C4 e−C1 η + C5 ,
Z
λ
−
dζ − C2 C4 ec1 (ξ−η) − C3 C4 e−c1 η − C2 C5 ec1 ξ + C6
4
v = ue
11
(4.19)
and
2aa00 − a02 = −16C12 C2 C4 eC1 (ξ−η) .
(4.20)
Let us differentiate Eq. (4.20), eliminating the constants. We obtain the
equation (2.9) from Ovsyaniikov’s classification.
Applying the formula for the change of variables (4.19) to the operators of
the canonical equation (2.5), we obtain the operators of the characteristic
equation (2.7):
X1 = C2 + C3 e−C1 ξ ∂ξ + C4 + C5 eC1 η ∂η ,
λ
X2 = e−C1 ξ ∂ξ − e−C1 ξ u∂u ,
4
λ
C1 η
X3 = −e ∂η − eC1 η u∂u ,
4
X4 = u∂u .
4.2
Transformation of Eq. (2.7) to Eq. (1.2)
In the case, when q 6= 0 we know the symmetries of the characteristic equation
(2.7). Applying the change of variables to these symmetries, we get the
operators of the original equation (1.2):
R
−1
R
−1
X1 = e−C1 (t− a dx) (∂t − a∂x ) + eC1 (t+ a dx) (∂t + a∂x ) ,
Z
R −1
−1
X2 = C1 t − a dx + C2 e−C1 (t− a dx) (∂t − a∂x ) −
Z
R −1
−1
− C3 t + a dx + C4 eC1 (t+ a dx) (∂t + a∂x ) ,
2
Z
R −1
λ
−1
X3 = C1 t − a dx + C2 e−C1 (t− a dx) u∂u −
4
2
Z
R −1
λ
− C3 t + a−1 dx + C4 eC1 (t+ a dx) u∂u ,
4
X4 = u∂u .
12
Similarly, in the case of q = 0, we obtain the symmetries of the original Eq.
(1.2):
R −1
R
X1 = C2 + C3 e−C1 (t− a dx) + C4 + C5 eC1 (t+ 1/adx) ∂t +
R 1
R
+ −C2 − C3 e−C1 (t− a dx) + C4 + C5 eC1 (t+ 1/adx) a∂x ,
R 1
λ
a∂x − e−C1 (t− a dx) u∂u ,
4
R −1
R −1
R −1
λ
X3 = −eC1 (t+ a dx) ∂t − eC1 (t+ a dx) a∂x − eC1 (t+ a dx) u∂u ,
4
X4 = u∂u .
X2 = e−C1 (t−
5
R
a−1 dx)
∂t − e−C1 (t−
R
a−1 dx)
Some invariant solutions
We construct invariant solutions for Eq. (1.2), where the form of a(x) is
taken from [7]. Let us consider the equation
utt − A2 e2Bx uxx = 0
admitting the operator
1
e−2Bx
1
2
X2 = At∂x +
ABt +
∂t + ABtu∂u .
2
AB
2
The invariants of the system are written
C
J1 = u(Ax + B)− 2 , J2 = t2 (Ax + B)C−1 −
(Ax + B)C−1
.
A2 (1 − C)2
Then the invariant solution is sought for in the form
C
(Ax + B)C−1
2
C−1
u = F t (Ax + B)
− 2
(Ax + B)− 2 ,
2
A (1 − C)
and for the function F we obtain the equation
F 00 λ2 + 2F 0 λ + F
C(C − 2)
= 0,
4(C − 1)2
(Ax + B)C−1
.
A2 (1 − C)2
The solution of this equation is
where λ = t2 (Ax + B)C−1 −
C
C −2
F = C1 λ 2 − 2C + C2 λ 2 − 2C .
13
Then the invariant solution can be written as
−Bx
C1 + C2 ln t2 eBx − Ae 2 B 2 B
q
u=
e 2 x.
−Bx
e
t2 eBx − A2 B 2
Now let us consider the equation
utt − (Ax + B)2C uxx = 0
admitting the operator
1
(Ax + B)2−2C
1
2
X2 = t(Ax + B)∂x +
A(1 − C)t +
∂t + ACtu∂u .
2
A(1 − C)
2
The invariants of the system are written
J1 = ue−
Bx
2
, J2 = t2 eBx −
e−Bx
.
A2 B 2
Then the invariant solution is sought for in the form
Bx
e−Bx
2 Bx
u = F t e − 2 2 e− 2 ,
AB
and for the function F we obtain the equation
F 00 λ2 + 2F 0 λ +
F
= 0,
4
e−Bx
.
A2 B 2
The solution of this equation is
where λ = t2 eBx −
C1
C2 ln(λ)
.
F =√ + √
λ
λ
Then the invariant solution can be written as
u=
C1
(Ax + B)1−C
t2 (Ax + B)C−1 −
A2 (1 − c)2
C
2 − 2C
+
C
c−2 !
1−C
(Ax
+
B)
2
−
2C
+C2 t2 (Ax + B)C−1 −
(Ax + B) 2 .
A2 (1 − c)2
14
6
Symmetry properties of linear wave equation utt − a2(x)b2(t)uxx = 0
Now we consider the linear wave equation of the form
utt − a2 (x)b2 (t)uxx = 0.
(1.1)
In the characteristic variables
Z
Z
Z
Z
1
1
ξ = bdt −
dx, η = bdt +
dx
a
a
this equation becomes
1 b0
1 b0
0
0
uξη +
− a uξ +
+ a uη = 0.
4 b2
4 b2
For arbitrary functions a(x), b(t), the Lie algebra is infinite and is spanned
by X0 = u∂u .
Let us carry out a group classification of this equation by Ovsyannikov’s
method.
The Laplase invariants are
1
2bb00 − b02
00
0 2
h=k=
2aa − (a ) +
.
16
b4
1. q = 0.
In this case we obtain the equations in unknown functions a(x), b(t) :
(a0 )2
aa00
+
C
,
1
a2
a2
2C2 b00 C2 (b0 )2
b000 =
−
.
b2
b3
The second equation admits the operators
a000 = 2C1
X1 = ∂t ,
X2 = t∂t − b∂b ,
X3 =
t2
∂t + tb∂b .
2
2. q 6= 0.
In this case, the equations become
2C1 (2aa00 − (a0 )2 ) 2aa000
4C1 (a000 )2
−
+
,
a3
a
2aa00 − (a0 )2
C2 b(2bb00 − (b0 )2 ) 8bb0 b00 b000 − 4(b0 )3 − 2b2 (b000 )2
=
−
.
b4
2bb00 − (b0 )2
a(IV ) =
b(IV )
15
The second equation admits the operators
X1 = ∂t ,
X2 = t∂t − b∂b ,
X3 =
t2
∂t + tb∂b .
2
Acknowledgements
This work was partially supported by Grant of the Government of Russian
Federation Resolution No. 220, Agreement No. 11.G34.31.0042.
First and foremost, I would like to thank to our supervisor of the laboratory
”GAMMETT” Prof. N. H. Ibragimov for the opportunity to participate in
the project and to work in the area of group analysis. The author is also
very grateful to Prof. R. K. Gazizov and A. A. Kasatkin for the teaching
on the group analysis. Besides, I would like to thank the researchers A.
A. Gainetdinova,E. D. Avdonina and L. R. Galiakberova for the valuable
guidance and advice.
References
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[4] L.V.
Ovsyannikov, Group analysis of differential equations, Nauka,
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16
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17