GROUP CLASSIFICATION OF LINEAR SECOND-ORDER DELAY ORDINARY DIFFERENTIAL EQUATION | Pue-on | Proceedings of ICMSA 2979 5742 1 SM
Proceedings of the 2nd IMT-GT Regional Conference on Mathematics, Statistics and Applications
Universiti Sains Malaysia
GROUP CLASSIFICATION OF LINEAR SECOND-ORDER DELAY
ORDINARY DIFFERENTIAL EQUATION
Prapart Pue-on
School of Mathematics, Institute of Science, Suranaree University of Technology, Thailand
e-mail: [email protected]
Abstract. The linear delay ordinary differential equation
y”( x) + a( x) y’ ( x) + b( x) y’ ( x − τ ) + c( x) y ( x) + d ( x) y ( x − τ ) = g ( x)
is studied, where the coefficients a ( x), b( x), c( x) and d ( x) and function g ( x) are arbitrary. In this manuscript,
group analysis is applied to find equivalent symmetries of the equation.
1 Introduction
Let us consider a linear second-order delay ordinary differential equation
y”( x) + a ( x) y’ ( x) + b( x) y’ ( x − τ ) + c( x) y ( x) + d ( x) y ( x − τ ) = g ( x).
(1)
For brevity, the symbol yτ will be used to denote y ( x − τ ) , y to denote y( x) and y’, yτ ’ will mean the first
derivatives of y at point x and x − τ , respectively. Then equation (1) can be simply written as
y” + a ( x) y’ + b( x) yτ ’ + c( x) y + d ( x) yτ = g ( x).
(2)
Here it is assumed that b2 + c 2 ≠ 0 . Application of this equation can be found in biology, physics and
engineering, where it is used to model natural phenomena.
2 Lie Group of Transformations
In [6], a definition of admitted Lie group of transformations for delay differential equations was developed. A
theory of equivalence Lie groups can be considered similar to that of admitted Lie groups. Here this analysis is
developed.
Let ϕ : Ω× Δ → Ω be a transformation, where Ω is a set of variables ( x, y, φ ), φ = (a, b, c, d , g ) and Δ ⊂ R is a
symmetric interval with respect to zero. The variable ε ∈ Δ is considered as a parameter of the transformation
ϕ . This transform maps the variables ( x, y, φ ) to variables (x , y , φ ) . Let ϕ (x, y, φ ; ε ) be denoted by
ϕε ( x, y, φ ) . The set of functions
ϕε
forms a one-parameter transformation Lie group of the space Ω if it
contains the identity transformation as well as inverse of its elements and their composition [4,5,6].
Alternatively, the notation x = ϕ x ( x, y, φ ; ε ), y = ϕ y ( x, y, φ ; ε ), φ = ϕ φ ( x, y, φ ; ε ) is used instead of
ϕε = (x , y , φ ) . The transformed variable y with delay term, it’s derivatives and it’s derivatives with delay term
dy
dy
and y τ’ =
are defined by y τ = y ( x − τ ) , y’ =
( x − τ ) , respectively.
dx
dx
Let us consider a delay differential equation
F ( x, y, yτ , y’, yτ ’, y”, φ ) = 0.
33
(3)
A Lie group of transformations is called admitted if each transformation maps a solution of the differential
equation to a solution of the same equation. Such transformation are called symmetries. For this reason,
equations for defining symmetries were constructed under the assumption that a Lie group of transformations
maps a solution of a delay differential equation into a solution of the same equation. This assumption leads to
∂F ( x , y , y τ , y’, y τ’, y”, φ )
( x, y, y , y’, y ’, y”, φ ) |
|ε = 0 = XF
τ
τ
!1.2) = 0.
∂ε
(4)
Here, the operator X is defined by
φ
X = (η − y’ξ )∂ y + (ητ − yτ ’ξτ )∂ yτ + η y’ ∂ y’ + η yτ ’ ∂ yτ ’ + η y” ∂ y” + (ζ − φxξ )∂ φ + ζ y ∂φ y .
where
∂ϕ x
∂ϕ y
( x, y, φ ; 0), η ( x, y, φ ) =
( x, y, φ ; 0),
∂ε
∂ε
∂ϕ φ
ζ ( x, y, φ ) =
( x, y, φ ; 0), ξτ ( x, y, φ ) = ξ ( x − τ , yτ , φτ ),
∂ε
η y’ = Dx (η − y’ξ ), η y” = Dx (η y’ ),
ξ ( x, y, φ ) =
ητ ( x, y, φ ) = η ( x − τ , yτ , φτ ), η yτ ’ = Dx (ητ − yτ ’ξτ ), ζ
φy
= Dy (ζ − φxξ ),
Dx = ∂ x + y’ ∂ y + y”∂ y’ + …+ yτ ’ ∂ yτ
+ (φx +φy y ′)∂φ +(φxx +φxy y’ )∂φx ,
Dy = ∂ y + φ y ∂φ + φxy ∂φx + …
cklund infinitesimal operator of a symmetry. Lie’s theory [4,5,6]
The operator X is called a canonical Lie-B a
shows that there is a one-to-one correspondence between generators and symmetries. This operator is also
equivalent to an infinitesimal generator [6]
X = ξ∂ x + η ∂ y + ζ ∂ φ .
Equation (4) gives a definition of an action of the infinitesimal generator of a Lie group onto a delay differential
equation.
1.
Determining Equations
Definition 1. A Lie group of transformations is called admitted if it satisfies the equation
( x, y, y , y’, y ’, y”, φ ) |
XF
(2.1) = 0.
τ
τ
(5)
for any solution of (3).
Equation (5) is called a determining equation. For solving the determining equation one can use the theory of
existence of a solution of an initial value problem for delay equation (2) [1]. This problem is formulated as
follows. Let a function χ ( x), x ∈ ( x0 − τ , x0 ) be given. Find a solution y ( x), x ∈ [ x0 , x0 + ε ) which satisfies the
condition
y ( x ) = χ ( x), x ∈ ( x0 − τ , x0 ).
Because the initial values are arbitrary, the variables y, yτ and their derivatives can be considered as arbitrary
elements. Thus, if the determining equation (5) is written as a polynomial of of variables and their derivatives,
the coefficients of these variables in the equations must vanish. The method for obtaining the overdetermined
system of equations is called splitting the determining equation. This gives an overdetermined system of partial
differential equations for the coefficients of the infinitesimal generator. The unknown functions ξ ,η and ζ can
be found by solving this system.
34
2. Equivalence Problem
The problem of finding all equations, which are equivalent to a given equation is called an equivalence problem.
If the given equation is a linear equation, then the equivalence problem is called a linearization problem. In this
section the importance of an equivalence Lie group of transformations is shown. Consider a linear second order
ordinary differential equation
y” + a ( x ) y’ + c ( x ) y = g ( x).
(6)
A Lie group of transformations of the independent variables, dependent variables and coefficients, which
conserves the differential structure of the equation is called an equivalent Lie group. This group allows
simplifying the coefficients of equations. For example, S. Lie showed that any linear second order ordinary
differential equation (6) is equivalent to the equation
y” = 0.
(7)
Equation (7) admits the eight-dimensional Lie algebra spanned by the generators
X1 =
∂
∂
∂
∂
∂
, X2 = , X3 = x , X4 = y , X5 = x ,
∂x
∂y
∂x
∂x
∂y
X6 = y
∂
∂
∂
∂
∂
, X 7 = x2
+ xy , X 8 = xy + y 2 .
∂y
∂y
∂x
∂y
∂x
If one tries to find an admitted Lie group for equation (6), then the system of determining equations consists of
four second-order ordinary differential equations. In general, this system cannot be solved. The purpose of this
manuscript is to do group classification of equation (2).
3 Equivalence Symmetries of (2)
Let y p be a particular solution of equation (2). Considering the change x = x, y = y − y p , equation (2) is
reduced to the equation
y” + a ( x) y’ + b( x) yτ ’ + c( x) y + d ( x) yτ = 0.
Similar to a second-order ordinary differential equation the coefficient a( x) can be reduced by change
y = v ( x)q ( x) with q satisfying the equation 2q’ + aq = 0 . We will consider equivalence symmetries of
equation
y” + b( x) yτ ’ + c( x) y + d ( x) yτ = 0.
(8)
instead of (2). Letting F = y” + b( x) y’τ + c( x) y + d ( x) yτ , then equation (5) becomes
X ( y” + b( x ) yτ ’ + c( x) y + d ( x) yτ ) |(3.1) = 0.
(9)
Splitting this equation with respect to y’, y τ ’, y’2τ , b”, c”, d”, b’, c’, d’ and later with respect to y, yτ , one finds
ξ = ξτ = α ,η = β y + γ , ζ b = b(−α’ + β − βτ ),
ζ c = − β” − 2cα’, ζ d = −bβτ ’ − 2dα’ + dα − dατ ,
where α ( x ), β ( x) are arbitrary periodic functions with period τ , and γ ( x) is an arbitrary solution of (3). Thus
the equivalence symmetry is
x = α ( x), y = β ( x) y + γ ( x).
(10)
35
4 Group Classification of Equation (3)
A Lie group of transformations admitted by equation (3) has to satisfy the following determining equation [3]
Y ( y” + b( x) yτ ’ + c( x) y + d ( x) yτ ) |(3.1) = 0.
(11)
−
Here, the operator Y is defined by
(
)
Y = (η - y'ξ ) ∂ y + ητ - yτ' ξτ ∂ yτ + η y' ∂ y' + η yτ ∂ y' + η y" ∂ y" .
'
τ
where
∂ϕ 4
∂ϕ y
( x, y; 0 ) , η ( x, y ) =
( x, y; 0 ) ,
∂ε
∂ε
ξτ ( x, y ) = ξ ( x - τ , yτ ) , ητ ( x, y ) = η ( x - τ , yτ ) ,
ξ ( x, y ) =
η y′ = Dx (η - y'ξ ) , η yτ′ = Dx (ητ - yτ′ξτ ) , η y′′′ = Dx (η y′ ) ,
Dx = ∂ x + y' ∂ y + y ′′∂ y' + " + yτ′ ∂ yτ + yτ′′∂ y' .
τ
Splitting this equation with respect to y ′, yτ′ , y2′τ and later with respect to y, yτ , one finds
ξ = ξτ , η = β y + γ ,
ξ xx = 2β , β" = -c'ξ - 2cξ x ,
γ " = -bγ τ′ - cγ − d γ τ ,
b ( β - βτ ) = b′ξ + ξ x b ,
(15)
d ( β - βτ ) = d ′ξ + bβτ′ + 2ξ x d.
(16)
'
(12)
(13)
(14)
By integrating (13), one finds β = ξ x / 2 + C1 , where C1 is an arbitrary constant. Since ξ = ξτ , it implies
β = βτ . Hence, integrating equation (15) one has
bξ = C2 ,
(17)
C2 is an arbitrary constant. Equation (16) is written as
b
d ′ξ + 2ξ x d = - ξ xx .
2
(18)
1. Case b ≠ 0, d ≠ 0
Substituting β into the second equation (13), one gets
ξξ xx −
ξ x2
2
+ 2cξ 2 = C3 ,
(19)
C3 is an arbitrary constant. If C2 ≠ 0 , then from equations (17), (18) and (19), one obtains
ξ=
⎛
⎞
C2
⎜ C
⎟
,η = y ⎜ 2 + C1 ⎟ + γ ,
⎜ 2b’
⎟
b
⎝
⎠
b” ⎤
b’
1⎡
3 b’
C5b 2 − ( ) 2 + ⎥ , d = + C4 b 2 ,
⎢
2⎣
2 b
2b ⎦
2
where C4 is an arbitrary constants, C5 = C3 /C2 , and γ ( x) is an arbitrary solution of (3). Since ξ = ξτ , the the
coefficient b has to satisfy the same property b = bτ . The infinitesimal generator obtained is
c=
36
⎛
y ⎛ 1 ⎞ ⎞⎟
⎜1
X = C1 y∂ y + C2 ⎜⎜ ∂ x + ⎜ ⎟’ ∂ y ⎟⎟ + γ∂ y .
2⎝b⎠ ⎠
⎝b
If C2 = 0 , then ξ = 0,η = C1 y + γ and all coefficients are arbitrary. The infinitesimal generator is
X = (C1 y + γ )∂ y .
(20)
(21)
2. Case b ≠ 0, d = 0
Solving equations (16), (17), (18) and the second equation of (13), one obtains
βτ = C6 , bξ = C2 , ξ = C7 x + C8 , cξ 2 = C9 , where C6 , C7 , C8 , C9 are arbitrary constants. Since ξ = ξτ , then
C7 = 0. If C8 ≠ 0 , then
c=
C9
C
,b = 2 .
2
C8
C8
(22)
The infinitesimal generator of the admitted Lie group is
X = C8 ∂ x + (C6 y + γ )∂ y .
(23)
If C8 = 0 , then ξ = 0,η = C6 y + γ , b and c are arbitrary, γ ( x) is an arbitrary solution of (8). The infinitesimal
generator is
X = (C 6 y + γ ) ∂ y .
1. Case b = 0, d ≠ 0
From equation (18) one finds d ξ 2 = C10 , where C10 is an arbitrary constant. Hence,
⎛C ⎞
1/ 2
⎛
⎜
C
d′
⎞
⎟
ξ = ⎜ 10 ⎟ ,η = − ⎜⎜ 10 3 / 2 + C1 ⎟⎟ y + γ .
⎜ 4 d
⎟
⎝ d ⎠
⎝
⎠
(24)
If C10 ≠ 0 , then from equation (19) one finds
c=
d ′ 1 d ′ 2 ⎤⎥
1 ⎡⎢ C3
⎢
d
+
+ ( ) ⎥.
2 ⎢⎢⎣ C10
2d 8 d 2 ⎥⎥⎦
(25)
The infinitesimal generator obtained is
X =
⎛ C101/ 2 d ′
⎞
C10
∂
+
+ C1 + γ ⎟ ∂ y .
⎜−
x
1/ 2
3/ 2
d
⎝ 2d
⎠
If C10 = 0 , then ξ = 0, β = C1 ,η = C1 y + γ and the coefficients c and
infinitesimal generator is
X = (C1 y + γ )∂ y .
(26)
d are arbitrary functions. Hence, the
(27)
5 Conclusion
The linear second-order delay ordinary differential equation is classified into three cases as the followings:
•
b ≠ 0 and d ≠ 0 . The infinitesimal generator admitted by the equation of this case is (20)
•
b ≠ 0 and d = 0 . The infinitesimal generator admitted by the equation of this case is (23)
•
b = 0 and d ≠ 0 . The infinitesimal generator admitted by the equation of this case is (26).
Acknowledgements
This work is supported by Ministry of University Affairs of Thailand (MUA) and Nakhon Phanom University.
Author is deeply indebted to S. V. Meleshko for his frequent help.
37
References
[1] R. D. Driver, Ordinary and Delay Differential Equations, New York: Springer-Verlag, 1977.
[2] J. Thanthanuch, Application of group analysis to functional differential equations, Ph.D. Thesis, Nakhonratchasima,
Thailand: Suranaree University of Technology, 2003.
[3] J. Thanthanuch and S. V. Meleshko, On definition of an admitted Lie group for functional differential equations,
Communication in Nonlinear Science and Numerical Simulation, 9(2004), 117–125.
[4] N. H. Ibragimov, Elementrary Lie Group Analysis and Ordinary Differential Equations, London: John Wiley & Sons
Ltd, 1999.
[5] N. H. Ibragimov (Ed.), Lie Group Analysis of Differential Equations, Vol. 1-3., Florida: CRC Press., 1994.
[6] L. V. Ovsiannikov, Group Analysis of Differential Equations, New York: Academic Press, 1982.
38
Universiti Sains Malaysia
GROUP CLASSIFICATION OF LINEAR SECOND-ORDER DELAY
ORDINARY DIFFERENTIAL EQUATION
Prapart Pue-on
School of Mathematics, Institute of Science, Suranaree University of Technology, Thailand
e-mail: [email protected]
Abstract. The linear delay ordinary differential equation
y”( x) + a( x) y’ ( x) + b( x) y’ ( x − τ ) + c( x) y ( x) + d ( x) y ( x − τ ) = g ( x)
is studied, where the coefficients a ( x), b( x), c( x) and d ( x) and function g ( x) are arbitrary. In this manuscript,
group analysis is applied to find equivalent symmetries of the equation.
1 Introduction
Let us consider a linear second-order delay ordinary differential equation
y”( x) + a ( x) y’ ( x) + b( x) y’ ( x − τ ) + c( x) y ( x) + d ( x) y ( x − τ ) = g ( x).
(1)
For brevity, the symbol yτ will be used to denote y ( x − τ ) , y to denote y( x) and y’, yτ ’ will mean the first
derivatives of y at point x and x − τ , respectively. Then equation (1) can be simply written as
y” + a ( x) y’ + b( x) yτ ’ + c( x) y + d ( x) yτ = g ( x).
(2)
Here it is assumed that b2 + c 2 ≠ 0 . Application of this equation can be found in biology, physics and
engineering, where it is used to model natural phenomena.
2 Lie Group of Transformations
In [6], a definition of admitted Lie group of transformations for delay differential equations was developed. A
theory of equivalence Lie groups can be considered similar to that of admitted Lie groups. Here this analysis is
developed.
Let ϕ : Ω× Δ → Ω be a transformation, where Ω is a set of variables ( x, y, φ ), φ = (a, b, c, d , g ) and Δ ⊂ R is a
symmetric interval with respect to zero. The variable ε ∈ Δ is considered as a parameter of the transformation
ϕ . This transform maps the variables ( x, y, φ ) to variables (x , y , φ ) . Let ϕ (x, y, φ ; ε ) be denoted by
ϕε ( x, y, φ ) . The set of functions
ϕε
forms a one-parameter transformation Lie group of the space Ω if it
contains the identity transformation as well as inverse of its elements and their composition [4,5,6].
Alternatively, the notation x = ϕ x ( x, y, φ ; ε ), y = ϕ y ( x, y, φ ; ε ), φ = ϕ φ ( x, y, φ ; ε ) is used instead of
ϕε = (x , y , φ ) . The transformed variable y with delay term, it’s derivatives and it’s derivatives with delay term
dy
dy
and y τ’ =
are defined by y τ = y ( x − τ ) , y’ =
( x − τ ) , respectively.
dx
dx
Let us consider a delay differential equation
F ( x, y, yτ , y’, yτ ’, y”, φ ) = 0.
33
(3)
A Lie group of transformations is called admitted if each transformation maps a solution of the differential
equation to a solution of the same equation. Such transformation are called symmetries. For this reason,
equations for defining symmetries were constructed under the assumption that a Lie group of transformations
maps a solution of a delay differential equation into a solution of the same equation. This assumption leads to
∂F ( x , y , y τ , y’, y τ’, y”, φ )
( x, y, y , y’, y ’, y”, φ ) |
|ε = 0 = XF
τ
τ
!1.2) = 0.
∂ε
(4)
Here, the operator X is defined by
φ
X = (η − y’ξ )∂ y + (ητ − yτ ’ξτ )∂ yτ + η y’ ∂ y’ + η yτ ’ ∂ yτ ’ + η y” ∂ y” + (ζ − φxξ )∂ φ + ζ y ∂φ y .
where
∂ϕ x
∂ϕ y
( x, y, φ ; 0), η ( x, y, φ ) =
( x, y, φ ; 0),
∂ε
∂ε
∂ϕ φ
ζ ( x, y, φ ) =
( x, y, φ ; 0), ξτ ( x, y, φ ) = ξ ( x − τ , yτ , φτ ),
∂ε
η y’ = Dx (η − y’ξ ), η y” = Dx (η y’ ),
ξ ( x, y, φ ) =
ητ ( x, y, φ ) = η ( x − τ , yτ , φτ ), η yτ ’ = Dx (ητ − yτ ’ξτ ), ζ
φy
= Dy (ζ − φxξ ),
Dx = ∂ x + y’ ∂ y + y”∂ y’ + …+ yτ ’ ∂ yτ
+ (φx +φy y ′)∂φ +(φxx +φxy y’ )∂φx ,
Dy = ∂ y + φ y ∂φ + φxy ∂φx + …
cklund infinitesimal operator of a symmetry. Lie’s theory [4,5,6]
The operator X is called a canonical Lie-B a
shows that there is a one-to-one correspondence between generators and symmetries. This operator is also
equivalent to an infinitesimal generator [6]
X = ξ∂ x + η ∂ y + ζ ∂ φ .
Equation (4) gives a definition of an action of the infinitesimal generator of a Lie group onto a delay differential
equation.
1.
Determining Equations
Definition 1. A Lie group of transformations is called admitted if it satisfies the equation
( x, y, y , y’, y ’, y”, φ ) |
XF
(2.1) = 0.
τ
τ
(5)
for any solution of (3).
Equation (5) is called a determining equation. For solving the determining equation one can use the theory of
existence of a solution of an initial value problem for delay equation (2) [1]. This problem is formulated as
follows. Let a function χ ( x), x ∈ ( x0 − τ , x0 ) be given. Find a solution y ( x), x ∈ [ x0 , x0 + ε ) which satisfies the
condition
y ( x ) = χ ( x), x ∈ ( x0 − τ , x0 ).
Because the initial values are arbitrary, the variables y, yτ and their derivatives can be considered as arbitrary
elements. Thus, if the determining equation (5) is written as a polynomial of of variables and their derivatives,
the coefficients of these variables in the equations must vanish. The method for obtaining the overdetermined
system of equations is called splitting the determining equation. This gives an overdetermined system of partial
differential equations for the coefficients of the infinitesimal generator. The unknown functions ξ ,η and ζ can
be found by solving this system.
34
2. Equivalence Problem
The problem of finding all equations, which are equivalent to a given equation is called an equivalence problem.
If the given equation is a linear equation, then the equivalence problem is called a linearization problem. In this
section the importance of an equivalence Lie group of transformations is shown. Consider a linear second order
ordinary differential equation
y” + a ( x ) y’ + c ( x ) y = g ( x).
(6)
A Lie group of transformations of the independent variables, dependent variables and coefficients, which
conserves the differential structure of the equation is called an equivalent Lie group. This group allows
simplifying the coefficients of equations. For example, S. Lie showed that any linear second order ordinary
differential equation (6) is equivalent to the equation
y” = 0.
(7)
Equation (7) admits the eight-dimensional Lie algebra spanned by the generators
X1 =
∂
∂
∂
∂
∂
, X2 = , X3 = x , X4 = y , X5 = x ,
∂x
∂y
∂x
∂x
∂y
X6 = y
∂
∂
∂
∂
∂
, X 7 = x2
+ xy , X 8 = xy + y 2 .
∂y
∂y
∂x
∂y
∂x
If one tries to find an admitted Lie group for equation (6), then the system of determining equations consists of
four second-order ordinary differential equations. In general, this system cannot be solved. The purpose of this
manuscript is to do group classification of equation (2).
3 Equivalence Symmetries of (2)
Let y p be a particular solution of equation (2). Considering the change x = x, y = y − y p , equation (2) is
reduced to the equation
y” + a ( x) y’ + b( x) yτ ’ + c( x) y + d ( x) yτ = 0.
Similar to a second-order ordinary differential equation the coefficient a( x) can be reduced by change
y = v ( x)q ( x) with q satisfying the equation 2q’ + aq = 0 . We will consider equivalence symmetries of
equation
y” + b( x) yτ ’ + c( x) y + d ( x) yτ = 0.
(8)
instead of (2). Letting F = y” + b( x) y’τ + c( x) y + d ( x) yτ , then equation (5) becomes
X ( y” + b( x ) yτ ’ + c( x) y + d ( x) yτ ) |(3.1) = 0.
(9)
Splitting this equation with respect to y’, y τ ’, y’2τ , b”, c”, d”, b’, c’, d’ and later with respect to y, yτ , one finds
ξ = ξτ = α ,η = β y + γ , ζ b = b(−α’ + β − βτ ),
ζ c = − β” − 2cα’, ζ d = −bβτ ’ − 2dα’ + dα − dατ ,
where α ( x ), β ( x) are arbitrary periodic functions with period τ , and γ ( x) is an arbitrary solution of (3). Thus
the equivalence symmetry is
x = α ( x), y = β ( x) y + γ ( x).
(10)
35
4 Group Classification of Equation (3)
A Lie group of transformations admitted by equation (3) has to satisfy the following determining equation [3]
Y ( y” + b( x) yτ ’ + c( x) y + d ( x) yτ ) |(3.1) = 0.
(11)
−
Here, the operator Y is defined by
(
)
Y = (η - y'ξ ) ∂ y + ητ - yτ' ξτ ∂ yτ + η y' ∂ y' + η yτ ∂ y' + η y" ∂ y" .
'
τ
where
∂ϕ 4
∂ϕ y
( x, y; 0 ) , η ( x, y ) =
( x, y; 0 ) ,
∂ε
∂ε
ξτ ( x, y ) = ξ ( x - τ , yτ ) , ητ ( x, y ) = η ( x - τ , yτ ) ,
ξ ( x, y ) =
η y′ = Dx (η - y'ξ ) , η yτ′ = Dx (ητ - yτ′ξτ ) , η y′′′ = Dx (η y′ ) ,
Dx = ∂ x + y' ∂ y + y ′′∂ y' + " + yτ′ ∂ yτ + yτ′′∂ y' .
τ
Splitting this equation with respect to y ′, yτ′ , y2′τ and later with respect to y, yτ , one finds
ξ = ξτ , η = β y + γ ,
ξ xx = 2β , β" = -c'ξ - 2cξ x ,
γ " = -bγ τ′ - cγ − d γ τ ,
b ( β - βτ ) = b′ξ + ξ x b ,
(15)
d ( β - βτ ) = d ′ξ + bβτ′ + 2ξ x d.
(16)
'
(12)
(13)
(14)
By integrating (13), one finds β = ξ x / 2 + C1 , where C1 is an arbitrary constant. Since ξ = ξτ , it implies
β = βτ . Hence, integrating equation (15) one has
bξ = C2 ,
(17)
C2 is an arbitrary constant. Equation (16) is written as
b
d ′ξ + 2ξ x d = - ξ xx .
2
(18)
1. Case b ≠ 0, d ≠ 0
Substituting β into the second equation (13), one gets
ξξ xx −
ξ x2
2
+ 2cξ 2 = C3 ,
(19)
C3 is an arbitrary constant. If C2 ≠ 0 , then from equations (17), (18) and (19), one obtains
ξ=
⎛
⎞
C2
⎜ C
⎟
,η = y ⎜ 2 + C1 ⎟ + γ ,
⎜ 2b’
⎟
b
⎝
⎠
b” ⎤
b’
1⎡
3 b’
C5b 2 − ( ) 2 + ⎥ , d = + C4 b 2 ,
⎢
2⎣
2 b
2b ⎦
2
where C4 is an arbitrary constants, C5 = C3 /C2 , and γ ( x) is an arbitrary solution of (3). Since ξ = ξτ , the the
coefficient b has to satisfy the same property b = bτ . The infinitesimal generator obtained is
c=
36
⎛
y ⎛ 1 ⎞ ⎞⎟
⎜1
X = C1 y∂ y + C2 ⎜⎜ ∂ x + ⎜ ⎟’ ∂ y ⎟⎟ + γ∂ y .
2⎝b⎠ ⎠
⎝b
If C2 = 0 , then ξ = 0,η = C1 y + γ and all coefficients are arbitrary. The infinitesimal generator is
X = (C1 y + γ )∂ y .
(20)
(21)
2. Case b ≠ 0, d = 0
Solving equations (16), (17), (18) and the second equation of (13), one obtains
βτ = C6 , bξ = C2 , ξ = C7 x + C8 , cξ 2 = C9 , where C6 , C7 , C8 , C9 are arbitrary constants. Since ξ = ξτ , then
C7 = 0. If C8 ≠ 0 , then
c=
C9
C
,b = 2 .
2
C8
C8
(22)
The infinitesimal generator of the admitted Lie group is
X = C8 ∂ x + (C6 y + γ )∂ y .
(23)
If C8 = 0 , then ξ = 0,η = C6 y + γ , b and c are arbitrary, γ ( x) is an arbitrary solution of (8). The infinitesimal
generator is
X = (C 6 y + γ ) ∂ y .
1. Case b = 0, d ≠ 0
From equation (18) one finds d ξ 2 = C10 , where C10 is an arbitrary constant. Hence,
⎛C ⎞
1/ 2
⎛
⎜
C
d′
⎞
⎟
ξ = ⎜ 10 ⎟ ,η = − ⎜⎜ 10 3 / 2 + C1 ⎟⎟ y + γ .
⎜ 4 d
⎟
⎝ d ⎠
⎝
⎠
(24)
If C10 ≠ 0 , then from equation (19) one finds
c=
d ′ 1 d ′ 2 ⎤⎥
1 ⎡⎢ C3
⎢
d
+
+ ( ) ⎥.
2 ⎢⎢⎣ C10
2d 8 d 2 ⎥⎥⎦
(25)
The infinitesimal generator obtained is
X =
⎛ C101/ 2 d ′
⎞
C10
∂
+
+ C1 + γ ⎟ ∂ y .
⎜−
x
1/ 2
3/ 2
d
⎝ 2d
⎠
If C10 = 0 , then ξ = 0, β = C1 ,η = C1 y + γ and the coefficients c and
infinitesimal generator is
X = (C1 y + γ )∂ y .
(26)
d are arbitrary functions. Hence, the
(27)
5 Conclusion
The linear second-order delay ordinary differential equation is classified into three cases as the followings:
•
b ≠ 0 and d ≠ 0 . The infinitesimal generator admitted by the equation of this case is (20)
•
b ≠ 0 and d = 0 . The infinitesimal generator admitted by the equation of this case is (23)
•
b = 0 and d ≠ 0 . The infinitesimal generator admitted by the equation of this case is (26).
Acknowledgements
This work is supported by Ministry of University Affairs of Thailand (MUA) and Nakhon Phanom University.
Author is deeply indebted to S. V. Meleshko for his frequent help.
37
References
[1] R. D. Driver, Ordinary and Delay Differential Equations, New York: Springer-Verlag, 1977.
[2] J. Thanthanuch, Application of group analysis to functional differential equations, Ph.D. Thesis, Nakhonratchasima,
Thailand: Suranaree University of Technology, 2003.
[3] J. Thanthanuch and S. V. Meleshko, On definition of an admitted Lie group for functional differential equations,
Communication in Nonlinear Science and Numerical Simulation, 9(2004), 117–125.
[4] N. H. Ibragimov, Elementrary Lie Group Analysis and Ordinary Differential Equations, London: John Wiley & Sons
Ltd, 1999.
[5] N. H. Ibragimov (Ed.), Lie Group Analysis of Differential Equations, Vol. 1-3., Florida: CRC Press., 1994.
[6] L. V. Ovsiannikov, Group Analysis of Differential Equations, New York: Academic Press, 1982.
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