Sound & Vibration

DOI: 10.32604/sv.2022.014547

ARTICLE

An Alternative Algorithm for the Symmetry Classification of Ordinary Differential Equations

Yi Tian1,2,* and Jing Pang3,4

1College of Data Science and Application, Inner Mongolia University of Technology, Hohhot, 010080, China
2Inner Mongolia Autonomous Region Engineering and Technology Research Center of Big Data Based Software Service, Hohhot, 010080, China
3College of Science, Inner Mongolia University of Technology, Hohhot, 010051, China
4Inner Mongolia Key Laboratory of Statistical Analysis Theory for Life Data and Neural Network Modeling, Inner Mongolia University of Technology, Hohhot, 010051, China
*Corresponding Author: Yi Tian. Email: ttxsun@163.com
Received: 07 October 2020; Accepted: 10 June 2021

1  Introduction

In the past decades, a wealth of methods have been developed to deal with exact solutions of differential equations (DEs), which include partial differential equations (PDEs) and ordinary differential equations (ODEs). Some of the most important methods are the homotopy perturbation method [1–3], variational iteration method [4–6], Taylor series method [7], and the exp-function method [8–10], etc. At present, symmetries of PDEs are widely used in mechanics, mathematics and physics fields, from symmetries of a PDEs, one can obtain more important information on solving PDEs, such as exact solutions, conservation laws, and integral factors, etc. Hence the topics finding symmetries of a PDEs have being brought about the interest of more and more people. We have done some work on symmetries of PDEs, in reference [11] the Wu’s method is used to complete symmetry classification of PDEs, in reference [12] the traditional Lie algorithm is used to complete symmetry classification of the diffusion-convection equation, in reference [13] the Wu’s method is used to simplify the symmetry computation of PDEs and a special symmetry reduction approach is used for a class of wave equations.

In my memory, researchers always pay attention to PDEs, well there are not enough research results on ODEs, especially on the symmetries of DEs in literature. In this paper, we mainly consider a class of ODEs appeared in the fields of physics and engineering, named the generalizations of the Kummer-Schwarz equations [14]:

y′′′y′+n(y′′y′)2=f(y)(y′)2, (1)

and

y′′′y′+n(y′′y′)2=f(y), (2)

where n ∈ R is a constant, not necessarily integer.

In symmetry analysis of DEs, the first step is determining symmetries of the given equations, this process can be come down to solving a determining equations, which is however sometimes large and not easy to solve directly, in this paper, differential form Wu’s method [11,15] is used to decompose the determining equations into a series of equations, which are easy to be solved relatively.

The Wu’s method (also named characteristic set algorithm) [16] established by the Chinese mathematician Wu Wen Tsun in the 1970s. It also has become a fundamental algorithmic theory in algebraic geometry together with the Gröbner base algorithm [17]. The method has been applied in a wide range of science fields, such as mechanical theorem proving [18], optimization problems, surface-fitting problems in CAGD, Bar Linkage Design, ⋅⋅⋅ , etc. [16]. The differential analogue of Wu’s method was proposed in the 1980s [19]. The method is more especially on target to deal with the zero set of a differential polynomial system (dps) and efficient differential elimination without directly involving the concept of an algebra ideal. As far as i know, this is the first paper on symmetry classification of ODEs based on Wu’s method.

2  Lie Point Symmetries of Eq. (1)

To begin with, we note that Eq. (1) can be written as

y′′′y′+n(y′′)2−f(y)(y′)4=0, (3)

The point symmetry

x∗=x+εξ(x,y)+O(ε2),

y∗=y+εη(x,y)+O(ε2), (4)

is admitted by Eq. (3) if and only if it satisfies the determining equation

X(3)(y′′′y′+n(y′′)2−f(y)(y′)4)=0, (5)

for any y that solves Eq. (3).

X=ξ(x,y)∂∂x+η(x,y)∂∂y, (6)

is the infinitesimal generator of the point symmetry (4).

X(3)=X+η(1)∂∂y′+η(2)∂∂y′′+η(3)∂∂y′′′, (7)

with

η(i)=Dxη(i−1)−y(i)Dxξ,i=1,2,3,η(0)=η, (8)

is the three-order extension (prolongation) of X.

Dx=∂x+y′∂y+y′′∂y′+⋯+y(k)∂y(k−1)⋯ (9)

is the total derivative operator.

The determining Eq. (5) simplify to

DTE={ηx=0,−3ξy−nξy=0,−6ξyy−2nξyy=0,−f(y)ξy−ξyyy=0,−3ξxx−2nξxx=0,−3ξxxy=0,−ξxxx=0,−9ξxy−4nξxy+3ηyy+2nηyy=0,−3ξxyy−2f(y)ηy+ηyyy−ηf′(y)=0, (10)

taking left hand side for each equation, we have following corresponding differential polynomial system

DPS={ηx,−3ξy−nξy,−6ξyy−2nξyy,−f(y)ξy−ξyyy,−3ξxx−2nξxx,−3ξxxy,−ξxxx,−9ξxy−4nξxy+3ηyy+2nηyy,−3ξxyy−2f(y)ηy+ηyyy−ηf′(y), (11)

Step 1 Compute Zero(DPS).

Under the rank x≺y≺ξ≺η , using the differential form Wu’s method, we obtain characteristic set

DCS1={ξy,ξxx,η}, (12)

with IS products I1∗I2∗I3≠0 , and

I1= 3f′(y)2−2f(y)f′′(y), (13)

I2= 3 + n, (14)

I3= 3 + 2n, (15)

then, we get

Zero(DPS)=Zero(DCS1/I1∗I2∗I3)+Zero(DPS,I1)+Zero(DPS,I2)+Zero(DPS,I3), (16)

in which

Zero(DCS1/I1∗I2∗I3)= {ξ= c1x+c2,η=0}, (17)

and c1, c2 are arbitrary constants.

Step 2 I1 = 0, compute Zero(DPS, I1).

Obtain characteristic set

DCS2={ξy,ξxx,ηx,2f(y)ηy+ηf′(y)}, (18)

with IS products I2∗I3≠0 , then, we get

Zero(DPS,I1)= Zero(DCS2/I2∗I3) (19)

in which

Case 1. Zero(DCS2/I2∗I3)= {ξ= c3x+c4,η=c5(e6y)1/4c1e6y+c2} , with the corresponding function f(y)=c1e6y2+c2e−6y2 , and c1, c2, ⋅ ⋅ ⋅ c5 are arbitrary constants.

Case 2. Zero(DCS2/I2∗I3)= {ξ= c1x+c2,η=F(y)} , with the corresponding function f(y) = 0, F(y) is an arbitrary function of y, and c1, c2 are arbitrary constants.

Step 3 I2 = 0, compute Zero(DPS, I2).

The determining Eq. (5) simplify to

DTE1={ηx=0,ξxx=0,−f(y)ξy−ξyyy=0,3ξxy−3ηyy=0,8pt−3ξxyy−2f(y)ηy+ηyyy−ηf′(y)=0, (20)

taking left hand side for each equation, we have following corresponding differential polynomial system:

DPS1={ηx,ξxx,−f(y)ξy−ξyyy,3ξxy−3ηyy,−3ξxyy−2f(y)ηy+ηyyy−ηf′(y), (21)

under the rank x≺y≺ξ≺η , using the differential form Wu’s method, we obtain characteristic set

DCS3={f(y)ξy+ξyyy,ξxy,ξxx,η}, (22)

with IS products I4 ≠ 0, and

I4=27f′(y)4−18f(y)f′(y)2f′′(y)−56f′′(y)3+72f′(y)f′′(y)f(3)(y)−18f′(y)2f(4)(y), (23)

then, we get

Zero(DPS,I2)= Zero(DCS3/I4)+ Zero(DPS,I2,I4), (24)

in which

Case 1. Zero(DCS3/I4) = {ξ = c1x + c2, η = 0}, with the corresponding function f(y) is an arbitrary function of y, and c1, c2 are arbitrary constants.

Case 2. Zero(DCS3/I4)={ξ=c1x+12c2u2+c3u+c4,η=0} , with the corresponding function f(y) is an arbitrary function of y, and c1, c2, ⋅ ⋅ ⋅ , c4 are arbitrary constants.

Case 3. Zero(DCS3/I4) = {ξ = c1x + F(y), η = 0}, with the corresponding function f(y) = 0, F(y) is an arbitrary function of y, and c1 is an arbitrary constant.

Step 3.1 I4 = 0, compute Zero(DPS, I2, I4).

Obtain characteristic set

DCS4={f(y)ξy+ ξyyy,ξxx,ηx,3ηyf′(y)+ηf′′(y),9ξxyf′(y)2−4ηf′′(y)2+3ηf′(y)f(3)(y)}, (25)

with IS products is empty set, then, we get

Zero(DPS,I2,I4)= Zero(DCS4), (26)

in which

Case 1. Zero(DCS4)= {ξ= c2x+c3+ c4sin⁡(c1y)+c5cos⁡(c1y),η=0} , with the corresponding function f(y) = c1 ≥ 0, and c1, c2, ⋅ ⋅ ⋅ c5 are arbitrary constants.

Case 2. Zero(DCS4)= {ξ= c1x+c2,η=F(y)} , and with the corresponding function f(y) = c3, F(y) is an arbitrary function of y, and c1, c2, c3 are arbitrary constants.

Case 3. Zero(DCS4)= {ξ= (c4x+c6)e−c1y+(c3x+c7)ec1y+c2x+c5,η=F(y)} , with the corresponding function f(y)=−c12 , F(y) is an arbitrary function of y, and c1, c2, ⋅ ⋅ ⋅ , c7 are arbitrary constants.

Case 4. Zero(DCS4)= {ξ= c1x+c2,η=0} , with the corresponding function f(y) = c3, and c1, c2, c3 are arbitrary constants.

Case 5. Zero(DCS4)= {ξ= (c4x+c6)e−c1y+(c3x+c7)ec1y+c2x+c5,η=0} , with the corresponding function f(y)=−c12 , and c1, c2, ⋅ ⋅ ⋅ , c7 are arbitrary constants.

Case 6. Zero(DCS4)= {ξ= c1x+c2,η=c3y+c4} , with the corresponding function f(y)=c5(c3y+c4)2 , and c1, c2, ⋅ ⋅ ⋅ , c5 are arbitrary constants.

Case 7. Zero(DCS4)= {ξ= c1x+12c2y2+c3y+c4,η=c5y+c6} , with the corresponding function f(y) = 0, and c1, c2, ⋅ ⋅ ⋅ , c6 are arbitrary constants.

Step 4 I3 = 0, compute Zero(DPS, I3).

The determining Eq. (5) simplify to

DTE2={ηx=0,ξy=0,−2ξxxx=0,−4f(y)ηy+2ηyyy−2ηf′(y)=0, (27)

taking left hand side for each equation, we have following corresponding differential polynomial system

DPS2={ηx,ξy,−2ξxxx,−4f(y)ηy+2ηyyy−2ηf′(y), (28)

under the rank x≺y≺ξ≺η , using the differential form Wu’s method, we obtain characteristic set

DCS5={ξy,ξxxx,ηx,2f(y)ηy−ηyyy+ηf′(y)}, (29)

with IS products is empty set, and we get

Zero(DPS,I3)= Zero(DCS5), (30)

in which

Case 1. Zero(DCS5)= {ξ= 12c1x2+c2x + c3,η=0} , with the corresponding function f(y) is an arbitrary function of y, and c1, c2, c3 are arbitrary constants.

Case 2. Zero(DCS5)= {ξ= 12c1x2+c2x+ c3,η=F(y)} , with the corresponding function f(y) = −F′(y)22+F(y)F′′(y)+c4F(y)2 , and c1, c2, ⋅ ⋅ ⋅ , c4 are arbitrary constants.

At the end of this section, we give the symmetry classification for Eq. (1) with n = 0 based on differential form Wu’s method.

As stated before, the determining Eq. (5) simplify to

DTE3={ξy=0,ηyy=0,−3ξxx+3ηxy=0,−3f(y)ηx+3ηxyy=0,−ξxxx+3ηxxy=0,ηxxx=0,−2f(y)ηy+ηyyy−ηf′(y)=0, (31)

taking left hand side for each equation, we have following corresponding differential polynomial system:

DPS3={ξy,ηyy,−3ξxx+3ηxy,−3f(y)ηx+3ηxyy,−ξxxx+3ηxxy,ηxxx,−2f(y)ηy+ηyyy−ηf′(y), (32)

Step 1 Compute Zero(DPS3).

Under the rank x≺y≺ξ≺η , using the differential form Wu’s method, we obtain characteristic set

DCS6={ξy,ξxx,η}, (33)

with IS products I5 ≠ 0, and

I5=15f′(y)3−18f(y)f′(y)f′′(y)+ 4f(y)2f(3)(y), (34)

then, we get

Zero(DPS3)= Zero(DCS6/I5)+ Zero(DPS3,I5), (35)

in which

Zero(DCS6/I5)={ξ=c1x+c2,η=0}, (36)

and c1, c2 are arbitrary constants.

Step 2 I5 = 0, compute Zero(DPS3, I5).

Obtain characteristic set DCS6with IS products I6 ≠ 0, and

I6=3f′(y)2−2f(y)f′′(y), (37)

then, we get

Zero(DPS3,I5)= Zero(DCS6/I6)+ Zero(DPS3,I5,I6), (38)

in which

Zero(DCS6/I6)= Zero(DCS6/I5), (39)

Step 2.1 I6 = 0, compute Zero(DPS3, I5, I6).

Obtain characteristic set

DCS7={ξy,ξxx,ηx,2f(y)ηy+ηf′(y)} (40)

with IS products is empty set, and we get

Zero(DPS3,I5,I6)= Zero(DCS7), (41)

in which

Case 1. Zero(DCS7)= {ξ= c1x+c2,η=c5(c3y+c4)} , with the corresponding function f(y)=4(c3y+c4)2 , and c1, c2, ⋅ ⋅ ⋅ , c5 are arbitrary constants.

Case 2. Zero(DCS7)= {ξ= c1x+ c2,η=F(y)} ,with the corresponding function f(y) = 0, F(y) is an arbitrary function of y, and c1, c2 are arbitrary constants.

3  Lie Point Symmetries of Eq. (2)

The second generalized Kummer-Schwarz equation Eq. (2) can be written as

y′′′y′+n(y′′)2−f(y)(y′)2=0. (42)

Consider the third-order prolongation

X(3)=X+η(1)∂∂y′+η(2)∂∂y′′+η(3)∂∂y′′′, (43)

of a Lie point symmetry generator

X=ξ(x,y)∂∂x+η(x,y)∂∂y, (44)

of Eq. (42), then the invariance condition is

X(3)(y′′′y′+n(y′′)2−f(y)(y′)2)|(3)=0, (45)

the determining equations from (45) simplify to

DTE¯={ηx=0,−3ξy−nξy=0,−6ξyy−2nξyy=0,−ξyyy=0,−3ξxx−2nξxx=0,−3f(y)ξy−3ξxxy=0,−9ξxy−4nξxy+3ηyy+2nηyy=0,−3ξxyy+ηyyy=0,−2f(y)ξx−ξxxx−ηf′(y)=0, (46)

taking left hand side for each equation, we have following corresponding differential polynomial system

DPS¯={ηx,−3ξy−nξy,−6ξyy−2nξyy,−ξyyy,−3ξxx−2nξxx,−3f(y)ξy−3ξxxy,−9ξxy−4nξxy+3ηyy+2nηyy,−3ξxyy+ηyyy=0,−2f(y)ξx−ξxxx−ηf′(y)=0, (47)

Step 1 Compute Zero(DPS¯) .

Under the rank x≺y≺ξ≺η , using the differential form Wu’s method, we obtain characteristic set

DCS1¯={ξy,ξx,η}, (48)

with IS products I1¯∗I2¯∗I3¯≠0 , and

I1¯= f′(y)2f′′(y)−2f(y)f′′(y)2+ f(y)f′(y)f(3)(y), (49)

I2¯= 3 + n, (50)

I3¯= 3 + 2n, (51)

then, we get

Zero(DPS¯)=Zero(DCS1¯/I1¯∗I2¯∗I3¯)+Zero(DPS¯,I1¯)+Zero(DPS¯,I2¯)+Zero(DPS¯,I3¯), (52)

in which

Zero(DCS1¯/I1¯∗I2¯∗I3¯)= {ξ= c1,η=0}, (53)

and c1 is an arbitrary constant.

Step 2 I1¯= 0 , compute Zero(DPS¯,I1¯) .

Obtain characteristic set

DCS2¯={ξy,ηx,2f(y)ξx+ηf′(y),ηyf(y)f′(y)−ηf′(y)2+ηf(y)f′′(y)}, (54)

with IS products I2¯∗I3¯≠0 , then, we get

Zero(DPS¯,I1¯)= Zero(DCS2¯/I2¯∗I3¯), (55)

in which

Case 1. Zero(DCS2¯/I2¯∗I3¯)= {ξ= c3x+c4,η=c1y+c2} , with the corresponding function f(y)=c5(c1y+c2)−2c3c1 , and c1, c2, ⋅ ⋅ ⋅ c5 are arbitrary constants.

Case 2. Zero(DCS2¯/I2¯∗I3¯)= {ξ= c1,η=0} , with the corresponding function f(y)=(c3−u)c2c4 , and c1, c2, ⋅ ⋅ ⋅ c4 are arbitrary constants.

Case 3. Zero(DCS2¯/I2¯∗I3¯)= {ξ= c1,η=F(y)} , with the corresponding function f(y) = c2, F(y) is an arbitrary function of y, and c1, c2 are arbitrary constants.

Case 4. Zero(DCS2¯/I2¯∗I3¯)= {ξ= F(x),η=G(y)} , with the corresponding function f(y) = 0, F(x) is an arbitrary function of x,G(y)is an arbitrary function of y.

Step 3 I2¯= 0 , compute Zero(DPS¯,I2¯) .

The determining Eq. (45) simplify to

DTE1¯={ηx=0,−ξyyy=0,3ξxx=0,−3f(y)ξy−3ξxxy=0,3ξxy−3ηyy=0,−3ξxyy+ηyyy=0,−2f(y)ξx−ξxxx−ηf′(y)=0, (56)

taking left hand side for each equation, we have following corresponding differential polynomial system

DPS1¯={ηx,−ξyyy,3ξxx,−3f(y)ξy−3ξxxy,3ξxy−3ηyy,−3ξxyy+ηyyy,−2f(y)ξx−ξxxx−ηf′(y), (57)

under the rank x≺y≺ξ≺η , using the differential form Wu’s method, we obtain characteristic set

DCS3¯={ξx,ξy,η}, (58)

with IS products I1¯≠0 , then, we get

Zero(DPS¯,I2¯)= Zero(DCS3¯/I1¯)+ Zero(DPS¯,I2¯,I1¯), (59)

in which

Zero(DCS3¯/I1¯)={ξ=c1,η=0}, (60)

and c1is an arbitrary constant.

Step 3.1 I1¯= 0 , compute Zero(DPS¯,I2¯,I1¯) .

Obtain characteristic set DCS2¯ with IS products is empty set, then, we get

Zero(DPS¯,I2¯,I1¯)= Zero(DCS2¯)= Zero(DCS2¯/I2¯∗I3¯), (61)

Step 4 I3¯= 0 , compute Zero(DPS¯,I3¯) .

The determining Eq. (45) simplify to

DTE2¯={ηx=0,ξy=0,2ηyyy=0,−4f(y)ξx−2ξxxx−2ηf′(y)=0, (62)

taking left hand side for each equation, we have following corresponding differential polynomial system

DPS2¯={ηx,ξy,2ηyyy,−4f(y)ξx−2ξxxx−2ηf′(y), (63)

under the rank x≺y≺ξ≺η , using the differential form Wu’s method, we obtain characteristic set

DCS4¯={ξy,ξx,η}, (64)

with IS products I4¯≠0 , and

I4¯=−3f′(y)2f′′(y)2+6f(y)f′′(y)3+2f′(y)3f(3)(y),−6f(y)f′(y)f′′(y)f(3)(y)+ f(y)f′(y)2f(4)(y), (65)

then, we get

Zero(DPS¯,I3¯)= Zero(DCS4¯/I4¯)+ Zero(DPS¯,I3¯,I4¯), (66)

in which

Zero(DCS4¯/I4¯)={ξ=c1,η=0}, (67)

and c1 is an arbitrary constant.

Step 4.1 I4¯= 0 , compute Zero(DPS¯,I3¯,I4¯) .

Obtain characteristic set DCS2¯ with IS products I5¯≠0 , and

I5¯ = −3f′′(y)2+ 2f′(y)f(3)(y), (68)

then, we get

Zero(DPS¯,I3¯,I4¯)=Zero(DCS2¯/I5¯)+Zero(DPS¯,I3¯,I4¯,I5¯), (69)

in which

Case 1. Zero(DCS2¯/I5¯)= {ξ = c4x+c5,η=12c1y2+c2y+c3} , with the corresponding function f(y)=e−4c4arctan⁡(c1y+c22c1c3−c22)2c1c3−c22c6 , and c1, c2, ⋅ ⋅ ⋅ c6 are arbitrary constants.

Case 2. Zero(DCS2¯/I5¯)= {ξ= c1,η=F(y)} , with the corresponding function f(y) = c2, F(y) is an arbitrary function of y, and c1, c2 are arbitrary constants.

Case 3. Zero(DCS2¯/I5¯)= {ξ= c1,η=0} , with the corresponding function f(y)=ec5c2c3+2arctan⁡h(c3(c4+y)c2c3)c2c3 , and c1, c2, ⋅ ⋅ ⋅ , c5 are arbitrary constants.

Case 4. Zero(DCS2¯/I5¯)= {ξ= c1,η=0} , with the corresponding function f(y) = c2, and c1, c2 are arbitrary constants.

Case 5. Zero(DCS2¯/I5¯)= {ξ= F(x),η=G(y)} , w ith the corresponding function f(y) = 0, and F(x) is an arbitrary function of x,G(y) is an arbitrary function of y.

Step 4.2 I5¯= 0 , compute Zero(DPS¯,I3¯,I4¯,I5¯) .

Obtain characteristic set DCS2¯ with IS products is empty set, and we get

Zero(DPS¯,I3¯,I4¯,I5¯)= Zero(DCS2¯)= Zero(DCS2¯/I5¯). (70)

At the end of this section, we give the symmetry classification for Eq. (2) with n = 0 based on differential form Wu’s method.

As stated before, the determining Eq. (45) simplify to

DTE3¯={ξy=0,ηyy=0,−3ξxx+3ηxy=0,−f(y)ηx+ηxxx=0,−2f(y)ξx−ξxxx+3ηxxy−ηf′(y)=0, (71)

taking left hand side for each equation, we have following corresponding differential polynomial system:

DPS3¯={ξy,ηyy,−3ξxx+3ηxy,−f(y)ηx+ηxxx,−2f(y)ξx−ξxxx+3ηxxy−ηf′(y), (72)

Step 1 Compute Zero(DPS3¯) .

Under the rank x≺y≺ξ≺η , using the differential form Wu’s method, we obtain characteristic set

DCS5¯={ξy,ξx,η}, (73)

with IS products I1¯≠0 , then, we get

Zero(DPS3¯)= Zero(DCS5¯/I1¯)+ Zero(DPS3¯,I1¯), (74)

in which

Zero(DCS5¯/I1¯)={ξ=c1,η=0}, (75)

and c1 is an arbitrary constant.

Step 2 I1¯= 0 , compute Zero(DPS3¯,I1¯) .

Obtain characteristic set DCS2¯ with IS products is empty set, then, we get

Zero(DPS3¯,I1¯)= Zero(DCS2¯)= Zero(DCS2¯/I2¯∗I3¯). (76)

4  Conclusion

In this paper, Lie algorithm combined with differential form Wu’s method is used to complete the symmetry classification of ODEs containing arbitrary parameter. This process can be reduced to solve a system of determining equations, then the differential form Wu’s method is used to decompose the determining equations into a series of equations, which are easy to solve relatively. To illustrate the usefulness of this method, we apply it to the generalizations of the Kummer-Schwarz equations, and the results show the performance of the present work.

In addition, the second-order nonlinear ODE [20]

2(K(y)y′)′+xy′=0, (77)

and the following ODE [21]:

y′′=f(x)y2, (78)

are studied by many researchers. The algorithm of Wu’s method is performed on computer in this paper, we try to give the symmetry classification of Eqs. (77) and (78), after four days, the program is still running, only part of the results are obtained. In the next, we will make some improvements to the differential form Wu’s method, and expect to get the complete symmetry classification results.

Acknowledgement: The author is very thankful to Professor Jihuan He, who is work in national engineering laboratory for modern silk, college of textile and engineering, Soochow University, for his help.

Funding Statement: The authors received no specific funding for this study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.