An Alternative Algorithm for the Symmetry Classification of Ordinary Differential Equations

This is the first paper on symmetry classification for ordinary differential equations (ODEs) based on Wu’s method. We carry out symmetry classification of two ODEs, named the generalizations of the Kummer-Schwarz equations which involving arbitrary function. First, Lie algorithm is used to give the determining equations of symmetry for the given equations, which involving arbitrary functions. Next, differential form Wu’s method is used to decompose determining equations into a union of a series of zero sets of differential characteristic sets, which are easy to be solved relatively. Each branch of the decomposition yields a class of symmetries and associated parameters. The algorithm makes the classification become direct and systematic. Yuri Dimitrov Bozhkov, and Pammela Ramos da Conceição have used the Lie algorithm to give the symmetry classifications of the equations talked in this paper in 2020. From this paper, we can find that the differential form Wu’s method for symmetry classification of ODEs with arbitrary function (parameter) is effective, and is an alternative method.


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][2][3], variational iteration method [4][5][6], Taylor series method [7], and the exp-function method [8][9][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 000 y 0 þ n y 00 y 0 and y 000 y 0 þ n y 00 y 0 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, surfacefitting 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.
is the total derivative operator.
Under the rank x 0 y 0 n 0 g, using the differential form Wu's method, we obtain characteristic set with IS products I 1 ÃI 2 ÃI 3 6 ¼ 0, and then, we get and c 1 , c 2 are arbitrary constants.
Obtain characteristic set with IS products I 2 ÃI 3 6 ¼ 0, then, we get in which with the corresponding function is an arbitrary function of y, and c 1 , c 2 are arbitrary constants.
with the corresponding function f(y) is an arbitrary function of y, and c 1 , c 2 , ⋅ ⋅ ⋅ , c 4 are arbitrary constants.
with the corresponding function f(y) = 0, F(y) is an arbitrary function of y, and c 1 is an arbitrary constant.
Case 2. ZeroðDCS 4 Þ¼ fn¼ c 1 x þ c 2 ; g ¼ FðyÞg, and with the corresponding function f(y) = c 3 , F(y) is an arbitrary function of y, and c 1 , c 2 , c 3 are arbitrary constants.
with the corresponding function f(y) = c 3 , and c 1 , c 2 , c 3 are arbitrary constants.
The determining Eq. (5) simplify to taking left hand side for each equation, we have following corresponding differential polynomial system under the rank x 0 y 0 n 0 g, using the differential form Wu's method, we obtain characteristic set DCS 5 ¼ fn y ; n xxx ; g x ; 2f ðyÞg y À g yyy þ gf 0 ðyÞg; with IS products is empty set, and we get in which with the corresponding function f(y) is an arbitrary function of y, and c 1 , c 2 , c 3 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.
Obtain characteristic set DCS 2 with IS products is empty set, then, we get Step 4 I 3 ¼ 0, compute ZeroðDPS; I 3 Þ.
The determining Eq. (45) simplify to taking left hand side for each equation, we have following corresponding differential polynomial system under the rank x 0 y 0 n 0 g, using the differential form Wu's method, we obtain characteristic set with IS products I 4 6 ¼ 0, and and c 1 is an arbitrary constant.
Case 5. ZeroðDCS 2 =I 5 Þ¼ fn¼ FðxÞ; g ¼ GðyÞg, 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.
Obtain characteristic set DCS 2 with IS products is empty set, and we get ZeroðDPS; I 3 ; I 4 ; I 5 Þ¼ ZeroðDCS 2 Þ¼ ZeroðDCS 2 =I 5 Þ: 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 taking left hand side for each equation, we have following corresponding differential polynomial system: À2f ðyÞn x À n xxx þ 3g xxy À gf 0 ðyÞ; Step 1 Compute ZeroðDPS 3 Þ.
Under the rank x 0 y 0 n 0 g, using the differential form Wu's method, we obtain characteristic set DCS 5 ¼ fn y ; n x ; gg; with IS products I 1 6 ¼ 0, then, we get ZeroðDPS 3 Þ¼ ZeroðDCS 5 =I 1 Þþ ZeroðDPS 3 ; I 1 Þ; (74) in which and c 1 is an arbitrary constant.

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] 2ðKðyÞy 0 Þ 0 þ xy 0 ¼ 0; (77) and the following ODE [21]: y 00 ¼ f ðxÞy 2 ; (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.