Group Classification of Second-Order Linear Neutral Differential Equations

: In this paper, we shall extend the method of obtaining symmetries of ordinary differential equations to second-order non-homogeneous functional differential equations with variable coefficients. The existing research for delay differential equations defines a Lie-Bäcklund operator and uses the invariant manifold theorem to obtain the infinitesimal generators of the Lie group. However, we shall use a different approach that requires Taylor’s theorem for a function of several variables to obtain a Lie invariance condition and the determining equations for second-order functional differential equations. Certain standard results from the theory of ordinary differential equations have been employed to simplify the equation under study. The symmetry analysis of this equation was found to be non-trivial for arbitrary variable coefficients. In such cases, by selecting certain specific functions, arising in most practical models, we find the symmetries that are seen to be in terms of Bessel’s functions, Mathieu functions, etc. We then make a complete group classification of the second-order linear neutral differential equation, for which there is no existing literature.


Introduction
Functional differential equations (FDEs) are differential equations in which the unknown function appears with deviating arguments.One type of FDE of particular interest is called neutral differential equations (NDEs), which are differential equations in which the derivative term appears with delay.These equations are a more powerful and accurate way to model systems involving flip-flop circuits, the controlled motion of a rigid body, the human postural balance model, classical electrodynamics, etc.More information on the applications and methods of solving NDEs can be found in [1][2][3][4], and in general, information on FDEs can be found in [5].
As pointed out in [6], symmetries are transformations that leave an object unchanged or invariant and are very useful in the formation and study of laws of nature.In the past, the solvability of algebraic equations of a high degree posed a challenge to mathematicians.It was Galois who then associated a group with such equations, called the Galois group, to determine the solvability of such equations.It turned out that the solvability of the Galois group, an easier task, led to determining if the corresponding algebraic equation was solvable.This motivated Sophus Lie to study the solvability of differential equations by associating with them a group called the Lie group.
In [7,8], symmetries of delay differential equations are obtained by defining a certain operator equivalent to the canonical Lie-Bäcklund operator.In [9], equivalent symmetries of a second-order delay differential equation are obtained.However, in [9], an operator equivalent to the canonical Lie-Bäcklund operator and suitable other operators are defined.In [10], the authors exhaustively describe the Lie symmetries of systems of second-order linear ordinary differential equations with constant coefficients over both real and complex fields.They propose an algebraic approach to obtain bounds for the dimensions of the maximal Lie invariance algebras possessed by such systems.Further, such systems are thoroughly provided with their group classification in [11,12], with extensions to linear systems of second-order ordinary differential equations with more than two equations.Higher-order symmetries for ordinary differential equations are studied in [13].In [14], the author suggests a group method to study FDEs based on a search of symmetries of underdetermined differential equations by methods of classical and modern group analysis, using the principle of factorization.The method therein encompasses the use of a basis of invariants consisting of universal and differential invariants.Then, in [15], an admitted Lie group for first-order delay differential equations with constant coefficients is defined, and corresponding generators of the Lie group for this equation are obtained.The approach in [15] consists of using Lie-Bäcklund operators to obtain the determining equations.Lie symmetries of first-order NDEs with a general time delay have been found using a Lie type invariance condition obtained from Taylor's theorem for a function of several variables in [16] and for the case with constant delay in [17].While a thorough group classification for delay differential equations with applications is seen in [18][19][20], there has been no work for second-order NDEs with variable coefficients.As these equations would more accurately model several systems, this present paper attempts to bridge the gap and extend the results to second-order NDEs with variable coefficients.Recent studies include a complete classification of FDEs with constant coefficients to solvable Lie algebras that can be found in [21,22].
In this paper, we do a symmetry analysis of the second-order FDE.
(  d x dt evaluated at t -r, respectively.Our focus is to obtain the equivalent symmetries and the corresponding generators of the Lie group under which NDEs are invariant.We shall first find the admitted Lie group for equation (1).We then use this group to obtain the desired symmetries.In the absence of the term ( ) x t r ′′ − , equation (1) reduces to a second-order delay differential equation, the group classification of which can be found in [23].By following a completely different approach from the existing literature for delay differential equations, we, in this paper, extend the results of obtaining symmetries of ordinary differential equations found in [24] to obtain a complete group classification of second-order non-homogeneous linear NDEs with twice differentiable variable coefficients.
We shall use Taylor's theorem for a function of several variables to obtain a Lie-type invariance condition for NDEs.Using this, we obtain our determining equations.These equations are then split with respect to the independent variables to obtain an overdetermined system of partial differential equations, which are then solved to obtain the most general generator of the Lie group and the corresponding equivalent symmetries.It may be noted that a pro of our approach is that it does not lead to any magnification of the delay terms in the determining equations as compared to the existing literature.However, as a con of the approach, it is seen that in most cases, we do not get an explicit solution due to the arbitrariness of the variable coefficients.As such, we do not get explicit infinitesimal generators.By then choosing particular values of the variable coefficients or restricting our equation by choosing certain values of the obtained constants (which does not alter the symmetries obtained), we illustrate the infinitesimal generators of the admitted group, which are explicitly obtained, for these special cases.We finally make a thorough group classification of this second-order NDE.It is noteworthy to point out here that there is no existing literature on the group classification of NDEs that will aid in studying the properties of solutions of several models involving the use of such equations. (1) The rest of the paper is organized as follows: (i) The following section gives the necessary preliminaries.(ii) Section 3 sets up the Lie invariance condition using Taylor's theorem, a novel approach, and proves certain results from the theory of ordinary differential equations required in the simplification of our equation under study.(iii) Section 4 makes a thorough group classification of the second-order NDE by taking various cases that are of importance.(iv) In Section 5, the results are suitably illustrated.(v) The concluding section summarizes the results obtained and brings forth some research questions for interested researchers to continue with further work.

Preliminaries
We begin this section by stating the definition of Lie groups.,  , , ,

Lie invariance condition for second-order FDEs
In this section, we establish the Lie invariance condition for second-order FDEs.In order to determine this FDE completely, we need to specify the delay term.Otherwise, the problem is not fully determined.
)), It may be noted that the coefficients of the infinitesimal transformations ω(t, x) and ϒ (t, x) are as defined earlier in the previous section.We then naturally define Considering the second-order extended infinitesimals, we see that ϒ contains t, x, and x′, we need to extend the definition of D t .Hence, we have ) ) ( ).
Comparing the coefficient of δ, we get (2 Equation ( 4) is a Lie invariance condition.
We then, naturally define the extended operator, for equation ( 2) as The operator given by equation ( 5) will be acted upon the NDE under study to get the determining equations, from which the symmetries of the equation can be found.
The following lemmas shall be used for simplifying our analysis.Lemma 3.2 If x 1 (t) is an arbitrary solution of Proof.For invariance, we must have Using the given change of variables, we get

, x t a t x t b t x t r c t x t d t x t r k t x t r h t
which by the hypothesis on x 1 (t) reduces the above equation to equation (7).Lemma 3.3 By employing a suitable transformation, the NDE with twice differentiable variable coefficients can be reduced to a one in which the first-order ordinary derivative term is missing.Proof.By employing the transformation, ( ) ( ), ∫ is an arbitrary constant, equation (8), can be reduced to

Symmetries of the non-homogeneous linear second-order NDE
In this section, we shall obtain symmetries of the second-order non-homogeneous NDE with continuously differentiable variable coefficients given by Because of Lemma 3.3, we shall consider equivalent symmetries of
Splitting this equation with respect to ( ) x t r ′ − and using the fact that ( ) 0, k t ≠ we get 0.
x ω = This with equation ( 14) Splitting equation ( 13) with Substituting equations ( 15) and ( 16) into the determining equation ( 13), we get From equation ( 12), we have Splitting equation (17) with respect to x(t), we get Splitting equation (17) with respect to x'(t), we get Using equation ( 18), we get Splitting equation (17) with respect to the constant terms, we get That is, ρ(t) satisfies the homogeneous NDE of second-order given by equation (7).Splitting equation ( 17) with respect to x''(t − r), and using equations ( 18) and ( 21), we get

Group classification of equation (10)
Theorem 4. 1 The NDE (10) for which k(t) is not a constant, admits a two-dimensional group generated by 23) having to be true for an arbitrary β(t) and k(t) implies that for a nonconstant k(t), we must have, β(t) = 0 and consequently, ω(t, x) = 0 and The infinitesimal generator of the Lie group is given by where c 1 is an arbitrary constant and ρ(t) satisfies equation (7).Remark 4.2 Theorem 4.1 is useful in studying the symmetry properties of solutions of NDEs arising in the modeling of networks containing lossless transmission lines.
Having obtained the infinitesimal generator for the case when k(t) is non-constant, we now perform symmetry analysis and a complete group classification of the second-order NDE (9), for which where c 2 is an arbitrary constant.
Splitting equation ( 17) with respect to x(t − r) and using equation ( 21), we get Splitting equation (17) with respect to ( ) x t r ′ − and using equations ( 18) and ( 21), we get Equation ( 27) can be easily integrated to give where c 3 is an arbitrary constant.
In this case, we have obtained the coefficients of the infinitesimal transformation as The infinitesimal generator in this case is given by ( ) 2 ( ) where ρ(t) is an arbitrary solution of equation (10).If c 3 = 0, then The infinitesimal generator is given by Remark 4.4 Theorem 4.3 finds applications in the symmetry analysis of models involving the study of vibrating masses attached to an elastic bar.
Theorem 4. 5 The NDE (10) with 2 ( ) 0, ( ) 0, ( ) b t d t k t c ≠ = = admits the infinitesimal generator given by * 1 ( ) for ω(t) and A is a root (or zero) of Proof.If 3 0, c ≠ then substituting equation (34) into equation (32), we get This is a non-linear third-order differential equation, the solution ω(t) of which is given by where A is a root (or zero) of for y with

E c B = +
It is to be noted that the expression in equation ( 45) may be complex valued, and we are finding the zeroes for y.In this solution, c 7 , c 8 , and c 9 are arbitrary constants.To obtain the corresponding infinitesimal generator, we have to solve equation (45) for ω(t).The infinitesimal generator in this case is given by where Φ 1 (t) solves equation (45) for ω(t) and [ ] By considering a very special case in which c 2 = 1 = c 3 , we obtain from equation ( 32) Equation ( 48) yields a solution for which some infinitesimal generators can be explicitly found.This solution, in which c 11 , c 12 , and c 13 are arbitrary constants, is given by ( ) where G is a root (or zero) of for y.
The infinitesimal generator in this case is If c 15 = 0, then the infinitesimal generator is given by equation ( 47).Finally, if c 3 = 0, then the infinitesimal generator is given by equation ( 43).It may be noted that, using equation (39), Equation ( 52) can be integrated once to obtain where c 16 is an arbitrary constant.Equation ( 53) is extremely difficult to solve for an arbitrary d(t).If where Φ 3 (t) solves equation (53) for ω(t) and admits the five-dimensional Lie group generated by where The infinitesimal generator is given by The infinitesimal generator in this case is explicitly given by ( ) ( ) . 4 2 13 Corollary 4.12 is extremely important in studying the properties of solutions of models arising in the postural balance models for humans described in [26].
Corollary 4. 14 The NDE given by equation (10) for which b(t) = 0, d(t) = t m where m is any constant, k(t) = c 2 admits the five-dimensional Lie group generated by where c 28 and c 29 are arbitrary constants.From equation (29), we get The infinitesimal generator is given by Remark 4.15 Corollary 4.14 is useful in the analysis of the Cauchy-Euler differential equation with neutral delay appearing in modeling time-harmonic vibrations of a thin elastic rod, problems on annual and solid discs, wave mechanics, etc.

Illustrative examples
Example 5.1 Consider the second order NDE given by ( ) ( ) 0.
x t x t π ′′ ′′ + − = A solution of this differential equation is ( ) sin .
x t t = Following the procedure given in the previous section, we can show that ω(t, x) = c 38 , a constant and

Conclusion
The approach presented in this paper has the potential to expand the current methods for studying second-order non-homogeneous FDEs with variable coefficients.By using Taylor's theorem for a function of several variables, we have derived a Lie invariance condition and the determining equations for these equations, which can lead to new insights and techniques for solving them.Additionally, our complete group classification of the second-order linear NDE can provide a foundation for further research in this area.Our study has presented a novel approach for studying secondorder non-homogeneous FDEs with variable coefficients.Overall, our findings can contribute to the advancement of the field of differential equations.The research findings can be extended to include more choices of variable coefficients.This classification can be used in studying the properties of solutions arising in several models of practical importance, like those found in [27].

2
we get ( ) ( ). b t b t r = − Using equation (33) in equation (30), we get It can be easily seen that the generators corresponding to c 11 = 0 can be explicitly obtained.In this case, ω(t,x) = c 14 t + c 15 is a solution to equation (48), where c 14 and c 15 are arbitrary constants.The condition ω = Ω implies c 14 = 0. c ≠ then infinitesimal generator is given by

Theorem 4 . 8 2 (
The NDE given by equation(10) for which ) 0, ( ) 0, ( ) b solves equation (53), then the infinitesimal generator in this case is given by Definition 2.1([25]) Let t = (t 1 , t 2 , ..., t n ) lie in a region .definedforeach t in E, depending on the parameter δ lying in the set , S ⊂  with ϕ(δ, ) defining a law of composition of parameters δ and  in S, forms a one-parameter Lie group of transformations on E if:1.For each parameter δ in S the transformations are one-one onto E, in particular, t lies in E.2. S with the law of composition ϕ forms a group G. is a continuous parameter, that is, S is an interval in R. Without loss of generality, δ = 0 corresponds to the identity element e.Consider the Lie group of rotation matrices denoted by SO(2, R).These form a subgroup of the group (under multiplication) of 2 × 2 real invertible matrices denoted by GL(2, R).Let the parameter δ denote the rotation angle.Then we can parametrize this group as follows: 3. t = t when δ = e, that is f(t, e) = t.6.f is infinitely differentiable with respect to t in E and an analytic function of δ in S.7.φ(δ, ) is an analytic function of δ and , ( ) As the infinitesimal generator cannot always be explicitly solved due to the arbitrariness of d(t), we shall choose a few explicit values of d(t). ) 22and c 23 are arbitrary constants.From equation (29), we get 11 Corollary 4.10 finds applications in the theory of automatic control, in which the attached coefficients are variables.
The generators of the Lie group (or vector fields of the symmetry algebra) corresponding to this NDE are given by This is equivalent to the second order delay differential equation (special case of the NDE with ( ) 0) Following the procedure in the previous section, we get ω(t, x) = c 39 , where 39The generators of the Lie group (or vector fields of the symmetry algebra) corresponding to this delay differential