Lie Symmetry Analysis of Burgers Equation and the Euler Equation on a Time Scale

: As a powerful tool that can be used to solve both continuous and discrete equations, the Lie symmetry analysis of dynamical systems on a time scale is investigated. Applying the method to the Burgers equation and Euler equation, we get the symmetry of the equation and single parameter groups on a time scale. Some group invariant solutions in explicit form for the trafﬁc ﬂow model simulated by a Burgers equation and Euler equation with a Coriolis force on a time scale are studied.


Introduction
A time scale is an arbitrary nonempty closed subset of the real number set [1,2], which was initiated by Hilger to unify the continuous and discrete analysis [3]. Unification and extension are two main features of time scale calculus.
Some practical problems possess both continuous and discrete cases. Simple continuous or discrete analysis is not enough to solve problems in some compound problems. Thanks to the time scale theory, unifying results can be produced for complicated models under the frame of time scale. With the wide application and rapid development of the theory, the study of solution to dynamical equations on a time scale has raised more and more attention [4][5][6][7][8][9][10][11][12]. Peterson et al. studied the boundedness and uniqueness of solutions to dynamical equations on a time scale by defining suitable Lyapunov-type functions [6]. Hoffacker et al. investigated the stability and instability of the first-order system of dynamical equations with Lyapunov function on a time scale [7]. Amster et al. proved the existence of solutions to boundary value problem by Leray-Schauder and Brouwer degree theory on time scales [8]. Sun et al. obtained the existence of positive solutions to one-dimensional p-Laplacian boundary value problems by a fixed point theorem on a time scale [9].
The first purpose of writing this paper is to give the general Lie symmetry analysis method to dynamical equations on a time scale. It is well-known that group theory is a universal and convenient tool for analysis of partial differential equations (PDEs) and symmetry properties of PDEs have been extensively studied. As a powerful tool that can be used to solve both continuous and discrete equations, a Lie symmetry analysis method provides an effective way to solve the dynamical equations on a time scale [13][14][15][16][17][18]. To the best of our knowledge, the study of the method on this topic is still new but meaningful in solving practical problems.
The second purpose of writing this paper is to study the exact solutions to the Burgers equation and Euler equation with important physical significance on a time scale. As a nonlinear partial differential equations simulating shock wave propagation and reflection, the Burgers equation is applied widely in traffic flow, shock wave, turbulence problem, and a continuous stochastic The forward jump operator σ : T → T is defined by

4.
If σ(t) > t, then t is right-scattered. If ρ(t) < t, then t is left-scattered. Points that are both right scattered and left-scattered are called isolated. 5.
If t < sup T and σ(t) = t, then t is right-dense. If t > inf T and ρ(t) = t, then t is left-dense. Points that are both right-dense and left-dense are called dense. 6.
We set σ(t) = t if T has a maximum t, and ρ(t) = t if T has a minimum t.
Consider the set T κ that is derived from the time scale T. If T has a left-scattered maximum m, We call f ∆ (t) the delta derivative of f at t.
In this paper, we denote ∆ t f := f ∆ (t) for ease of expression.

Proposition 1 ([2]). (Leibnitz formula on a time scale) Let S
(n) k be the set consisting of all possible strings of length n, containing exactly k times σ and n − k times ∆. If f Λ exists for all Λ ∈ S (n) k , then holds for all n ∈ N, where f Λ denotes all possible permutations of k times σ and n − k times ∆ acting on f .

Symmetry Analysis on a Time Scale
.., m, for j = 1, 2, ..., n corresponding to all jth-order partial derivatives of u with respect to x.
Proof. We prove Theorem 1 by induction on n ∈ N * . Let us first initialize the proof for n = 1. As n = 1, .
where 1 means identity transformation. By where I is an identity matrix. Then, Then, Symmetry 2020, 12, 10

of 15
The coefficient of ∆ t u in Pr (1) which means that the theorem holds for n = 1. Assuming the conclusion holds for n − 1, we prove that it also holds for n. As M (n) can be considered as the subspace of 1-order prolongation (M (n−1) ) (1) of M (n−1) , then Pr (n) V − can be obtained by the 1-order prolongation of Pr (n−1) V − . Thus, the coefficient of denotes the number that satisfies j i = t for i = 1, ..., n.

Lie Symmetry Analysis of the Burgers Equation on a Time Scale
For a (1+1)-dimensional Burgers equation with constant coefficients on a time scale, Let v = u x ; we obtain the potential form of the Burgers equation The Lie algebra of Equation (3) is spanned by the following vector field The infinitesimal invariance criterion of Equation (3) can be written as The operator Pr (2) V has the following form: The determining equation is Thus, the determining Equation (5) is converted to from which the Lie point symmetry group can be ascertained. Firstly, from the fact that the coefficients of ∆ t u x , (∆ t u x )u x , u xx u x in Equation (6) being 0, we have Furthermore, substituting Equation (7) into Equation (6), we get The coefficients of u xx , u 2 x , u x , 1 in Equation (8) should be 0, Solving Equation (9), we have where C i (i = 1, ..., 5) are arbitrary constants. According to vector field (4) and Equation (10), we obtain the corresponding vector field The Lie algebra with infinitesimal symmetry of Equation (3) is spanned by the following vector fields We construct the one-dimensional optimal system of the subgroups of Equation (3). The construction of the one-dimensional optimal system of subgroups is equivalent to that of constructing an optimal system of subalgebras [29]. Table 1.
Utilizing the commutators obtained in Table 1, we can get the adjoint representations generated by V 1 ∼ V 5 by Based on the adjoint representations of the vector fields obtained in Table 2, Table 2. Adjoint representations generated by V 1 ∼ V 5 .
In order to have an intuitive understanding of the above solution to the Burgers equation, we give the corresponding Figure 1 in some cases.

Remark 1.
From the solutions and respective figures obtained above, we get 1.
As A > 0, B > 0, the tanh-type smooth kink solution of the Burgers equation is obtained.

2.
As A < 0, B > 0, the singular kink solution of the Burgers equation is obtained, and the shock wave appears, which corresponds to the local worst traffic jam.

3.
By data fitting and changing model parameters, the models for specific practical problems can be built, which can provide a theoretical basis for the prediction of traffic congestion.

Exact Solutions to the Euler Equation with Coriolis force on a Time Scale
If f (t, x, y) = (u(t, x, y), v(t, x, y), P(t, x, y)) is the solution to Equation (11), then the following are also solutions to Equation (11) x, y), P(t, x, y) + β(t)ε), ε ∈ R is an arbitrary constant.
Adding the result of partial derivative onξ for Equation (18) with the result of partial derivative onη for Equation (17), we have (20) Substituting Equation (19) into Equation (20), Equations (17)- (19) can be reduced to Different forms of analytical solutions to Euler equation can be derived from Equations (21)-(23), e.g., We give the respective Figure 2 to obtain an intuitive understanding of the solution (24).
which shows the Euler flow with Coriolis force is a rotational flow. 2.
Substituting the group invariants into Equations (11), we obtain the following reduction equations: Fξ Gξ + FGξξ + Gξ Gη + GGξη + Qξη = −2ω(t)Fξ, If F depends onη and G depends onξ only, the third equation of (25) is satisfied naturally, then Equations (25) can be further reduced as Solving Equations (26), the exact solutions to the Euler equation can be obtained. We give one form of exact solutions as follows: u = cos y, v = cos x, P = sin x sin y + 2ω(t) sin x − 2ω(t) sin y.
We give the respective Figure 3 to have a more intuitive understanding of the solution (28).
(b) v with ε = 2. this shows that the Euler flow with Coriolis force is a rotational flow with periodic oscillation.

Conclusions
As a powerful tool that can be used to derive the exact solutions for both continuous and discrete equations, the Lie symmetry analysis method to general dynamical system is generalized