Abstract
Lie symmetry analysis of differential equations proves to be a powerful tool to solve or at least reduce the order and nonlinearity of the equation. Symmetries of differential equations is the most significant concept in the study of DE’s and other branches of science like physics and chemistry. In this present work, we focus on Lie symmetry analysis to find symmetries of some general classes of KS-type equation. We also compute transformed equivalent equations and some invariant solutions of this equation.
1. Introduction
Symmetry has been a source of inspiration as a powerful tool in the formulation of the laws of the universe. A great number of physical phenomena is transformed into differential equations. Lie symmetry analysis can change the given differential equation into an equivalent form which is easier to solve. In the analysis of differential equations, the symmetry group approach is quite useful. Galois’s use of finite groups to solve algebraic equations of degrees two, three, and four, as well as to prove that the general polynomial equation of degrees larger than four could not be solved by radicals, served as the paradigm for this application [1–7]. The symmetry group approach is well-known for its importance in the field of differential equations analysis. Sophus Lie is credited with the invention of group categorization methods and the theoretical basis for the Lie groups.
There are many different methods for computing the symmetries of differential equations. But Lie symmetry analysis is the best because it is a systematic and algorithmic procedure that does not take into account any guesses or approximations. The principal paper on Lie symmetry is [1], in which Lie demonstrated that a linear 2D, 2nd-order PDE admits at most three boundary invariance group. He processed the maximal invariance group of the one-dimensional heat conductivity and used this analysis to compute its explicit solutions. Symmetry reduction is a leading strategy for resolving nonlinear PDEs. Ovisiannikov made a substantial contribution in persisting with these techniques. He presented the strategy of partially invariant solutions [2, 3]. In this work, he gave a methodology that is based on the idea of group called the equivalence group. Gazizov and Ibragimov [8] tracked down the total symmetry analysis of the one-dimensional Black-Scholes model. Shu-Yong and Feng-Xiang, [9] discussed about the connection between the form invariance and Lie symmetry of nonholonomic framework. Buckwar and Luchko [4] initiated the study of symmetry group of scaling transformation for PDEs of fractional order. Yan et al. [6] performed Lie symmetry analysis and fundamental similarity reductions for the coupled Kuramoto-Sivashinsky(KS) equations. Bozhkov and Dimas [10] computed the conversation laws and group classification for generalized 2D KS equation. Nadjafikhah and Ahangari [7] determined the Lie symmetries and reduction for the two-dimensional damped Kuramoto-Sivashinsky ((2D) DKS) equation. Najafikhah and Ahangari also computed Lie symmetry of 2D generalized Kuramoto-Sivashinsky (KS) equation in [11]. The one-dimensional modified KS-type equation is
Chou [12] determined the solution of the cauchy problem for the MKS equation and also computed the solvability with the help of the blow up theorem.
In the present paper, we deal with the generalized modified one-dimensional Kuramoto-Sivashinsky (GMKS) type equation and determine the symmetry algebra by using Lie symmetry analysis. In particular, we want to find the optimal system and similarity solutions corresponding to some special cases of GMKS equation. The GMKS type equation is given as
We seek the Lie symmetry algebras for this GMKS equations for , , and where and are arbitrary constant and For , equation (2) is proposed in [13]. Its second derivative satisfies an equation of Cahn-Hilliard type in [14]. This equation has various applications as physical models in biofluids, mechanics, and liquids. In equation (2), is the velocity function, is space parameter, and is time variable. This equation can also be derived from a model in the continuity equation by fitting a suitable function [15]. Actually, the Kuramoto-Sivashinsky equation gives the change of the position of a flame front (Figure 1). It shows the flame front position against time for horizontally propagating methane flame, the movement of a fluid going down a vertical wall, or a spatially uniform oscillating chemical reaction in a homogeneous medium [16]. This equation is also helpful to display solitary pulses in a falling slender film [17]. Figure 2 shows the schematic representation of the flow showing a film flowing vertically down, subjected to an electric field imposed across electrodes separated by a distance .
2. Lie Symmetry of Generalized Modified Kuramoto-Sivashinsky Equation
In this part, we compute our main results.
Consider one parameter local Lie group of transformation for the independent factors , and dependent factor as follows: in which is the parameter.
Proposition 1. For all , the algebra of symmetries of is 2-dimensional Abelian Lie algebra.
Proof. The general infinitesimal generator (symmetries) is
The derivation of th prolongation of interprets the relating jet space , where is a dependent variables, and with , and
where is an independent variable and and .
The fourth-order prolongation of is
We can calculate ,, , and from equation (7) such that
where and are total derivatives.☐
Putting all the above in equation (9) and eliminating by using the relation , we get a polynomial equation containing the different differentials of . Equating the coefficient of to zero, which are some derivatives of , , and , it gives the total set of determining equations.
This gives
This implies that the Lie group (algebra) of infinitesimal generators of equaution (2) is comprised of two vector fields:
The commutator table of the Lie group for equation (2) is given as in Table 1,
The adjoint table of infinitesimal symmetries for equation (2) is given as in Table 2,
In this case, we have only two different basis for a Lie algebra of symmetries.
Hence, this shows that the group of symmetries of equation (2) is two dimensional and tables ensure that it is abelian.
Proposition 2. For all and , the group of symmetries of is two-dimensional abelian.
Proof. The infinitesimal generator is In order to find the symmetry group of equation (14), we have to apply invariance condition that is on equation .(14) where is the fourth-order prolongation of given as After applying an invariance condition on equation (14), we get We can calculate , , , and from equation (7) such that where and are total derivatives.☐
After putting all the above in equation (17), and eliminating by using the relation , we get a polynomial equation containing the different differentials of . Equating the coefficient of to zero, which are some derivatives of , , and , it gives the total set of determining equations, as given by
, ,
that gives
This implies that the Lie group (algebra) of infinitesimal generators of equation (9) comprises two vector fields. Following, Table 3 gives the commutator table as
The commutator table of the Lie group for equation (14) is given in Table 3,
The adjoint table of infinitesimal symmetries for equation (14) is given in Table 4,
In this case, we have only two different basis for Lie algebra.
Hence, this shows that the group of symmetries of equation (14) is two-dimensional abelian.
2.1. Symmetry Algebra for When
For , the equation is
The general infinitesimal generator is
In order to find the symmetry algebra, we have to apply invariance condition that is on equation (21) where is the fourth-order prolongation of such that
After applying invariance condition (23) on equation (21) where ,, , and from equation (7) such that where and are total derivative.
Putting all the above in equation (25), we eliminate by using the relation and get a polynomial equation containing the different differentials of . Equating the coefficient of to zero, which are some derivatives of , , and , it gives the total set of determining equations.
, ,
that gives
This implies that the Lie group (algebra) of infinitesimal generators of equation (21) is comprised of three vector fields:
The commutator table of the Lie group for equation (21) is given in Table 5,
The adjoint table of infinitesimal symmetries for equation (21) is given in Table 6,
In this case, we have three different Lie algebras.
Theorem 3. The algebra of symmetries of Kuramoto-Sivashinsky type equation is where it is two-dimensional abelian for all , and three-dimensional nonabelian for
Proof. The proof follows easily using Propositions 1 and 2.☐
Theorem 4. If be the one parameter group generated by equation (28) then There will be a family of solutions to each one parameter subgroups of the full symmetry group of a system called group invariant solutions.
Theorem 5. If is a solution of equation (21), so are the functions where , and is any positive number.
Proof. The one parameter Lie group of equation (21) is
with the infinitesimal generator
if is any function then it transformed by as
now
therefore,
The graph for is given in Figure 3.
The one parameter Lie group of equation (21) is
with the infinitesimal generator
if is any function then it transformed by as
now
therefore
The graph for is given in Figure 4.
The one parameter Lie group of equation (21) is
with the infinitesimal generator
if is any function then it transformed by as
now
therefore
The graph for as a solution is given in Figure 5.☐
3. Optimal System of Subalgebras
This is remarkable that the Lie symmetry technique assumes a significant part to determine the solutions of PDEs as well as performing the symmetric reductions. Every combination (should be linear) of infinitesimal symmetries(generators) is a result of another infinitesimal symmetry(generator). As any transformation in the full symmetry groups plot a solution to another, it is sufficient to determine the invariant solution which are not related by transformations in the full symmetry group; this prompted the Optimal system [18, 19].
Theorem 6. A 1D optimal system of equation (21) is given by those generated by
Proof. Since the combination of vector field (infinitesimal generator) is also a vector field. Consider a linear combination of , , and , a nonzero vector field. Here, for proof, we will improve as many of the coefficient as possible by using adjoint application on V.☐
Case 1. Firstly assume that then
acting on with by using the adjoint table (adjoint Table 3)
When , then we get .
When , then we get .
When , then we get .
Case 2. Let and , when then we get , Let , and , then we get There is no any more cases for consultation and the proof is complete.
4. Lie Invariants and Similarity Solutions
We can discover that the invariants correlate with the infinitesimal symmetries (28); they can be determined by solving the equations (by using characteristic method). For , the characteristic equation is and the corresponding invariants of this system and .
We obtain a similar solution of the form , and we put it into equation (21) to obtain the form of the function , and then, we conclude that solution of the following differential equation as similarity reduce equation: For other example, take ; the characteristic equation for this has the form so the corresponding invariants are and .
Taking into account the last invariants, the following similarity solution is obtained where the solution satisfied the similarity reduce equation:
5. Conclusions
The present paper addresses Lie symmetries for some general cases of modified one-dimensional Kuramoto-Sivashinky equation (MKS) as well as its similarity solutions using a symmetry operator. In Section 2, we discussed general results for Lie algebras for some general cases of MKS and provide a comparison between them and obtained some general results. In Section 3, we find the optimal system for (MKS). In the last section, we obtained similarity solutions and Lie invariants.
Remarks. It is worthmentioning that can be any arbitrary function. For other similar functions chosen as , the procedure for symmetry analysis can be very tedious and symmetry algebra can be different.
Data Availability
Data sharing is not applicable to this article as no dataset was generated or analyzed during the current study
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research is partially supported by the Higher Education Commission, Pakistan.