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Comment published on 1 June 2020, see Symmetry 2020, 12(6), 900.
Open AccessArticle

Lie Symmetry Analysis, Explicit Solutions and Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation

1
Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan
2
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City 72915, Vietnam
3
Department of Mathematics, Cankaya University, 06790 Ankara, Turkey
4
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40447, Taiwan
5
Institute of Space Sciences, 077125 Magurele-Bucharest, Romania
6
Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, Prince Sattam bin Abdulaziz University, Al-Kharj 11991, Saudi Arabia
*
Author to whom correspondence should be addressed.
Symmetry 2020, 12(1), 170; https://doi.org/10.3390/sym12010170
Received: 4 December 2019 / Revised: 28 December 2019 / Accepted: 8 January 2020 / Published: 16 January 2020

Abstract

In this paper, we investigate a spatially two-dimensional Burgers–Huxley equation that depicts the interaction between convection effects, diffusion transport, reaction gadget, nerve proliferation in neurophysics, as well as motion in liquid crystals. We have used the Lie symmetry method to study the vector fields, optimal systems of first order, symmetry reductions, and exact solutions. Furthermore, using the power series method, a set of series solutions are obtained. Finally, conservation laws are derived using optimal systems.
Keywords: Burgers–Huxley equation; Lie symmetry method; conservation laws; Exact solutions Burgers–Huxley equation; Lie symmetry method; conservation laws; Exact solutions

1. Introduction

Mathematical modeling of dynamical systems results in nonlinear partial differential equations (PDEs). In reality, most complicated phenomena—namely diffusion, reaction, conservation, and many more—can be illustrated by means of partial differential equations. Due to their quintessence, PDEs are studied profusely in science and engineering. Various peculiar methods are designed for obtaining their exact and approximate solutions, which, in turn, help us in quantitative and qualitative analysis of these PDEs. The interested reader can see some of these methods in [1,2].
Lie symmetry analysis is a powerful and influential tool for mathematically analyzing partial differential equations. It can be used in securing analytic solutions or in switching PDEs into solvable ordinary differential equations (ODEs). Diverse symmetry vectors are also discovered for the considered system in some cases, however, sometimes there emerges a chance of linear combination of these vectors. To avoid this, an optimal system is constructed. Each member of this system is used in lessening independent variables of the system until analytic solutions are obtained or PDEs are switched to solvable ODEs [3,4,5,6,7]. The system also analyzes solutions of PDEs of different kinds as well as opens many fields [8].
The generalized Burgers–Huxley equation, which has many utilizations in the fields of biology, metallurgy, chemistry, mathematics, and engineering is of the following type,
u t 2 u x 2 + a u n u x b u ( 1 u n ) ( u n k ) = 0 .
This is a non-linear equation that has secured much importance due to its appearance in many physical phenomena and its scientific utilization. The parameters a , b 0 are real constants, n is a positive integer and k [ 0 , 1 ] . When a = 0 and n = 1 , Equation (1) reduces to the Huxley equation and with b = 0 , n = 1 , it becomes the Burgers equation. Some exact numerical and traveling wave solutions to (1) were reported in [9,10,11]. However, the spline collocation method for the Burgers–Huxley equation was discussed in a book by Schiesser [12]. In addition, many other analytical and numerical methods for generalized Burgers–Huxley equations have been developed in the past, see for example [13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29].
In this paper, we will analyze the Burgers–Huxley equation in two spatial dimensions, which is of the following form,
u t 2 u x 2 2 u y 2 u u x u u y + u 3 + k u ( k + 1 ) u 2 = 0 .
Equation (2) couples both the assets of the Burgers equation (this is one of the basic models in fluid mechanics and is used to catch some of the properties of turbulent flow in a channel, which occurs due to the interaction of the reverse outcome of convection and diffusion and also describes the format of shock waves, traffic flow, and acoustic transmission [30]) and the Huxley equation (which is used for nerve proliferation in neurophysics and wall proliferation in liquid crystals [31]). So, we will designate this equation as simply a two dimensional Burgers–Huxley equation. The combined Burgers–Huxley equation shows a prototypical imitation that specifies the interaction between the reaction gadget, diffusion transport and convection effects, nerve proliferation, and motion in liquid crystals [32].
In Section 2, the vector fields and optimal systems for (2) are obtained by using the Lie symmetry method. In Section 3, we computed the similarity diminution for one and two dimensional subalgebras and hence obtained the group invariant solutions for (2). Employing the power series method, certain power series solutions are achieved in Section 4. Finally, in Section 5, conservation laws are derived using optimal systems.

2. Lie Symmetry Analysis

In this section, we will study the Lie symmetries and optimal systems of the Burgers–Huxley equation. Consider the one-parameter Lie group of transformation:
x x + ϵ ξ ( x , y , t ; u ) + O ( ϵ 2 ) , y y + ϵ η ( x , y , t ; u ) + O ( ϵ 2 ) , t t + ϵ ζ ( x , y , t ; u ) + O ( ϵ 2 ) , u u + ϵ φ ( x , y , t ; u ) + O ( ϵ 2 ) ,
where ϵ is the group parameter. The infinitesimal operator associated with the above transformations is:
X = ξ ( x , y , t ; u ) x + η ( x , y , t ; u ) y + ζ ( x , y , t ; u ) t + φ ( x , y , t ; u ) u .
The coefficient functions ξ ( x , y , t ; u ) , η ( x , y , t ; u ) , ζ ( x , y , t ; u ) , and φ ( x , y , t ; u ) are to be determined and the vector field X satisfies the Lie symmetry condition,
p r ( 2 ) X ( Δ ) | Δ = 0 = 0 ,
where
Δ = u t 2 u x 2 2 u y 2 u u x u u y + u 3 + k u ( k + 1 ) u 2 = 0 .
The second prolongation of the infinitesimal generator is given by
X ( 2 ) = X + φ x ( x , y , t ; u ) u x + φ y ( x , y , t ; u ) u y + φ t ( x , y , t ; u ) u t + φ x x ( x , y , t ; u ) u x x + φ x y ( x , y , t ; u ) u x y + φ x t ( x , y , t ; u ) u x t + φ y y ( x , y , t ; u ) u y y + φ y t ( x , y , t ; u ) u y t + φ t t ( x , y , t ; u ) u t t .
with
φ t = D t ( φ ) u t D t ( ζ ) u x D t ( ξ ) u y D t ( η ) , φ x = D x ( φ ) u t D x ( ζ ) u x D x ( ξ ) u y D x ( η ) , φ y = D y ( φ ) u t D y ( ζ ) u x D y ( ξ ) u y D y ( η ) , φ x x = D x ( φ x ) u x t D x ( ζ ) u x x D x ( ξ ) u x y D x ( η ) , φ y y = D y ( φ y ) u y t D y ( ζ ) u x y D y ( ξ ) u y y D y ( η ) , φ t t = D t ( φ t ) u t t D t ( ζ ) u x t D t ( ξ ) u y t D t ( η ) , φ x y = D x ( φ y ) u y t D x ( ζ ) u x y D x ( ξ ) u y y D x ( η ) , φ x t = D x ( φ t ) u t t D x ( ζ ) u x t D x ( ξ ) u y t D x ( η ) , φ y t = D y ( φ t ) u t t D y ( ζ ) u x t D y ( ξ ) u y t D y ( η ) ,
where the the operator D i is defined as:
D i = x i + u i u + u i j u j + , i , j = 1 , 2 , 3 .
Coupling Equations (5) and (6), we can obtain the compatible condition for Equation (5). Substituting extended transformations into the obtained compatible conditions and making the coefficients of several monomials in partial derivatives and numerous powers of u equal, we get the following over determining system of PDEs:
η y ξ x = 0 , ξ x x + ξ y y ξ t 2 φ x u φ = 0 , η t η x x η y y + 2 φ y u + φ = 0 , η x u + ξ y u = 0 , ζ x u = 0 , η u = 0 , ξ u = 0 , ζ u = 0 , ζ x x + ζ y y ζ t + 2 η y = 0 , φ u u 2 ξ x u = 0 , ξ u u = 0 , ζ u u = 0 , η u u = 0 , φ u u 2 η y u = 0 , ζ x + ζ y + k ζ u = 0 , ξ x + ξ y + k ξ u 2 η y = 0 , η y η x = 0 , η x + ξ y = 0 , ζ x = 0 , ζ y = 0 ,
φ x x + φ y y φ t k φ = 0 , φ x + φ y + k φ u + 2 k φ + 2 φ 2 k η y = 0 , ( k + 1 ) φ u 3 φ + ( 2 k + 2 ) η y = 0 , φ u 2 η y = 0 .
By solving the over determining system of PDEs (8), we obtain the coefficient functions ζ , ξ , and η as:
ζ = c 1 , ξ = c 2 , η = c 3 ,
where c 1 , c 2 , and c 3 are random constants. The Lie algebra of infinitesimal symmetry of Equation (2) with k 0 is given by,
X 1 = t , X 2 = x , X 3 = y .
.

2.1. Transformed Solutions

One can acquire the group transformation initiated by the Lie point symmetry operator X i ( i = 1 , 2 , 3 ) by solving the following ODEs
d x d ϵ = ξ ( x , t , y , u ) , x | ϵ = 0 = x , d t d ϵ = ζ ( x , t , y , u ) , t | ϵ = 0 = t , d y d ϵ = φ ( x , t , y , u ) , y | ϵ = 0 = y , d u d ϵ = η ( x , t , y , u ) , u | ϵ = 0 = u .
The one-parameter Lie symmetry groups generated by infinitesimals X 1 , X 2 , and X 3 are given by
r 1 : ( x , y , t , u ) ( x + ϵ 1 , y , t , u ) ,
r 2 : ( x , y , t , u ) ( x , y + ϵ 2 , t , u ) ,
r 3 : ( x , y , t , u ) ( x , y , t + ϵ 3 , u ) ,
where ϵ 1 , ϵ 2 , and ϵ 3 are group parameters.
Depending on the values of r 1 , r 2 , and r 3 , if f ( x , y , t ) is any confidential solutions of Equation (2), then the new solutions can be given by
r 1 ( ϵ ) . u ( x , y , t ) = f ( x ϵ 1 , y , t ) , r 2 ( ϵ ) . u ( x , y , t ) = f ( x , y ϵ 2 , t ) , r 3 ( ϵ ) . u ( x , y , t ) = f ( x , y , t ϵ 3 ) .

2.2. Optimal System of Subalgebras

For the optimal system, we will first construct the tables for commutation relations and adjoint action of the obtained symmetries. For the sake of obtaining the adjoint representation we will use the Lie series in the form:
A d ( exp ( ϵ X i ) ) X j = X j ϵ [ X i , X j ] + ϵ 2 2 ! [ X i , [ X i , X j ] ] ,
The commutation relations between basis elements satisfies:
[ X i , X j ] = 0 , i , j = 1 , 2 , 3 ,
whereas, the adjoint representation Table 1 is given as:
Following the method given in [8], consider the vector X with random coefficients a 1 , a 2 , a 3 , such that
X = a 1 X 1 + a 2 X 2 + a 3 X 3 ,
suppose a 3 0 and set up a 3 = 1 , so that
X = a 1 X 1 + a 2 X 2 + X 3 .
The scheme involves simplifying the coefficients as much as possible. To abolish the coefficient of X 2 , we will use it to act on X. It is easy to see that the vector form cannot be reduced much more because commutation relations are zero.
Further, suppose a 3 = 0 and establish a 2 = 1 so that,
X = a 1 X 1 + X 2 .
Repeating the same process and normalizing the coefficients we have the following one-dimensional optimal system of subalgebras,
{ X 1 , X 2 , X 3 , a X 1 + X 2 , a X 1 + b X 2 + X 3 } ,
where a and b are arbitrary constants.

3. Symmetry Reduction

Symmetry Reductions for Optimal System

Symmetry diminution and explicit solutions have numerous utilizations in the connection of differential equations. Solutions of partial differential equations asymptotically tend to the solutions of lower dimensional equations, which are obtained by symmetry diminution. Especially, exact solutions originating from symmetry technique can be utilized effectively to study properties such as asymptotics etc. In the previous section, we obtained the point symmetries and optimal system for two-dimensional Burgers–Huxley equation, now we will interrogate the symmetry reductions and exact solutions using one-dimensional subalgebras.
(i)
For the linear combination a X 1 + X 2 , the invariants can be obtained by solving the characteristic equation d t a = d x 1 = d y 0 = d u 0 , giving
s = a x t , r = y , Ψ = u .
Using,
u = Ψ ( s , r ) ,
we obtained the following PDE with two independent variables,
a 2 2 Ψ s 2 + 2 Ψ r 2 + ( 1 + a Ψ ) Ψ s + Ψ Ψ r + Ψ ( k Ψ ) ( Ψ 1 ) = 0 .
By the application of the similarity transformation method on reduced Equation (11) again, we have
ξ = c 1 , ζ = c 2 , φ = 0 ,
where c 1 and c 2 are random constants and the characteristic Equation for (12) is
d s 1 = d r 1 = d Ψ 0 .
Hence, Ψ can be written as:
Ψ = β ( α ) , α = s r .
Substituting Equation (14) into (11), we have the following ODE:
( a 2 + 1 ) d 2 β d α 2 + ( 1 + a β β ) d β d α + β ( β 1 ) ( k β ) = 0 .
(ii)
For the linear combination a X 1 + b X 2 + X 3 , the invariants can be obtained by solving the characteristic equation
d t a = d x b = d y 1 = d u 0 ,
giving
s = a x b t , r = a y t , Ψ = u .
Using,
u = Ψ ( s , r ) ,
we obtained the following PDE with two independent variables:
a 2 2 Ψ s 2 + a 2 2 Ψ r 2 + a Ψ Ψ s + a Ψ Ψ r + b Ψ s + Ψ r + Ψ ( k Ψ ) ( Ψ 1 ) = 0 .
By the application of the similarity transformation method on reduced Equation (17) again, we obtain
ξ = c 1 , ζ = c 2 , φ = 0 ,
where c 1 and c 2 are random constants. Choosing c 1 and c 2 equal to 1 , the characteristic Equation for (18) is:
d s 1 = d r 1 = d Ψ 0 .
Therefore, Ψ can be written as:
Ψ = β ( α ) , α = s r .
Substituting Equation (20) into (17), we have following ODE:
2 a 2 d 2 β d α 2 + ( b 1 ) d β d α + β ( β 1 ) ( k β ) = 0 .
Reductions corresponding to the remaining vectors occuring in the optimal system can be obtained in a similar way, hence, we omit the details here.

4. Explicit Power Series Solution

After reducing partial differential equations into ordinary differential equations in the previous section, we will now explore the exact analytic solution of the reduced equations using the power series method [8].

4.1. Series Solution of Reduced Equation (15)

Consider the power series solution of the form:
β ( α ) = n = 0 b n α n .
Incorporating (22) into Equation (15), we get
0 = ( a 2 + 1 ) n = 1 b n + 2 ( n + 1 ) ( n + 2 ) α n + ( k + 1 ) n = 1 m = 0 n b m b n m α n k n = 1 b n α n n = 1 m = 0 n j = 0 m b j b m j b n m α n + n = 1 b n + 1 ( n + 1 ) α n k b 0 ( k + 1 ) b 0 2 b 0 3 + ( a 1 ) n = 1 m = 0 n b m b n m + 1 ( n m + 1 ) α n + 2 ( a 2 + 1 ) b 2 + ( a 1 ) b 0 b 1 + b 1 .
Inspecting the coefficients for n = 0 and n 1 , we have
b 2 = ( k + 1 ) b 0 2 + b 0 3 + k b 0 ( a 1 ) b 0 b 1 b 1 2 ( a 2 + 1 ) ,
b n + 2 = 1 ( a 2 + 1 ) ( n + 1 ) ( n + 2 ) ( ( k + 1 ) m = 0 n b m b n m + k b n + m = 0 n j = 0 m b j b m j b n m ( a 1 ) m = 0 n b m b n m + 1 ( n m + 1 ) b n + 1 ( n + 1 ) ) .
So, we find the solution of Equation (15) in the form of the power series given by
β ( α ) = b 0 + b 1 α + b 2 α 2 + n = 1 b n + 2 α n + 2 .
Back substituting the values of α and β , we have
Ψ = b 0 + b 1 ( s r ) + b 2 ( s r ) 2 + n = 1 b n + 2 ( s r ) n + 2 .
Thus, in terms of original variables, we obtain the following solution of the two dimensional Burgers–Huxley equation:
u = b 0 + b 1 ( a x t y ) + b 2 ( a x t y ) 2 + n = 1 b n + 2 ( a x t y ) n + 2 ,
where b 0 and b 1 are random constants and the remaining constants can be calculated using Equations (24) and (25), respectively. The convergence of Equation (28) is shown in the Appendix A.

4.2. Series Solution of Reduced Equation (21)

Now, consider the power series solution of (21) in the form (22). These equations together yield
2 a 2 n = 1 b n + 2 ( n + 2 ) ( n + 1 ) α n + ( b 1 ) n = 1 b n + 1 ( n + 1 ) α n + ( b 1 ) b 1 k n = 1 b n α n b 0 3 + 4 a 2 b 2 + ( k + 1 ) n = 1 m = 0 n b m b n m α n + ( k + 1 ) b 0 2 k b 0 n = 1 m = 0 n j = 0 m b j b m j b n m α n = 0 .
Comparing the coefficients of like powers of α , we obtain
b 2 = k b 0 ( b 1 ) b 1 + b 0 3 ( k + 1 ) b 0 2 4 a 2 ,
b n + 2 = 1 2 a 2 ( n + 1 ) ( n + 2 ) k b n ( k + 1 ) m = 0 n b m b n m ( b 1 ) ( n + 1 ) b n + 1 + m = 0 n j = 0 m b j b m j b n m .
Next, we use values of α and β from (20) into (22) to have
Ψ = b 0 + b 1 ( s r ) + b 2 ( s r ) 2 + n = 1 b n + 2 ( s r ) n + 2 ,
where b 0 and b 1 are random constants. Hence, we have the following solution in terms of original variables:
u = b 0 + b 1 a x a y + t ( 1 b ) + b 2 a x a y + t ( 1 b ) 2 + n = 1 b n + 2 a x a y + t ( 1 b ) n + 2 ,
with b 2 is given in (29) and b n + 2 is given in (30).

5. Conservation Laws

Conservation laws are of fundamental importance in the study of partial differential equations as they provide conserved quantities for all the solutions, can detect integrability and linearization, and can be used to check the precision of numerical solution methods [33]. Conservation laws provide one of the basic rules in formulating models in mathematics and certain times a partial differential equation having a large number of conservation laws depicts a strong implication of its integrability.

5.1. Preliminaries

Here, we recall the Ibragimov scheme of constructing conservation laws corresponding to the given symmetries of any system of PDEs, provided that the number of equations in the system is equal to the number of dependent variables [34]. Consider a p-th order system of partial differential equations of p independent variables and q dependent variables, respectively, as x = ( x 1 , x 2 , , x p ) and u = ( u 1 , u 2 , , u q ) ,
G Λ ( x , u , u ( 1 ) , , u ( p ) ) = 0 , Λ = 0 , 1 , , q .
The infinitesimal generator for Equation (33) is:
X = ζ n x n + β Λ u Λ , n = 0 , 1 , , p , Λ = 1 , 2 , , q .
The p-th prolongation of the point symmetry operator is:
X p = X + β n ( 1 ) Λ u n Λ + + β n 1 n 2 n p ( p ) Λ u n 1 n 2 n p Λ , p 1 ,
with
β n ( 1 ) Λ = D n β Λ ( D n ζ m ) u m Λ ,
β n 1 n p ( p ) Λ = D n p β n 1 n ( p 1 ) ( p 1 ) Λ ( D n p ζ m ) u n 1 , , m n ( p 1 ) Λ ,
Here
m , n = 1 , 2 , , p , Λ = 1 , 2 , , q ,
and D n is the total derivative operator. A vector T = ( T 1 , , T n ) is a conserved vector of (33) if
D j T j = 0 ,
is satisfied for all solutions of (33). Equation (34) is called a local conservation law provided that T i are free of integral terms.
Theorem 1.
Every Lie point symmetry of Equation (33) results in a conservation law. The conserved vector components are
C n = ζ n L + W Λ L u n Λ + s 1 D n 1 D n s ( W Λ ) ( L u n 1 n s Λ ) ,
with,
W Λ = η Λ ζ m u m Λ .
and the formal Lagrangian is defined as
L = v Λ G Λ .
In order to find the adjoint equation we have,
G Λ ( x , u , v , u 1 , v 1 , , u p , v p ) = δ L δ u Λ ,
where δ δ u is the variational derivative and ( v Λ , Λ = 0 , 1 , , q ) is the adjoint variable.
Theorem 2.
Equation (2) is not strictly self-adjoint.
Proof. 
The formal Lagrangian for Equation (2) can be written as,
L = v ( u t 2 u x 2 2 u y 2 u u x u u y u ( k u ) ( u 1 ) ) .
The adjoint Equation of (2) is given by:
G ( x , u , v , v 1 , v 2 ) = δ ( v G ) δ u = 0 ,
where δ δ u is the Euler–Lagrange operator given by,
δ δ u = u D i L u i + D i D j L u i j .
Using the Euler–Lagrange operator in Equation (2), we have the following adjoint equation, after transforming in the original variable u,
u t + 2 u x 2 + 2 u y 2 + u u x u u y + 2 u u x + 2 u u y + 2 k u k 3 u 2 + 2 u = 0 ,
which is not strictly self-adjoint. □

5.2. Conservation Laws of a Spatially Two-Dimensional Burgers–Huxley Equation

Now, we will construct conservation laws for each element in the optimal system obtained in Section 2.
(i)
For X 1 , we have W = u t and ξ t = 1 . Substituting these values in Equation (35), we find
C t = v u x x v u y y v u u x v u u y k v u 2 + k u v + u 3 v v u 2 , C x = u v u t v x u t + v u x t , C y = u v u t v y u t + v u y t .
(ii)
For X 2 , W = u x and ξ x = 1 . Substituting in Equation (35), we obtain
C t = v u x , C x = v u t v u y y u v u y k v u 2 + k u v + u 3 v v u 2 u x v x , C y = u v u x v y u x + v u x y .
(iii)
For the generator X 3 , W = u y and ξ y = 1. So, in this case, we have the following conserved vector:
C t = v u y , C x = u y u v u y v x + v u x y , C y = v u t v u x x u v u x k v u 2 + k u v + v u 3 v u 2 u y v y .
(iv)
For the generator a X 1 + X 2 , W = a u t u x and ξ t = a, ξ x = 1. Substituting into (35), we have
C t = a v u x x a v u y y a u v u x a u v u y a k v u 2 + a k u v + a v u 3 a v u 2 v u x , C x = v u t v u y y u v u y u 2 k v + k u v + v u 3 v u 2 + a u v u t a u t v x u x v x + a v u x t , C y = a u v u t a u t v y + u v u x u x v y + a v u y t + v u x y .
(v)
For the generator a X 1 + b X 2 + X 3 , W = a u t b u x u y and ξ t = a, ξ x = b and ξ y = 1. Using these values in Equation (35), we have
C t = a v u x x a v u y y a u v u x a u v u y a k v u 2 + a k u v + a v u 3 a v u 2 b v u x v u y , C x = v u t b v u y y b u v u y b u 2 k v + b k u v + b v u 3 b v u 2 + a u v u t + b v u t + u v u y a u t v x b u x v x v x u y + a v u x t + v u x y , C y = v u t v u x x v u u x k v u 2 + k u v + v u 3 + a u v u t + b u v u x a v y u t b u x v y v y u y + a v u y t + b v u x y .
The conserved vectors comprise random solutions of the adjoint equation, thereby implying the interminable number of conservation laws. Conservation laws play a compelling role in the solution process of an equation or system of equations.

6. Concluding Remarks

In this work, we have investigated a spatially two-dimensional Burgers–Huxley equation, which combines both the properties of Burgers (convective phenomenon) and Huxley equation (nerve proliferation and motion in liquid crystals). We have computed the Lie point symmetries of (2), and constructed an optimal system of one and two dimensional subalgebras using these symmetries. Symmetry reductions are performed for one-dimensional subalgebras. Furthermore, some power series solutions are calculated using the power series method. Finally, conservation laws are derived using the Ibragimov theory and involvement of random solutions predicted that conservation laws consisted of an interminable number of conserved vectors.

Author Contributions

A.H. and S.B.; Writing—original draft, A.H. and S.B.; Conceptualization, D.B.; Methodology, I.K.; Software, K.S.N.; Validation, D.B.; Formal analysis, A.H. and S.B.; Writing—review and editing, Investigation, K.S.N. and A.H.; Resources, D.B.; Data curation, K.S.N. and I.K.; Visualization and supervision, D.B. and I.K.; Project administration and funding. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Conflicts of Interest

The authors declare no conflict of interest.

Appendix A

In this appendix, we will discuss the convergence of the power series solution given by (28) using the implicit function theorem [35]. For Equation (25) we have
| b n + 2 | 1 ( a 2 + 1 ) ( k + 1 m = 0 n b m b n m + k b n b n + 1 a 1 m = 0 n b m b n m + 1 + m = 0 n j = 0 m b j b m j b n m ) .
Choosing
M = max k + 1 ( a 2 + 1 ) , k ( a 2 + 1 ) , 1 ( a 2 + 1 ) , a 1 ( a 2 + 1 ) ,
we have
| b n + 2 | M ( m = 0 n b m b n m + b n + m = 0 n j = 0 m b j b m j b n m m = 0 n b m b n m + 1 b n + 1 ) .
Now define a power series γ ( α ) = n = 0 a n α n and consider a i = | b i | for i = 0 , 1 , 2 , . Then we have
a n + 2 M ( m = 0 n a m a n m + a n + m = 0 n j = 0 m a j a m j a n m m = 0 n a m a n m + 1 a n + 1 ) .
It can be easily seen that | b i | a i and hence the assumed series is majorant series. Now, we will show that the series γ ( α ) = n = 0 a n α n is convergent. Through some calculations, we have
γ ( α ) = a 0 + a 1 α + M n = 0 m = 0 n a m a n m α n + 2 + M n = 0 a n α n + 2 + M n = 0 m = 0 n j = 0 m a j a m j a n m α n + 2 + M n = 0 m = 0 n a m a n m + 1 α n + 2 M n = 0 a n + 1 α n + 2 .
Now, taking into account the implicit functional system with respect to α , we have
γ ( α , γ ) = γ a 0 a 1 α M ( γ ( α ) 2 α 2 + α 2 γ ( α ) α ( γ ( α ) a 0 ) + ( γ a 0 ) ( γ a 0 ) + ( γ a 0 a 1 α ) γ ) .
We consider the point ( 0 , a 0 ) , as γ is analytic in the neighbourhood of the point ( 0 , a 0 ) , as γ ( 0 , a 0 ) = 0 and γ ( 0 , a 0 ) 0 . Hence, by the implicit function theorem, the series γ ( α ) = n = 0 a n α n is analytic in the neighbourhood of the point ( 0 , a 0 ) and has a positive radius of convergence. Hence, convergence is proven.

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Table 1. Adjoint representation table.
Table 1. Adjoint representation table.
Ad X 1 X 2 X 3
X 1 X 1 X 2 X 3
X 2 X 1 X 2 X 3
X 3 X 1 X 2 X 3
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