Group Analysis Explicit Power Series Solutions and Conservation Laws of the Time-Fractional Generalized Foam Drainage Equation

. In this study, the classical Lie symmetry method is successfully applied to investigate the symmetries of the time-fractional generalized foam drainage equation with the Riemann–Liouville derivative. With the help of the obtained Lie point symmetries, the equation is reduced to nonlinear fractional ordinary diferential equations (NLFODEs) which contain the Erd´elyi–Kober fractional diferential operator. Te equation is also studied by applying the power series method, which enables us to obtain extra solutions. Te obtained power series solution is further examined for convergence. Conservation laws for this equation are obtained with the aid of the new conservation theorem and the fractional generalization of the Noether operators.


Introduction
Fractional calculus, which deals with fractional integrals and derivatives of arbitrary order, emerged towards the end of the 17th century. Since then, numerous researchers have dedicated their eforts to understanding, studying the properties, and applying fractional order diferential equations. Te interpretation, properties, and applications of such equations have garnered signifcant attention within the scientifc community [1][2][3]. In recent years, there has been a surge of interest in studying fractional diferential equations (FDEs) as they prove to be efective in describing various physical phenomena and processes across diverse felds. Tese equations have found applications in hydrology, viscoelasticity, mechanics, physics, fuid dynamics, biology, chemistry, control theory, electrochemistry, and fnance [4][5][6]. Numerous efcient methods have been developed to obtain both analytical and numerical solutions for fractional order diferential equations. Some prominent methods include homotopy perturbation method, subequation method, the frst integral method, and Lie group method; for more details see [7][8][9][10].
Te Lie symmetry method was initially introduced by Sophus Lie (1842-1899) in order to study the (DE) of integer order. Tis method is an algorithmic procedure to obtain the point symmetry which leaves the considered diferential equation invariant. Later, Gazizov proposed the generalization of the Lie symmetry method for fractional diferential equations (FDEs) by developing prolongation formulas for fractional derivatives. Since then, numerous studies have been conducted to investigate FDEs using the Lie symmetry method, see [11][12][13].
Conservation laws play a signifcant role in investigating various properties of nonlinear partial diferential equations (PDEs). Te relationship between the Lie symmetry group and conservation laws of PDEs was established by Noether's theorem [14] which provides a powerful framework for constructing conservation laws of diferential equations. In recent developments, Ibragimov [15] has introduced a new conservation theorem based on the concept of nonlinear self-adjoint equations to study the conservation laws for arbitrary diferential equations.
Te analysis on the equations of foam drainage is of considerable signifcance as foams omnipresent in daily activities, either naturel or industrial. Te generalized foam drainage equation describes the evolution of the vertical density of foam under the gravity, for more details see [16][17][18][19].
In this study, we are interesting in the following fractional generalized foam drainage equation: where z α t u is the Riemann-Liouville (R-L) fractional derivative of order α with respect to t. In [20], the generalized fractional foam drainage equation with α � 1 is reduced to the following classical generalized foam drainage equation: for m � (1/2), equation (1) becomes the well-known foam drainage equation which has been studied in both cases, fractional and integer order by using Lie symmetry analysis see ( [21,22]), also for the case and m � 1, 2 the equation is studied in [21]. Te paper is structured as follows. In Section 2, we provide some of the most important Lie symmetry analysis results in the context of fractional partial diferential equations FPDEs in general. In Section 3, we present Lie point symmetries and similarity reduction of generalized fractional foam drainage equation. In Section 4, we propose another type of solutions in the form of power series solution by using the power series method. By using the nonlinear self-adjointness method, the conservation laws of equation (1) are calculated in Section 5. Finally, some conclusions are given in Section 6.

A Review on Lie Symmetry Analysis
Te main idea of this section is to describe the Lie symmetry method for FPDEs; so, let us consider a general form of the FPDE expressed as follows: where the subscripts indicate the partial derivatives and (z α /zt α ) is R-L fractional derivative operator presented by where g(x, t) is a real valued function, and n ∈ N * (the set of natural numbers). Te Lie symmetry method is based on assuming that equation (3) is invariant under the following transformations introduced by where (5) is a one-parameter Lie group and ϵ is the group parameter and ξ, τ, and η are the infnitesimals and η x and η xx are the extended infnitesimals of order 1 and 2 and are given by the following explicit form: where D x , is the total derivative operator with respect to x, defned by Now, the explicit form of the extended infnitesimal η α t of order α is written as follows: where D α t is the total time fractional derivative, and μ is given by 2 International Journal of Diferential Equations We should mention here that μ vanishes when the infnitesimal η(x, t, u) is linear in the variable u, that is, One can now present the Lie algebra of the oneparameter Lie group (5), which is generated by the vector felds in the following form: Te prolonged X (α,2) operator of the infnitesimal generator X of order (α, 2) is written in the following form:

Theorem 1 (Infnitesimal criterion of invariance). Equation (1) is invariant under (5), if and only if equation (1) satisfes the following invariant condition described by
Remark 2 (Invariance condition). In equation (3), the lower limit of the integral must be invariant under (5), which means is an invariant solution of (5) if it satisfes the following conditions: is an invariant surface of (11), which is equivalent to

Application of the Proposed Method for Generalized Foam Drainage Equation
Suppose that equation (1) is invariant under equation (5), so we have that with u � u(t, x) satisfes equation (1). Now, we start by applying the second prolongation X (α,2) to (1), then the infnitesimal criterion (13) becomes Substituting the explicit expressions η x , η xx , and η α t into (17) and equating powers of derivatives up to zero, we obtain the determining equations, by analyzing the determining equations with the initial condition (14), and the infnitesimals are determined as follows: where C 1 and C 2 are arbitrary constants. Te corresponding Lie algebra is written as follows: If we set It is clear to show that the vector felds X 1 , X 2 are closed under the Lie bracket defned by [X i , X j ] � X i X j − X j X i ; thus, the Lie algebra X is generated by the vectors felds X i (1,2) and is rewritten as follows: In order to fnd the reduced form and exact solution, we should solve the characteristic equation corresponding of each infnitesimal generator, which is described by Case 4. Reduction with X 1 � (z/zx). By integrating the characteristic equation the symmetry, X 1 , leads to the group invariant solution and f(t) satisfes Terefore, the group invariant solution corresponding to X 1 , is given by with k 1 as an arbitrary constant. Figure 1 presents the graph of solution u 1 (x, t) for some diferent values of α, with k 1 as an arbitrary constant. Figure 1 presents the graph of solution u 1 (x, t) for some diferent values of α.

Theorem 6. Using the abovementioned similarity transformation (30) in (1), we fnd that the time fractional foam drainage equation is transformed into a nonlinear ODE of fractional order in the following form:
with (P δ,α λ f)(ζ) is the Erdélyi-Kober diferential operator given by where K δ,α λ is the Erdélyi-Kober fractional integral operator introduced by

International Journal of Diferential Equations
Proof. By using the Riemann-Liouville fractional derivative defnition for the similarity transformation, we have We have Let � (t/s), we have ds � (− t/v 2 ), so the abovementioned expression can be expressed as follows: On the other hand, we have We repeat this procedure n − 1 times, we get Continuing further by calculating u x and u xx for (31) and replacing in equation (1) we fnd that time-fractional generalized foam drainage equation is reduced to a FODE written as follows: which is equivalent to International Journal of Diferential Equations So, the proof becomes complete.

Conservation Laws
In this present section, some of the conservation laws of the foam drainage equation are derived. Te conservation laws of equation (1) are that each vector (C t , C x ) can be satisfed by the following conservation equation for all solutions of equation (1) where D t and D x are the total derivative operators with respect to t and x, respectively. So, by observing equation (1), we can easily see that equation (1) can be rewritten as follows: thus, C t � D α− 1 u and C x � − u m u x + u 2 is a conserved vector of equation (1). Now, let us introduce the formulation of formal Lagrangian which is written as follows: with ψ(x, t) as a new dependent variable. Te adjoint equation of the foam drainage equation is determined by where (δ/δu) is the Euler-Lagrange operator described by with and (D α t ) * is the right-side Caputo operator. Te construction of CLs for FPDEs is in the same way of PDEs; therefore, the fundamental Noether identity is given by where N x , N t are Noether operators, X (α,2) is defned by (12), and W i is the characteristic function represented as follows: For the x-component of the Noether operator, it is clear to defne N x as follows: substituting (63) and (64) into (62), we obtain . . .
n j�0 a n− j a j z n + k � 0, Comparing the coefcient of z, when n � 0, we get the second case is when n ≥ 0 thus, the power series solution of equation (1) can be expressed as where a 0 is an arbitrary constant, and by using (67) and (68), all the coefcients of sequence a n ∞ 2 can be calculated. It remains to prove the convergence of the power series solution.
We can see that a n+1 ≤ M a n + n i�0 a n− j a j ⎡ ⎣ ⎤ ⎦ , where M � max |υ/mC 2 a m 0 |, |1/mCa m 0 | . Now, we take another power series of the form with q 0 � a 0 , We can observe that |a n | ≤ q n , for n � 0, 1, 2, . . . , then Q(z) � ∞ n�0 q n z n is a majorant series of (63). We continue by showing that Q(z) has a positive radius of convergence We can conclude from (74) that B(z, δ) is analytic in the neighborhood of (0, q 0 ), with By using the implicit function theorem given in [23,24], Q(z) is analytic in a neighborhood of (0, q 0 ) with a positive radius, which shows that the power series Q � Q(z) � ∞ n�0 q n z n converges in a neighborhood of (0, q 0 ); therefore (63) is convergent in a neighborhood of (0, q 0 ).
Finally, we can present the graph of the power series solutions given in the following Figures 2 and 3

Conclusion
In this paper, the Lie symmetry method is used to study the fractional generalized foam drainage equation based in Riemann-Liouville derivative. Lie symmetries are calculated and used to reduce the foam drainage equation to an ordinary diferential equation of fractional order connected to the Erdélyi-Kober fractional operator, and an additional solution of equation is given by mean of the power series method. Some of CLs are obtained by using Ibragimov's method. Te power series method and the fractional Lie symmetry analysis technique ofer valuable and efective mathematical methods for researching other FDEs in mathematical physics and engineering.

Data Availability
No data were used to support the fndings of this study.    International Journal of Diferential Equations 9