Lie symmetry analysis and conservation laws for the time fractional simplified modified Kawahara equation

1 Introduction

Symmetry analysis has many applications in the field of science and engineering. Lie’s method is one of the global and efficient methods for investigating analytical solutions and symmetry properties of nonlinear partial differential equations (NLPDEs) [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Fractional calculus has been successfully used to explain many complex nonlinear phenomena and dynamic processes in physics, engineering, electromagnetics, viscoelasticity, and electrochemistry [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34].

Generally, physical phenomenon might depend on its current state and on its historical states, which can be modelled successfully by applying the theory of derivatives and integrals of fractional order [35, 36]. Due to this, several analytical techniques are used to derive exact, explicit, and numerical solutions of nonlinear fractional partial differential equations (FPDEs) [30, 31, 32, 33, 34]. We find very few studies of symmetry analysis for FPDEs and their group properties are not plainly understood [37, 38, 39, 40, 41].

In other words, Cls are universally known to possess an important role in the analysis of NLPDEs from a physical viewpoint [42] . If the considered system has Cls, then its integrability will be possible [43, 44]. Noether theorem supplies us with a strategic idea for constructing Cls of NLPDEs so long as the Noether symmetry associated with the Lagrangian is known for Euler-Lagrange equations [45]. Nevertheless, there are some techniques in the literature for obtaining the Cls of the NLPDEs, that do not have the Lagrangian [46, 47].

Time fractional NLPDEs come from classical NLPDEs by replacing its time derivative with a fractional derivative. In the present work, we study Lie symmetry analysis, explicit solution using the power series technique and Ibragimov’s nonlocal Cls [48] for the time fractional SMK equation given by

in Eq. (1), 0 < α ≤ 1, and β and γ are arbitrary constants, and α is the order of the fractional time derivative. If α = 1, Eq. (1) reduces to the classical SMK equation which was considered for exact travelling wave solutions and Cls in [49, 50, 51]. Moreover, one can find more details on the construction of analytical, exact, numerical solutions, and other information for classical NLPDEs, in [52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69].

2 Preliminaries

Consider the RL fractional derivative [70, 71] given by

where n is a natural number and I^μf(t) is the RL fractional integral of order μ given by

and Γ(z) represents Gamma function.

Consider time-fractional PDEs as below

Given a one-parameter Lie group of infinitesimal transformations of the form

where

In Eq. (5), D_x is the total differential operator defined by

The corresponding Lie algebra of symmetries consists of a set of vector fields of the form

The vector field Eq. (6) is a Lie point symmetry of Eq. (3) provided

Also, the invariance condition yields [72] gives

and the α^th extended infinitesimal related to RL fractional time derivative with Eq. (8) is given by [54, 55].

in Eq. (9),

It is worth noting that, μ = 0 if the infinitesimal η is linear in u, due to the presence of ∂ηk∂uk, where k ≥ 2 in Eq. (10).

3 Lie symmetries and reduction for Eq. (1)

Suppose that Eq. (1) is an invariant under Eq. (5), we have that

so that, u = u(x, t) satisfies Eq. (1). Using Eq. (5) in Eq. (11), we get the invariant equation

Putting the values of ηα0,η^x and η^xxxxx from Eq. (5) and Eq. (9) into Eq. (12) and isolating coefficients in partial derivatives with respect to x and power of u, we get

Solving these equations, we get:

where c₁ and c₂ are arbitrary constants. Thus infinitesimal symmetry group for Eq. (1) is spanned by the two vector fields

The similarity variables for the infinitesimal generator X₂ can be obtained by solving the following equations

Solving the above equations, we get

Hence, from the symmetry X₂, we get the group-invariant solution

in Eq. (15), f is an arbitrary function of ξ. Using Eq. (15), Eq. (1) is transformed to a special nonlinear ODE of fractional order.

Consider the following theorem

with the EK fractional differential operator [22]

where

is the EK fractional integral operator [74, 75].

4 Conservation laws

We now construct the Cls for Eq. (1). We start with some definitions. The RL left-sided time-fractional derivative given by

where D_t is the total differential operator with respect to t, n = [α] + 1, and oI^n−αu represents the left sided time-fractional integral of n − α order given by

In Eq. (30), Γ(z) represents Gamma function.

A Cls for Eq. (1) is represented as

where C^t = C^t(x, t, u,…), C^x = C^x (x, t, u, …), and Eq. (31) holds for all solutions u(x, t) of the Eq. (1).

We now apply Ibragimov method [48] for constructing the Cls of Eq. (1). Lagrangian for Eq. (1) can be presented as

where v(x, t) is another dependent variable. The Euler-Lagrange operator [45, 46] is

where (Dtα)∗ is the adjoint operator of (Dtα). The adjoint equation to Eq. (1) is given by [48]

Consider two independent variables x, t and one dependent variable u(x, t), we have that

in Eq. (35), l represent the identity operator, δδu is the Euler-Lagrangian operator, N^t and N^x represent the Noether operation, X̄ is defined by

and the Lie characteristic function W is given by

When RL time-fractional derivative is used in Eq. (1), N^t is defined by [45, 46]

With J given by

For Eq. (1), the operator N^x is given by

The invariance condition for any given generator X of Eq. (1) and its solutions reads

and consequently the Cls of Eq. (1) can be written as

Now, we present the Cls for Eq. (1) using the basic definitions presented above. We consider two cases corresponding to the order of α.

1. When α ∈ (0, 1), with the help of Eq. (38) and Eq. (39), the components of the conserved vectors are

   Cit=ξ2L+(−1)0oDtα−1(Wi)Dt0∂L∂(oDtαu)−(−1)1×J(Wi,Dt1∂L∂(oDtαu))=voDtα−1(Wi)+J(Wi,vt),

   Cix=ξ1l+Wi(∂∂ux−Dx∂∂uxx+Dx2∂∂uxxx−Dx3∂∂uxxxx+Dx4∂∂uxxxxx)+Dx(Wi)(∂∂uxx−Dx∂∂uxxx+Dx2∂∂uxxxx−Dx3∂∂uxxxxx)+Dx2(Wi)(∂∂uxxx−Dx∂∂uxxxx+Dx2∂∂uxxxxx)+Dx3(Wi)(∂∂uxxxx−Dx∂∂uxxxxx)+Dx4(Wi)∂∂uxxxxx

   =Wi(βu2v+Dx4γv)+Dx(Wi)(−Dx3γv)+Dx2(Wi)(Dx2γv)−γvxDx3(Wi)+γvDx4(Wi),

   where i = 1, 2 and the functions W_i are given by

   W1=−ux,W2=−2uα−5tut−αxux.

2. When α ∈ (1, 2), with the help of Eq. (38) and Eq. (39), the components of the conserved vectors are

   Cit=ξ2L+(−1)0oDtα−1(Wi)Dt0∂L∂(oDtαu)−(−1)1J(Wi,Dt1∂L∂(oDtαu))+(−1)1oDtα−2(Wi)Dt1∂L∂(oDtαu)−(−1)1J(Wi,Dt1∂L∂(oDtαu))

   =voDtα−1(Wi)+J(Wi,vt)−vtoDtα−2(Wi)−J(Wi,vtt)

   Cix=ξ1l+Wi(∂∂ux−Dx∂∂uxx+Dx2∂∂uxxx−Dx3∂∂uxxxx+Dx4∂∂uxxxxx)+Dx(Wi)(∂∂uxx−Dx∂∂uxxx+Dx2∂∂uxxxx−Dx3∂∂uxxxxx)+Dx2(Wi)(∂∂uxxx−Dx∂∂uxxxx+Dx2∂∂uxxxxx)+Dx3(Wi)(∂∂uxxxx−Dx∂∂uxxxxx)+Dx4(Wi)∂∂uxxxxx

   =Wi(βu2v+Dx4γv)+Dx(Wi)(−Dx3γv)+Dx2(Wi)(Dx2γv)−γvxDx3(Wi)+γvDx4(Wi),

   where i = 1, 2 and the functions W_i are given by

   W1=−ux,W2=−2uα−5tut−αxux.

5 Explicit power series solutions

Here, we investigate the exact analytic solutions via power series method [76] and symbolic computations [77] for Eq. (28). Set

from Eq. (43), we can have

Substituting Eqs. (44) into Eq. (28), we obtain

Comparing coefficients in Eq. (45) when n = 0, we obtain

when n ≥ 1, we have

Thus, the power series solution for Eq. (28) can be represented in the form:

Consequently, we acquire the exact power series solution for Eq. (28) as

6 Physical interpretation of the power series solution for Eqs. (47)

In order to have clear and proper understanding of the physical properties of the power series solution, the 3-D, 2-D, and contour plots for the solution Eqs. (47), are plotted in Figures 1-4 by using suitable parameter values.

Figure 1

3D plot of (47) a₀ = a₁ = a₂ = 1, a₃ = 0.5, a₄ = 1.7, β = 2, γ = 1, α = 0.5, Γ = 0.85

Figure 2

Contour plot of (47) a₀ = a₁ = a₂ = 1, a₃ = 0.5, a₄ = 1.7, β = 2, γ = 1, α = 0.5, Γ = 0.85

Figure 3

3D plot of (47) a₀ = a₁ = a₂ = 0.8, a₃ = a₄ = 1, β = 1.2, γ = 3, α = 0.9,Γ = 0.1

Figure 4

Contour plot of (47) a₀ = a₁ = a₂ = 0.8, a₃ = a₄ = 1, β = 1.2, γ = 3, α = 0.9,Γ = 0.1

7 Concluding remarks

In this research, we analyzed time fractional SMK by means of Lie symmetry analysis using the RL derivative. We reduced the governing equation to a nonlinear ODE of fractional order. The obtained fractional ODE was solved using a power series technique. Ibragimov’s nonlocal conservation theorem was applied to establish Cls for the governing equation. Some 3-D, 2-D, and contour plots were also presented.