Lie Symmetry Analysis of Partial Differential Equations

Abstract

In this article, we discussed the Lie symmetry analysis of fractional and integer order differential equations. The symmetry algebra of both differential equations is obtained and utilized to find the similarity reductions, invariant solutions, and conservation laws. In both cases, the symmetry algebra is of low dimensions.

1. Introduction

In the last century, fractional partial differential equations (FPDEs) have played important rules in the fields of science and engineering, for instance, physics, chemistry, biology, andcontrol theory. Recently, those class of differential equations has also attracted much more interest of mathematicians and physicists [1–6].

Finding the best methods of obtaining the exact solutions of differential equations remains one of the unanswered questions in the field. Many approaches have been developed by mathematicians to study the solutions of PFDEs, such as Adomian decomposition method, the fractional subequation method, numerical method, the first integral method, and Lie symmetry method [7–14]. In this article, we consider one of the powerful techniques of solving and analyzing differential equations, i.e., the Lie symmetry method. The Lie symmetry method is widely used to transformed partial differential equations (PDEs) into ordinary differential equations (ODEs), and the ODE is later solve numerically or analytically using similarity invariant [7, 9, 10, 12, 14–22]. Lie symmetry is also utilized in obtaining the conservation laws (Cls) [23]. The method developed by Noether theorem [24] and Ibraginov’s [25] is one of the best and simplest methods of evaluating Cls of differential equations.

Consider general forms of fractional differential equations: where denotes the unknown function, and is a known function. The fractional order is a real number and denotes Riemann-Liouville(R-L) derivative defined in [1, 3, 26] as where denotes the standard gamma function defined by

The partial differential equations where denotes the spatial dimension , have been introduced in [27]. In particular, if , it becomes which is equivalent to

Therefore, the fractional form of differential equation is define as where is given by (2).

2. Lie Symmetries of Eq. (6)

In this section, we first consider the Lie symmetry analysis of Eq.(6). To obtained the Lie symmetry analysis, we first consider a one-parameter Lie group of transformations with a small parameter . The vector field associated with the one-parameter group of transformation is

Thus, expanding the infinitesimals generator to include the transformation of the derivatives, we obtained the following third prolongation

In (10), , , , and are all undetermined functions, which are given by the following formulae where and represent the total derivatives with respect to and , respectively. refers the reader(s) to [28] for details of how to evaluate the prolongation formulae.

If the vector field (9) forms a symmetry of Eq.(6), the infinitesimal generator must satisfy the following invariance criterion for Eq.(6), given as where

Substituting Eqs. (11)–(14) into Eq. (15) and equating the coefficients of various powers of partial derivatives of to zero, an overdetermined system of equations known as determining equations is obtained. Solving the determining equations the following infinitesimals for has been derived where , , and are arbitrary constants, and throughout this paper, we denote

Therefore, we have the following conclusion.

Theorem 1. For the arbitrary parameters if , , the vector field admitted by the differential equation (6) is

The sketch of the proof has been stated above. In details, we should divide it into the following four cases. Obviously, the vector fields of each case are the special cases of (19) (i)If , , the vector field is (ii)If , , the vector field is the same as (i)(iii)If , , the vector field is (iv)If , , , the vector field is the same as (iii).

The vector fields , , and form a Lie algebra under the following Lie bracket where and stand for .

That is to say

By solving the following ordinary differential equations with the initial conditions:

We therefore obtain the group transformation which is generated by infinitesimal generators , respectively

Remark 2. In (25)–(27), an arbitrary element in can transfer one solution of Eq. (6) to another one, so do the products of the elements from , , and .

Remark 3. The Lie group is a normal Lie subgroup of . The Lie algebra generated by and is an ideal of .

Theorem 4. If is a solution of Eq. (1.3), then , , and as follows are solutions of Eq. (6) as well.

3. Similarity Reductions to Eq. (6)

In the preceding section, we obtained the group symmetry analysis of Eq. (6). In this section, the characteristic equations of vector fields are obtained by making use of (19) and utilized to perform the symmetry reduction.

For , the characteristic equation can be presented as follows:

Solving the characteristic equation (29), we have , where which yields a trivial solution.

For , the characteristic equation can be expressed as follows: from which we have , where which transformed (6) to an ODE

For , the characteristic equation can be written as follows:

Solving the characteristic equation, we have the following invariants , , which transformed (6) to an ODE

4. Nonlinear Self-Adjointness of Eq. (6)

In this section, we shall show that Eq. (6) is nonlinearly self-adjoint. Let it start by presenting the definitions of nonlinear self-adjointness of differential equations according to Ibragimov’s [13, 25].

Definition 5. Given a differential function and the new dependent variable known as the adjoint variable or local variable [13, 25], the formal Lagrangian for the differential equation is the differential function given by

Definition 6. [25]. The differential equation (6) is said to be nonlinearly selfadjoint if there exists a substitution such that for some undermine function λ.

Theorem 7. The differential equation (6) is self-adjoint.

Proof. The formal Lagrangian is Substituting into , we have the adjoint equation to Eq. (6) Let and the left hand of (38) be , we shall get , and Then, we prove that Eq. (6) is self-adjoint.☐

Generally speaking, we are now to calculate the conserved vectors. However, by using the method of Ibragimov [25], so we here omit the process.

5. Lie Symmetry and Reductions of Eq. (7)

In this section, we deal with all of the point symmetries of Eq. (7). We now assume that in this section. Thus, Eq. (1.4) is an FPDE with the infinitesimal generator given by (2.2). If the vector field (9) generates a symmetry of Eq. (7), then must satisfy the following Lie symmetry condition: where

Also, the invariant condition yields [29] and the αth extended infinitesimal related to Riemann-Liouville fractional time derivative with (42) is given by [30, 31]. where

The expression of μ is complicated; however, it should converges to zero when the infinitesimal φ is linear in u, because of the existence of the derivatives in the above expression (44).

Thus, the Lie group classification method for the FPDE leads to the following result.

Theorem 8. The infinitesimal symmetry group of the equation (7) is spanned by the two vector fields

Proof. Considering the invariance criterion (40), we have solving the determining equation (46), and we obtained the following infinitesimals which follows that the fractional differential equation (7) admitted two dimensional symmetries and is spanned by (45).☐

Next, we utilized the admitted Lie symmetry and perform similarity reductions, present the reduced nonlinear fractional ordinary differential equations (FODEs), and classify the corresponding group-invariant solutions of the fractional .

Case 9. For , the characteristic equation is and the following similarity variables are obtained by solving the characteristic equation , t, which yields the following reduced ODE The fractional ODE has a polynomial general group-invariant solution where is an arbitrary constant of integration [6, 15, 16].

Case 10. For , the characteristic equation is and by solving the above equation, we get the group-invariant solution where is an arbitrary function of ζ. Using these invariants, Eq. (7) transforms to a special nonlinear ODE of fractional order. Thus, we have the following theorem corresponding to this case.

Theorem 11. The transformation (52) reduces (7) to the following nonlinear ordinary differential equation of fractional order with the Erd’elyi-Kober fractional differential operator of order [5] where is the Erd’elyi-Kober fractional integral operator.

Proof. Let , according to the Riemann-Liouville fractional derivative, and one can obtain Let , and then ;so, (55) can be written as In view of the Erd’elyi-Kober fractional integral operator (54), one can get Therefore, the right hand side of (57) becomes Repeating the same procedure times, one can obtain Using the definition of the Erd’elyi-Kober fractional differential operator (53), we get Substituting (60) into (57), we obtain an expression for the time fractional derivative Thus, the fractional equation (7) can be reduced into a fractional order.
ODE: ☐

6. Conservation Law of Eq. (7)

In this section, we will construct the conservation laws of the fractional differential equation (7).

Let be the formal Lagrangian of Eq. (7) written as , where is the new introduced dependent variable; so, the adjoint equation of (1.4) is written as , where is the Euler-Lagrange operator with respect to u and defined by and is the adjoint operator of . For Riemann-Liouville fractional differential operators, we have where is the right-sided fractional integral operators of order .

In this case, the formal Lagrangian of Eq. (7) is written as

Every Lie point symmetry (45) admitted by the Eq. (7) leads to a conservation law where the components constructed by the following formula [5, 32]: where and are the formal Lagrangian which are written in the symmetric form, and the mixed derivatives are given by the integral: for . For other cases, the calculations of conserved vector components are similar. In this case,

The conserved vector components are as follows: where , .

Data Availability

All the data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that this article has no conflicts of interest.

Acknowledgments

The first and third authors are supported by the National Science Foundation of China [Grant No. 11801264], Hunan Provincial Natural Science Foundation of China [Grant Nos. 2019JJ50505, 2019JJ50490, 2019JJ40240], and Scientific Research Foundation of Hunan Province Education Department (Grant No. 18C0455).