Conserved vectors with conformable derivative for certain systems of partial differential equations with physical applications

Maysaa Mohamed Al Qurashi

doi:10.1515/phys-2020-0127

Article Open Access

Conserved vectors with conformable derivative for certain systems of partial differential equations with physical applications

Maysaa Mohamed Al Qurashi

Published/Copyright: June 2, 2020

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From the journal Open Physics Volume 18 Issue 1

Abstract

In this paper, we examine conservation laws (Cls) with conformable derivative for certain nonlinear partial differential equations (PDEs). The new conservation theorem is used to the construction of nonlocal Cls for the governing systems of equation. It is worth noting that this paper introduces for the first time, to our knowledge, the analysis for Cls to systems of PDEs with a conformable derivative.

Keywords: systems of partial differential equations; Lie symmetry; conformable derivative; conservation laws

1 Introduction

Fractional calculus is an emerging field of mathematics having important real-world applications in various branches of science and engineering [1,2,3,4,5,6,7,8,9,10]. This fascinating field had many useful transformations during more than 300 years, and it appears within various representations, namely, several fractional operators were developed and successfully applied to solve complicated dynamical systems [1,2,3,4,5,6,7,8,9,10]. In [11], some researchers tried to introduce the conformable derivative to use the classical Leibniz rule and composition rule as in the classical case but they discovered a new local derivative containing a parameter. Within the literature, many authors used this conformable derivative in some useful applications [12,13,14,15,16].

We recall that conservation laws (Cls) stemmed from the pragmatic phenomena of energy, mass, and momentum [17]. The Cls were used to develop numerical methods, to demonstrate the nature and uniqueness of solutions [18], and to examine internal characteristics such as recurrence operators and bi-Hamiltonian structures [19]. It should be noted that various generalizations of the Noether’s theorem and Euler–Lagrange [20] along with various fractional derivatives to define Cls for fractional nonlinear partial differential equations (PDEs) with fractional Lagrangians [21,22,23] have been studied. The literature included several studies for Cls to various equations using fractional derivatives from the sense of Riemann–Liouville. In this research, we aim to present a technique using symmetries from Lie points to create Cls for nonlinear PDEs with physical applications with conformable derivatives using new conservation theorem.

This paper has been organized as follows: in Section 2, some basic properties of the conformable derivative are provided; Section 3 deals with the conformable symmetry analysis; Section 4 is focussing on the conformable dispersive long-wave system; the conformable Whitham–Broer–Kaup system is scrutinized in Section 5; and conclusions are presented in Section 6.

2 Basic properties of the conformable derivative

Conformable derivative (CD) is an extended classical derivative that was proposed in [11,24,25,26]. This derivative has overcome the barriers with other derivatives. It is described as

Suppose f : ( 0 , ∞ ) → ℝ , then CD of f with order α is given by [11]

(1) T α [ f ( t ) ] = lim ε → 0 f ( t + ε t 1 − α − f ( t ) ) ε ,

for t > 0, α ∈ (0, 1). If T _α[ f(t)] exists for t in some interval (0, a) with a > 0, and lim t → 0 + T α [ f ( t ) ] also exists, then T α [ f ( 0 ) ] = lim t → 0 + T α [ f ( t ) ] . Moreover, if T _α[ f(t)] exists on [0, ∞), then f is said to be α-differentiable at t.

The following properties are associated with the conformable derivative [11]:

T _α(af + bg) = aT _α( f ) + bT _α( f ), a , b ∈ ℝ ,
T _α(t ^μ) = μt ^μ−α, μ ∈ ℝ ,
T _α( fg) = f T _α + gT _α(f),
T α ( f g ) = g T α ( f ) − f T α g 2 ,
If f is differentiable, then T α ( f ) ( t ) = t 1 − α d f d g .

The integral associated with the conformable derivative can be written as [11] I α [ f ( t ) ] = I [ t α − 1 f ( t ) ] = ∫ 0 t f ( τ ) τ 1 − α d τ , where the integral depicts the standard Riemann improper integral such that α ∈ (0, 1].

Besides, we can prove easily that [11]

T ^α I ^α[ f(t)] = f(t), for t ≥ 0, where f depicts any function that is continuous within the domain of I ^α. For every t > 0, we get I ^α T ^α[ f(t)] = f(t) − f(0). Also, we have T _α( fog)(t) = ( f ′(g(t)))(t)T _α g(t) [11].

3 Symmetry analysis

Consider the conformable system of PDEs [27]:

(2) ∂ α u ∂ t α = F ( x , t , u , u x 2 α , u x 3 α , … ) , ∂ α v ∂ t α = F ( x , t , v , v x 2 α , v x 3 α , … ) ,

where u = u(x, t), F(x, t, u, u _xx, u _xxx,…) depicts a nonlinear function and ∂ α u ∂ t α and ∂ α v ∂ t α depict the conformable fractional derivative. Given a one-parameter Lie group of infinitesimal transformations of the form

(3) t ¯ = t + ε ξ 1 ( t , x , u , v ) + O ( ε 2 ) , x ¯ = x + ε ξ 2 ( t , x , u , v ) + O ( ε 2 ) , u ¯ = u + ε η 1 ( t , x , u , v ) + O ( ε 2 ) , v ¯ = v + ε η 2 ( t , x , u , v ) + O ( ε 2 ) , ∂ α u ¯ ∂ t ¯ = ∂ α u ∂ t α + ε η 1 α , t ( t , x , u , v ) + O ( ε 2 ) , ∂ α v ¯ ∂ t ¯ = ∂ α v ∂ t α + ε η 2 α , t ( t , x , u , v ) + O ( ε 2 ) , ∂ α u ¯ ∂ x ¯ α = ∂ u ∂ x + ε η 1 α , x ( t , x , u , v ) + O ( ε 2 ) , ∂ α v ¯ ∂ x ¯ α = ∂ v ∂ x + ε η 2 α , x ( t , x , u , v ) + O ( ε 2 ) , ∂ 2 α u ¯ ∂ x ¯ 2 α = ∂ 2 α u ∂ x 2 α + ε η 1 α , x x ( t , x , u , v ) + O ( ε 2 ) , ∂ 2 α v ¯ ∂ x ¯ 2 α = ∂ 2 α u ∂ x 2 α + ε η 2 α , x x ( t , x , u , v ) + O ( ε 2 ) , ∂ 3 α u ¯ ∂ x ¯ 3 α = ∂ 3 α u ∂ x 3 α + ε η 1 α , x x x ( t , x , u , v ) + O ( ε 2 ) , ∂ 3 α v ¯ ∂ x ¯ 3 α = ∂ 3 α v ∂ x 3 α + ε η 2 α , x x x ( t , x , u , v ) + O ( ε 2 ) , .. .

where

(4) η 1 α , t = t 1 − α η 1 t + ( 1 − α ) ξ 1 t − α u t , η 2 α , t = t 1 − α η 2 t + ( 1 − α ) ξ 1 t − α v t , η 1 α , x = x 1 − α η 1 x + ( 1 − α ) ξ 2 x − α u x , η 2 α , x = x 1 − α η 2 x + ( 1 − α ) ξ 2 x − α v x , η 1 α , x x = x 2 − 2 α η 1 x x + ( 1 − α ) x 1 − 2 α η 1 x + ( 2 − 2 α ) x 1 − 2 α ξ 2 u x x + ( 1 − α ) ( 1 − 2 α ) x − 2 α ξ 2 u x , η 2 α , x x = x 2 − 2 α η 2 x x + ( 1 − α ) x 1 − 2 α η 2 x + ( 2 − 2 α ) x 1 − 2 α ξ 2 v x x + ( 1 − α ) ( 1 − 2 α ) x − 2 α ξ 2 v x , η 1 α , x x x = x 3 − 3 α η 1 x x x + ( 3 − 3 α ) x 2 − 3 α η 1 x x + ( 1 − α ) ( 1 − 2 α ) x 1 − 3 α η 1 x + ( 3 − 3 α ) ξ 2 x 2 − 3 α u x x x + ( 3 − 3 α ) ( 2 − 3 α ) ξ 2 x 1 − 3 α u x x + ( 1 − α ) ( 1 − 2 α ) ( 1 − 3 α ) ξ 2 x − 3 α u x , η 2 α , x x x = x 3 − 3 α η 2 x x x + ( 3 − 3 α ) x 2 − 3 α η 2 x x + ( 1 − α ) ( 1 − 2 α ) x 1 − 3 α η 2 x + ( 3 − 3 α ) ξ 2 x 2 − 3 α v x x x + ( 3 − 3 α ) ( 2 − 3 α ) ξ 2 x 1 − 3 α v x x + ( 1 − α ) ( 1 − 2 α ) ( 1 − 3 α ) ξ 2 x − 3 α v x , …

and

(5) η 1 t = D t ( η 1 ) − u x D t ( ξ 2 ) − u t D t ( ξ 1 ) , η 2 t = D t ( η 2 ) − v x D t ( ξ 2 ) − v t D t ( ξ 1 ) , η 1 x = D x ( η 1 ) − u x D x ( ξ 2 ) − u t D x ( ξ 1 ) , η 2 x = D x ( η 2 ) − v x D x ( ξ 2 ) − v t D x ( ξ 1 ) , η 1 x x = D t ( η 1 x ) − u x x D t ( ξ 2 ) − u x t D t ( ξ 1 ) , η 2 x x = D t ( η 2 x ) − v x x D t ( ξ 2 ) − v x t D t ( ξ 1 ) , η 1 x x x = D t ( η 1 x x ) − u x x x D t ( ξ 2 ) − u x x t D t ( ξ 1 ) , η 2 x x x = D t ( η 2 x x ) − v x x x D t ( ξ 2 ) − v x x t D t ( ξ 1 ) ,

The total differential operator D _x and D _t are defined by

(6) D t = ∂ ∂ t + u t ∂ ∂ u + u t t ∂ ∂ u t + u x t ∂ ∂ u x + ⋯ + v t ∂ ∂ v + v t t ∂ ∂ v t + v x t ∂ ∂ v x ⋯ D x = ∂ ∂ x + u x ∂ ∂ u + u x x ∂ ∂ u x + u x t ∂ ∂ u t + ⋯ + v x ∂ ∂ v + v x x ∂ ∂ v x + v x t ∂ ∂ v t ⋯

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

(7) X = ξ 1 ∂ ∂ x + ξ 2 ∂ ∂ t + η 1 ∂ ∂ u + η 2 ∂ ∂ v .

The vector field equation (7) is a Lie point symmetry of equation (2) provided that

(8) P α , i r X ( Δ ) | Δ = 0 = 0 ,

where P is the prolongation operator, Δ is the symbolize form of equation (2) and i is the order of the system in equation (2). Also, the invariance condition yields

(9) ξ 1 ( x , t , u , v ) | t = 0 = 0 .

4 The dispersive long-wave system

The conformable dispersive long-wave system is given by

(10) ∂ α u ∂ t α = ( ∂ α u ∂ x α ) 2 − ∂ 2 α u ∂ x 2 α + 2 ∂ α v ∂ x α , ∂ α v ∂ t α = 2 u ∂ α v ∂ x α + 2 v ∂ α u ∂ x α + ∂ 2 α v ∂ x 2 α ,

where 0 < α ≤ 1 and α describes the order of the conformable derivative. If α = 1, equation (10) reduces to the classical dispersive long-wave system which is given by

(11) ∂ u ∂ t = ( u 2 − u x + 2 v ) x , ∂ v ∂ t = ( 2 u v + v x ) x .

Equation (11) is the dispersive long-wave equation [28]. In hydrodynamics, it describes the evolution of the horizontal velocity portion of water waves, which propagates in both directions in an infinite narrow channel of constant depth.

4.1 Symmetry analysis for equation (10)

In accordance with the invariance of equation (10) via equation (3), imposing equation (8) into equation (10), we have that

(12) η 1 α , t − 2 u η 1 α , x + 2 η 2 α , x + η 1 α , x x = 0 , η 1 α , t − 2 u η 1 α , x + 2 v η 2 α , x + η 2 α , x x = 0 ,

which must hold whenever equation (10) holds. It is of great importance to note that, using the properties of conformable derivative, we get the following equivalent form of equation (10).

(13) t 1 − α u t − ( x 1 − α u x ) 2 + ( 1 − α ) x 1 − 2 α u x + x 2 − 2 α u x x − 2 x 1 − α v x = 0 , t 1 − α v t − 2 x 1 − α u v x − 2 x 1 − α v u x − ( 1 − α ) x 1 − 2 α v x − x 2 − 2 α v x x = 0 .

Plugging equations (4) and (5) into equation (12) and using equation (13) instead of u _xx and v _xx where ever they appear and equating the coefficients of the various monomials in partial derivatives of u and v, the determining equations is obtained. Solving the obtained determining equations, we acquire

(14) ξ 1 = − 2 t α c 1 α + c 2 , ξ 2 = t 1 − α c 3 , η 1 = c 1 , η 2 = 0 ,

and the vector field is obtained as follows:

(15) X 1 = ∂ ∂ x , X 2 = ∂ ∂ t , X 3 = − 2 t α α ∂ ∂ x + ∂ ∂ u .

4.2 Nonlocal conservation laws (Cls) for equation (10)

The new conservation theorem [29] will be used for building the non-local Cls of equation (10). The conservation equation

(16) D i ( T i ) | = 0 ,

needs to be satisfied by the obtained conserved vectors. Where

(17) T i = ξ i ℒ + W α [ ∂ ℒ ∂ u i α − D j ( ∂ ℒ ∂ u i j α ) + D j D k ∂ ℒ ∂ u i j k α ⋯ ] + D j ( W α ) [ ∂ ℒ ∂ u i j α − D j ( ∂ ℒ ∂ u i j k α ) + ⋯ ] + D j D k ( W α ) [ ∂ ℒ ∂ u i j k α ] + ⋯

where W α is the characteristics function given by W α = η − ξ i u i .

The nonlocal Cls for the governing equation will now be presented. We begin with the Lagrangian of equation (10) as

(18) ℒ = p ( t 1 − α u t − ( x 1 − α u x ) 2 + ( 1 − α ) x 1 − 2 α u x + x 2 − 2 α u x x − 2 x 1 − α v x ) + q ( t 1 − α v t − 2 x 1 − α u v x − 2 x 1 − α v u x − ( 1 − α ) x 1 − 2 α v x − x 2 − 2 α v x x )

The adjoint equations can be presented as follows:

(19) { δ ℒ δ u ≡ t − α x − 2 α ( − 2 t α x α ( − 1 + α ) q v − t x 2 α p t + 2 t α x 2 p t u t + p ( ( − 1 + α ) ( x 2 α + t α ( − 1 + 2 α ) ) + 2 t α x 2 u t t ) + 3 t α x p x − 3 t α x α p x + 2 t α x 1 + α v q x + t α x 2 p x x ) = 0 , δ ℒ δ v ≡ t − α x − 2 α ( − 2 t α x α ( − 1 + α ) p − ( − 1 + α ) q ( − x 2 α + t α ( − 1 + 2 α ) + 2 t α x α u ) − t x 2 α q t + 2 t α x 1 + α p x − 3 t α x q x + 3 t α x α q x + 2 t α x 1 + α u q x − t α x 2 q x x ) = 0 } .

Now, with the help of the obtained point symmetries equation (15), we use the Noether operator N [29,30] to obtain conserved vectors, (T ¹, T ²) as follows:

For the symmetry X 1 = ∂ ∂ x , we obtain
(20) T 1 t = t − α x − 2 α ( p ( t x 2 α u t − t α x 2 u t 2 − 2 t α x ( − 1 + α ) u x ) + q ( t x 2 α v t + 2 t α x ( − 1 + α ) v x ) + t α x 2 ( p x u x − q x v x ) ) , T 1 x = p ( − t 1 − α + 2 x 2 − 2 α u t ) u x − t 1 − α q ( x , t ) v x
For the symmetry X 2 = ∂ ∂ t , we obtain
(21) T 2 t = t 1 − α x 1 − 2 α ( x ( u t p x − v t q x ) + p ( − ( − 1 + α ) u t + 2 x α v t − x u x t ) + q ( 2 x α v ( x , t ) u t + ( − 1 + α + 2 x α u ( x , t ) ) v t + x v x t ) ) , T 2 x = t 1 − α x 1 − 2 α ( p ( x u t 2 − ( − 1 + α ) u x − 2 x α v x + x u x x ) − q ( 2 x α v u x + ( 1 − α + 2 x α u ) v x + x v x x ) )
For the symmetry X 3 = − 2 t α α ∂ ∂ x + ∂ ∂ u , we obtain

(22) T 3 t = 1 α ( x − 2 α ( p ( − 2 t x 2 α u t + 2 t α x 2 u t 2 + x ( − 1 + α ) ( α + 4 t α u x ) ) − 2 q ( x 1 + α α v + t x 2 α v t + 2 t α x ( − 1 + α ) v x ) − x 2 ( p x ( α + 2 t α u x ) − 2 t α q x v x ) ) ) , T 3 x = p ( t 1 − α − 2 x 2 − 2 α u t ) ( 1 + 2 t α u x α ) + 2 t q v x α

5 The Whitham–Broer–Kaup system

The conformable Whitham–Broer–Kaup Wilson system is given by

(23) ∂ α u ∂ t α + u ∂ α u ∂ x α + ∂ α v ∂ x α + μ ∂ 2 α u ∂ x 2 α = 0 , ∂ α v ∂ t α + u ∂ α v ∂ x α + v ∂ α u ∂ x α + β ∂ 3 α u ∂ x 3 α − μ ∂ 2 α v ∂ t 2 α = 0

If α = 1, equation (23) becomes the classical Whitham–Broer–Kaup system given by

(24) ∂ u ∂ t + u u x + v x + μ u x x = 0 , ∂ v ∂ t + ( u v ) x + β u x x x − μ v x x = 0 .

Equation (24) is a completely integrable model that describes a dispersive long wave in shallow water. In equation (23), β and μ are real constants that represent different dispersive powers.

5.1 Symmetry analysis for equation (28)

Considering the invariance of equation (23) under the group of transformations equation (3), applying equation (8) into equation (23), we obtain

(25) η 1 α , t + u η 1 x + η 2 x + μ η 1 x x = 0 , η 2 α , t + v η 1 x + u η 2 x − μ η 2 x x + β η 1 x x x = 0 ,

which must hold whenever equation (23) holds. It is of great importance to note that, using the properties of conformable derivative, we get the following equivalent form of equation (23) as

(26) t 1 − α u t + x 1 − α u u x + x 1 − α v x + μ ( 1 − α ) x 1 − 2 α u x + u x 2 − 2 α u x x = 0 , t 1 − α v t + x 1 − α u v x x 1 − α v u x − μ ( 1 − α ) x 1 − 2 α u x − μ x 2 − 2 α u x x + λ ( 1 − α ) ( 1 − 2 α ) x 1 − 3 α u x + 3 λ ( 1 − α ) x 2 − 3 α u x x + λ x 3 − 3 α u x x x = 0 .

Plugging equations (4) and (5) into equation (23) and using equation (26) and equating the coefficients of the various monomials in partial derivatives of u and v, the determining equations are obtained. Solving the obtained determining equations, we acquire

(27) ξ 1 = t α c 1 α + c 2 , ξ 2 = t 1 − α c 3 , η 1 = c 1 , η 2 = 0 ,

and the associated Lie algebra is generated by the vector field as follows:

(28) X 1 = ∂ ∂ x , X 2 = ∂ ∂ t , X 3 = t α α ∂ ∂ x + ∂ ∂ u .

5.2 Nonlocal Cls for equation (28)

Now we present the nonlocal Cls for the governing equation. We now begin with the Lagrangian of equation (23) as

(29) ℒ = p ( t 1 − α u t + x 1 − α u u x + x 1 − α v x + μ ( 1 − α ) x 1 − 2 α u x + u x 2 − 2 α u x x ) + q ( t 1 − α v t + x 1 − α u v x x 1 − α v u x − μ ( 1 − α ) x 1 − 2 α u x − μ x 2 − 2 α u x x + λ ( 1 − α ) ( 1 − 2 α ) x 1 − 3 α u x + 3 λ ( 1 − α ) x 2 − 3 α u x x + λ x 3 − 3 α u x x x ) .

The adjoint equations can be presented as follows:

(30) { δ ℒ δ u ≡ t − α x − 3 β ( x β p ( x 2 β ( − 1 + α ) + t α ( 1 − 3 β + 2 β 2 ) μ + t α x β ( − 1 + β ) u ) + t α ( − 1 + β ) q ( ( − 1 + 2 β ) ( ( − 1 + 3 β ) λ − x β μ ) + x 2 β v ) − t x 3 β p t + 3 t α x 1 + β μ p x − 3 t α x 1 + β β μ p x t α x 1 + 2 β u p x − 7 t α x λ q x + 18 t α x β λ q x − 11 t α x β 2 λ q x − 3 t α x 1 + β μ q x + 3 t α x 1 + β β μ q x − t α x 1 + 2 β v q x + t α x 2 + β μ p x x − 6 t α x 2 λ q x x + 6 t α x 2 β λ q x x − t α x 2 + β μ q x x − t α x 3 λ q x x x ) = 0 , δ ℒ δ v ≡ t − α x − β ( t α ( − 1 + β ) p [ x , t ] + q ( x β ( − 1 + α ) + t α ( − 1 + β ) u ) − t x β q t − t α x p x − t α x u q x ) = 0 } .

Now, with the help of the obtained point symmetries equation (28), we use the Noether operator N [29] to obtain conserved vectors, (T ¹,T ²).

For the symmetry X 1 = ∂ ∂ x , we get
(31) T 1 t = t − α x − 3 α ( x α p ( t x 2 α u t − 2 t α x ( − 1 + α ) μ u x ) + q ( t x 3 α v t + t α x ( − 1 + α ) ( 2 x α μ u x − 3 x λ u x x ) ) + t α x 2 ( x α μ p x u x − x λ u x q x x + q x ( ( 3 ( − 1 + α ) λ − x α μ ) u x + x λ u x x ) ) ) , T 1 x = − t 1 − α ( p ( x , t ) u x + q ( x , t ) v x )
For the symmetry X 2 = ∂ ∂ t , we have
(32) T 2 t = − t 1 − α x 1 − 3 α ( x α p ( ( ( − 1 + α ) μ + x α u ( x , t ) ) u t + x α v t + x μ u x t ) + x ( − x λ q x u x t + u t ( − x α μ p x + ( 3 λ − 3 β λ + x α μ ) q x + x λ q x x ) ) + q ( ( ( − 1 + α ) ( − λ + 2 α λ − x α μ ) + x 2 α v ) u t + x 2 α u v t + x ( − x α μ u x t + x λ u x x t ) ) ) ,
For the symmetry X 3 = t α α ∂ ∂ x + ∂ ∂ x , we acquire

(33) T 3 t = 1 α x − 3 α ( x α p ( x 1 + α α u ( x , t ) + t x 2 α u t + x ( − 1 + α ) μ ( α − 2 t α u x ) ) + x 2 ( − x α μ p x ( α − t α u x ) + x λ ( α − t α u x ) q x x + q x ( 3 α λ − 3 α 2 λ + x α α μ + t α ( 3 ( − 1 + α ) λ − x α μ ) u x + t α x λ u x x ) ) + q ( x 1 + 2 α α v + t x 3 α v t − x ( − 1 + α ) ( α ( λ − 2 α λ + x α μ ) − 2 t α x α μ u x + 3 t α x λ u x x ) ) ) , T 3 x = t ( p ( t − α α − u x ) − q v x ) α

6 Conclusion

Finding a suitable operator for a better description of the dynamical complex systems is a big issue nowadays for researchers from various fields of science and engineering. In this paper, we investigated Cls for some nonlinear PDEs with conformable derivative. The new conservation theorem is applied to construct nonlocal Cls for nonlinear PDEs possessing conformable derivative. It is worth noting that, the construction of Cls to nonlinear PDEs with conformable derivative is presented for the first time, to our knowledge, in this paper.

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Received: 2020-03-24

Revised: 2020-04-06

Accepted: 2020-04-07

Published Online: 2020-06-02

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/phys-2020-0127

Keywords for this article

systems of partial differential equations; Lie symmetry; conformable derivative; conservation laws

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BY 4.0