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

Maysaa Mohamed Al Qurashi 1
  • 1 Department of Mathematics, King Saud University, P.O.Box 22452, Riyadh, 11495, Saudi Arabia
Maysaa Mohamed Al Qurashi
  • Corresponding author
  • Department of Mathematics, King Saud University, P.O.Box 22452, Riyadh, 11495, Saudi Arabia
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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.

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,), then CD of f with order α is given by [11]

Tα[f(t)]=limε0f(t+εt1αf(t))ε,
for t > 0, α ∈ (0, 1). If Tα[ f(t)] exists for t in some interval (0, a) with a > 0, and limt0+Tα[f(t)] also exists, then Tα[f(0)]=limt0+Tα[f(t)]. Moreover, if Tα[ 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]:

  • Tα(af + bg) = aTα( f ) + bTα( f ), a,b,
  • Tα(tμ) = μtμα, μ,
  • Tα( fg) = f Tα + gTα(f),
  • Tα(fg)=gTα(f)fTαg2,
  • If f is differentiable, then Tα(f)(t)=t1αdfdg.

The integral associated with the conformable derivative can be written as [11] Iα[f(t)]=I[tα1f(t)]=0tf(τ)τ1αdτ, where the integral depicts the standard Riemann improper integral such that α ∈ (0, 1].

Besides, we can prove easily that [11]

TαIα[ f(t)] = f(t), for t ≥ 0, where f depicts any function that is continuous within the domain of Iα. For every t > 0, we get IαTα[ f(t)] = f(t) − f(0). Also, we have Tα( fog)(t) = ( f ′(g(t)))(t)Tαg(t) [11].

3 Symmetry analysis

Consider the conformable system of PDEs [27]:

αutα=F(x,t,u,ux2α,ux3α,),αvtα=F(x,t,v,vx2α,vx3α,),
where u = u(x, t), F(x, t, u, uxx, uxxx,…) depicts a nonlinear function and αutα and αvtα depict the conformable fractional derivative. Given a one-parameter Lie group of infinitesimal transformations of the form
t¯=t+εξ1(t,x,u,v)+O(ε2),x¯=x+εξ2(t,x,u,v)+O(ε2),u¯=u+εη1(t,x,u,v)+O(ε2),v¯=v+εη2(t,x,u,v)+O(ε2),αu¯t¯=αutα+εη1α,t(t,x,u,v)+O(ε2),αv¯t¯=αvtα+εη2α,t(t,x,u,v)+O(ε2),αu¯x¯α=ux+εη1α,x(t,x,u,v)+O(ε2),αv¯x¯α=vx+εη2α,x(t,x,u,v)+O(ε2),2αu¯x¯2α=2αux2α+εη1α,xx(t,x,u,v)+O(ε2),2αv¯x¯2α=2αux2α+εη2α,xx(t,x,u,v)+O(ε2),3αu¯x¯3α=3αux3α+εη1α,xxx(t,x,u,v)+O(ε2),3αv¯x¯3α=3αvx3α+εη2α,xxx(t,x,u,v)+O(ε2),...
where
η1α,t=t1αη1t+(1α)ξ1tαut,η2α,t=t1αη2t+(1α)ξ1tαvt,η1α,x=x1αη1x+(1α)ξ2xαux,η2α,x=x1αη2x+(1α)ξ2xαvx,η1α,xx=x22αη1xx+(1α)x12αη1x+(22α)x12αξ2uxx+(1α)(12α)x2αξ2ux,η2α,xx=x22αη2xx+(1α)x12αη2x+(22α)x12αξ2vxx+(1α)(12α)x2αξ2vx,η1α,xxx=x33αη1xxx+(33α)x23αη1xx+(1α)(12α)x13αη1x+(33α)ξ2x23αuxxx+(33α)(23α)ξ2x13αuxx+(1α)(12α)(13α)ξ2x3αux,η2α,xxx=x33αη2xxx+(33α)x23αη2xx+(1α)(12α)x13αη2x+(33α)ξ2x23αvxxx+(33α)(23α)ξ2x13αvxx+(1α)(12α)(13α)ξ2x3αvx,
and
η1t=Dt(η1)uxDt(ξ2)utDt(ξ1),η2t=Dt(η2)vxDt(ξ2)vtDt(ξ1),η1x=Dx(η1)uxDx(ξ2)utDx(ξ1),η2x=Dx(η2)vxDx(ξ2)vtDx(ξ1),η1xx=Dt(η1x)uxxDt(ξ2)uxtDt(ξ1),η2xx=Dt(η2x)vxxDt(ξ2)vxtDt(ξ1),η1xxx=Dt(η1xx)uxxxDt(ξ2)uxxtDt(ξ1),η2xxx=Dt(η2xx)vxxxDt(ξ2)vxxtDt(ξ1),

The total differential operator Dx and Dt are defined by

Dt=t+utu+uttut+uxtux++vtv+vttvt+vxtvxDx=x+uxu+uxxux+uxtut++vxv+vxxvx+vxtvt

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

X=ξ1x+ξ2t+η1u+η2v.

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

Pα,irX(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
ξ1(x,t,u,v)|t=0=0.

4 The dispersive long-wave system

The conformable dispersive long-wave system is given by

αutα=(αuxα)22αux2α+2αvxα,αvtα=2uαvxα+2vαuxα+2αvx2α,
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
ut=(u2ux+2v)x,vt=(2uv+vx)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

η1α,t2uη1α,x+2η2α,x+η1α,xx=0,η1α,t2uη1α,x+2vη2α,x+η2α,xx=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).
t1αut(x1αux)2+(1α)x12αux+x22αuxx2x1αvx=0,t1αvt2x1αuvx2x1αvux(1α)x12αvxx22αvxx=0.

Plugging equations (4) and (5) into equation (12) and using equation (13) instead of uxx and vxx 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

ξ1=2tαc1α+c2,ξ2=t1αc3,η1=c1,η2=0,
and the vector field is obtained as follows:
X1=x,X2=t,X3=2tααx+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

Di(Ti)|=0,
needs to be satisfied by the obtained conserved vectors. Where
Ti=ξi+Wα[uiαDj(uijα)+DjDkuijkα]+Dj(Wα)[uijαDj(uijkα)+]+DjDk(Wα)[uijkα]+
where Wα is the characteristics function given by Wα=ηξiui.

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

=p(t1αut(x1αux)2+(1α)x12αux+x22αuxx2x1αvx)+q(t1αvt2x1αuvx2x1αvux(1α)x12αvxx22αvxx)

The adjoint equations can be presented as follows:

{δδutαx2α(2tαxα(1+α)qvtx2αpt+2tαx2ptut+p((1+α)(x2α+tα(1+2α))+2tαx2utt)+3tαxpx3tαxαpx+2tαx1+αvqx+tαx2pxx)=0,δδvtαx2α(2tαxα(1+α)p(1+α)q(x2α+tα(1+2α)+2tαxαu)tx2αqt+2tαx1+αpx3tαxqx+3tαxαqx+2tαx1+αuqxtαx2qxx)=0}.

Now, with the help of the obtained point symmetries equation (15), we use the Noether operator N [29,30] to obtain conserved vectors, (T1, T2) as follows:

  1. For the symmetry X1=x, we obtain
    T1t=tαx2α(p(tx2αuttαx2ut22tαx(1+α)ux)+q(tx2αvt+2tαx(1+α)vx)+tαx2(pxuxqxvx)),T1x=p(t1α+2x22αut)uxt1αq(x,t)vx
  2. For the symmetry X2=t, we obtain
    T2t=t1αx12α(x(utpxvtqx)+p((1+α)ut+2xαvtxuxt)+q(2xαv(x,t)ut+(1+α+2xαu(x,t))vt+xvxt)),T2x=t1αx12α(p(xut2(1+α)ux2xαvx+xuxx)q(2xαvux+(1α+2xαu)vx+xvxx))
  3. For the symmetry X3=2tααx+u, we obtain
T3t=1α(x2α(p(2tx2αut+2tαx2ut2+x(1+α)(α+4tαux))2q(x1+ααv+tx2αvt+2tαx(1+α)vx)x2(px(α+2tαux)2tαqxvx))),T3x=p(t1α2x22αut)(1+2tαuxα)+2tqvxα

5 The Whitham–Broer–Kaup system

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

αutα+uαuxα+αvxα+μ2αux2α=0,αvtα+uαvxα+vαuxα+β3αux3αμ2αvt2α=0

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

ut+uux+vx+μuxx=0,vt+(uv)x+βuxxxμvxx=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

η1α,t+uη1x+η2x+μη1xx=0,η2α,t+vη1x+uη2xμη2xx+βη1xxx=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
t1αut+x1αuux+x1αvx+μ(1α)x12αux+ux22αuxx=0,t1αvt+x1αuvxx1αvuxμ(1α)x12αuxμx22αuxx+λ(1α)(12α)x13αux+3λ(1α)x23αuxx+λx33αuxxx=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

ξ1=tαc1α+c2,ξ2=t1αc3,η1=c1,η2=0,
and the associated Lie algebra is generated by the vector field as follows:
X1=x,X2=t,X3=tααx+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

=p(t1αut+x1αuux+x1αvx+μ(1α)x12αux+ux22αuxx)+q(t1αvt+x1αuvxx1αvuxμ(1α)x12αuxμx22αuxx+λ(1α)(12α)x13αux+3λ(1α)x23αuxx+λx33αuxxx).

The adjoint equations can be presented as follows:

{δδutαx3β(xβp(x2β(1+α)+tα(13β+2β2)μ+tαxβ(1+β)u)+tα(1+β)q((1+2β)((1+3β)λxβμ)+x2βv)tx3βpt+3tαx1+βμpx3tαx1+ββμpxtαx1+2βupx7tαxλqx+18tαxβλqx11tαxβ2λqx3tαx1+βμqx+3tαx1+ββμqxtαx1+2βvqx+tαx2+βμpxx6tαx2λqxx+6tαx2βλqxxtαx2+βμqxxtαx3λqxxx)=0,δδvtαxβ(tα(1+β)p[x,t]+q(xβ(1+α)+tα(1+β)u)txβqttαxpxtαxuqx)=0}.

Now, with the help of the obtained point symmetries equation (28), we use the Noether operator N [29] to obtain conserved vectors, (T1,T2).

  1. For the symmetry X1=x, we get
    T1t=tαx3α(xαp(tx2αut2tαx(1+α)μux)+q(tx3αvt+tαx(1+α)(2xαμux3xλuxx))+tαx2(xαμpxuxxλuxqxx+qx((3(1+α)λxαμ)ux+xλuxx))),T1x=t1α(p(x,t)ux+q(x,t)vx)
  2. For the symmetry X2=t, we have
    T2t=t1αx13α(xαp(((1+α)μ+xαu(x,t))ut+xαvt+xμuxt)+x(xλqxuxt+ut(xαμpx+(3λ3βλ+xαμ)qx+xλqxx))+q(((1+α)(λ+2αλxαμ)+x2αv)ut+x2αuvt+x(xαμuxt+xλuxxt))),
  3. For the symmetry X3=tααx+x, we acquire
T3t=1αx3α(xαp(x1+ααu(x,t)+tx2αut+x(1+α)μ(α2tαux))+x2(xαμpx(αtαux)+xλ(αtαux)qxx+qx(3αλ3α2λ+xααμ+tα(3(1+α)λxαμ)ux+tαxλuxx))+q(x1+2ααv+tx3αvtx(1+α)(α(λ2αλ+xαμ)2tαxαμux+3tαxλuxx))),T3x=t(p(tααux)qvx)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.

References

  • [1]

    Caputo M, Mainardi F. A new dissipation model based on memory mechanism. Pure Appl Geophys. 1971;91:134–47.

    • Crossref
    • Export Citation
  • [2]

    Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Prog Fract Differ Appl. 2015;1(2):73–85.

  • [3]

    Atangana A, Baleanu D. New fractional derivatives with nonlocal and non-singular kernel. Theory application heat Transf model, Therm Sci. 2016;20(2):763–9.

  • [4]

    Baleanu D, Avkar T. Lagrangians with linear velocities within Riemann-Liouville fractional derivatives. Nuovo Cim B. 2004;119(1):73–9.

  • [5]

    Yildiz TA, Jajarmi A, Yildiz B, Baleanu D. New aspects of time fractional optimal control problems within operators with nonsingular kernel. Discr Cont Dynam Syst S. 2020;13(3):407–28.

  • [6]

    Baleanu D, Jajarmi A, Mohammadi H, Rezapour S. A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative. Chaos Soliton Fract. 2020;134:109705.

    • Crossref
    • Export Citation
  • [7]

    Arshad S, Defterli O, Baleanu D. A second order accurate approximation for fractional derivatives with singular and non-singular kernel applied to a HIV model. Appl Math Comput. 2020;374:125061.

  • [8]

    Samko SG, Kilbas AA, Marichev OL. Fractional integrals and derivatives: theory and applications. New York: Gordon and Breach; 1993.

  • [9]

    Jajarmi A, Baleanu D, Sajjadi SS, Asad JH. A new feature of the fractional Euler-Lagrange equations for a coupled oscillator using a nonsingular operator approach. Front Phys 2019;7:196.

    • Crossref
    • Export Citation
  • [10]

    Baleanu D, Jajarmi A, Sajjadi SS, Mozyrska D. A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator. Chaos. 2019;29(8):083127.

    • Crossref
    • PubMed
    • Export Citation
  • [11]

    Khalil R, Al Horani M, Yousef A, Sababheh M. A new definition of fractional derivative. J Comput Appl Math. 2014;264:65–70.

    • Crossref
    • Export Citation
  • [12]

    Atangana A, Baleanu D, Alsaedi A. New properties of conformable derivative. Open Math. 2015;13:889–98.

  • [13]

    Zheng A, Feng Y, Wang W. The Hyers-Ulam stability of the conformable fractional differential equation. Math Aeterna. 2015;5:485–92.

  • [14]

    Iyiola OS, Nwaeze ER. Some new results on the new conformable fractional calculus with application sing D’Alambert approach. Progr Frac Differ Appl. 2016;2:115–22.

    • Crossref
    • Export Citation
  • [15]

    Michal P, Skripkov LP. Sturm's theorems for conformable fractional differential equations. Math Commun. 2016;21:273–281.

  • [16]

    Fuat U, Mehmet ZS. Explicit bounds on certain integral inequalities via conformable fractional calculus. Cogent Math. 2017;4:1277505.

  • [17]

    Bluman GW, Cheviakov AF, Anco SC. Applications of symmetry methods to partial differential equations. New York: Springer; 2010.

  • [18]

    Leveque RJ. Numerical methods for conservation laws. Lectures in mathematics. ETH Zurich: Birkhauser Verlag; 1992.

  • [19]

    Naz R. Conservation laws for some systems of nonlinear partial differential equations via multiplier approach. J Appl Math. 2012;2012:1–13.

  • [20]

    Noether E. Invariante variations probleme. Nachr Kn Gesell Wissen, Gttingen, Math-Phys KI, Heft. 1918;2:235–57, English translation in Transp Theor. Stat. Phys. 1971;1:186–207.

  • [21]

    Agrawal OP. Formulation of Euler-Lagrange equations for fractional variational problems. J Math Appl. 2002;272:368–79.

  • [22]

    Frederico GST, Delfim T. A formulation of Noether's theorem for fractional problems of the calculus of variations. J Math Anal. 2007;334:834–46.

    • Crossref
    • Export Citation
  • [23]

    Baleanu D. About fractional quantization and fractional variational principles. Commun Nonlin Sci Numer Simul. 2009;14:2520–3.

    • Crossref
    • Export Citation
  • [24]

    Abdeljawad T. On conformable fractional calculus. J Comput Appl Math. 2015;279:57–66.

    • Crossref
    • Export Citation
  • [25]

    Chung WS. Fractional Newton mechanics with conformable fractional derivative. J Comput Appl Math. 2015;290:150–8.

    • Crossref
    • Export Citation
  • [26]

    Cenesiz Y, Kurt A. The new solution of time fractional wave equation with conformable fractional derivative definition. J N Theor. 2015;7:79–85.

  • [27]

    Tayyan BA, Sakka AH. Lie symmetry analysis of some conformable fractional partial differential equations. Arab J Math. 2020;9:201–12.

    • Crossref
    • Export Citation
  • [28]

    Kaup DJ. A higher-order water-wave equation and the method for solving it. Prog Theor Phys 1975;54:396–408.

    • Crossref
    • Export Citation
  • [29]

    Ibragimov NH. A new conservation theorem. J Math Anal Appl. 2007;333:311–28.

    • Crossref
    • Export Citation
  • [30]

    Yomba E. The Extended Fan Sub-Equation Method and its Application to the (2+1)-Dimensional Dispersive Long Wave and Whitham-Broer-Kaup Equations. Chin J Phys. 2005;43(4):789–805.

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  • [1]

    Caputo M, Mainardi F. A new dissipation model based on memory mechanism. Pure Appl Geophys. 1971;91:134–47.

    • Crossref
    • Export Citation
  • [2]

    Caputo M, Fabrizio M. A new definition of fractional derivative without singular kernel. Prog Fract Differ Appl. 2015;1(2):73–85.

  • [3]

    Atangana A, Baleanu D. New fractional derivatives with nonlocal and non-singular kernel. Theory application heat Transf model, Therm Sci. 2016;20(2):763–9.

  • [4]

    Baleanu D, Avkar T. Lagrangians with linear velocities within Riemann-Liouville fractional derivatives. Nuovo Cim B. 2004;119(1):73–9.

  • [5]

    Yildiz TA, Jajarmi A, Yildiz B, Baleanu D. New aspects of time fractional optimal control problems within operators with nonsingular kernel. Discr Cont Dynam Syst S. 2020;13(3):407–28.

  • [6]

    Baleanu D, Jajarmi A, Mohammadi H, Rezapour S. A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative. Chaos Soliton Fract. 2020;134:109705.

    • Crossref
    • Export Citation
  • [7]

    Arshad S, Defterli O, Baleanu D. A second order accurate approximation for fractional derivatives with singular and non-singular kernel applied to a HIV model. Appl Math Comput. 2020;374:125061.

  • [8]

    Samko SG, Kilbas AA, Marichev OL. Fractional integrals and derivatives: theory and applications. New York: Gordon and Breach; 1993.

  • [9]

    Jajarmi A, Baleanu D, Sajjadi SS, Asad JH. A new feature of the fractional Euler-Lagrange equations for a coupled oscillator using a nonsingular operator approach. Front Phys 2019;7:196.

    • Crossref
    • Export Citation
  • [10]

    Baleanu D, Jajarmi A, Sajjadi SS, Mozyrska D. A new fractional model and optimal control of a tumor-immune surveillance with non-singular derivative operator. Chaos. 2019;29(8):083127.

    • Crossref
    • PubMed
    • Export Citation
  • [11]

    Khalil R, Al Horani M, Yousef A, Sababheh M. A new definition of fractional derivative. J Comput Appl Math. 2014;264:65–70.

    • Crossref
    • Export Citation
  • [12]

    Atangana A, Baleanu D, Alsaedi A. New properties of conformable derivative. Open Math. 2015;13:889–98.

  • [13]

    Zheng A, Feng Y, Wang W. The Hyers-Ulam stability of the conformable fractional differential equation. Math Aeterna. 2015;5:485–92.

  • [14]

    Iyiola OS, Nwaeze ER. Some new results on the new conformable fractional calculus with application sing D’Alambert approach. Progr Frac Differ Appl. 2016;2:115–22.

    • Crossref
    • Export Citation
  • [15]

    Michal P, Skripkov LP. Sturm's theorems for conformable fractional differential equations. Math Commun. 2016;21:273–281.

  • [16]

    Fuat U, Mehmet ZS. Explicit bounds on certain integral inequalities via conformable fractional calculus. Cogent Math. 2017;4:1277505.

  • [17]

    Bluman GW, Cheviakov AF, Anco SC. Applications of symmetry methods to partial differential equations. New York: Springer; 2010.

  • [18]

    Leveque RJ. Numerical methods for conservation laws. Lectures in mathematics. ETH Zurich: Birkhauser Verlag; 1992.

  • [19]

    Naz R. Conservation laws for some systems of nonlinear partial differential equations via multiplier approach. J Appl Math. 2012;2012:1–13.

  • [20]

    Noether E. Invariante variations probleme. Nachr Kn Gesell Wissen, Gttingen, Math-Phys KI, Heft. 1918;2:235–57, English translation in Transp Theor. Stat. Phys. 1971;1:186–207.

  • [21]

    Agrawal OP. Formulation of Euler-Lagrange equations for fractional variational problems. J Math Appl. 2002;272:368–79.

  • [22]

    Frederico GST, Delfim T. A formulation of Noether's theorem for fractional problems of the calculus of variations. J Math Anal. 2007;334:834–46.

    • Crossref
    • Export Citation
  • [23]

    Baleanu D. About fractional quantization and fractional variational principles. Commun Nonlin Sci Numer Simul. 2009;14:2520–3.

    • Crossref
    • Export Citation
  • [24]

    Abdeljawad T. On conformable fractional calculus. J Comput Appl Math. 2015;279:57–66.

    • Crossref
    • Export Citation
  • [25]

    Chung WS. Fractional Newton mechanics with conformable fractional derivative. J Comput Appl Math. 2015;290:150–8.

    • Crossref
    • Export Citation
  • [26]

    Cenesiz Y, Kurt A. The new solution of time fractional wave equation with conformable fractional derivative definition. J N Theor. 2015;7:79–85.

  • [27]

    Tayyan BA, Sakka AH. Lie symmetry analysis of some conformable fractional partial differential equations. Arab J Math. 2020;9:201–12.

    • Crossref
    • Export Citation
  • [28]

    Kaup DJ. A higher-order water-wave equation and the method for solving it. Prog Theor Phys 1975;54:396–408.

    • Crossref
    • Export Citation
  • [29]

    Ibragimov NH. A new conservation theorem. J Math Anal Appl. 2007;333:311–28.

    • Crossref
    • Export Citation
  • [30]

    Yomba E. The Extended Fan Sub-Equation Method and its Application to the (2+1)-Dimensional Dispersive Long Wave and Whitham-Broer-Kaup Equations. Chin J Phys. 2005;43(4):789–805.

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