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Lie symmetry analysis of conformable differential equations

  • In this paper, we construct a proper extension of the classical prolongation formula of point transformations for conformable derivative. This technique is illustrated and employed to construct a symmetry group admitted by a conformable ordinary and partial differential equations. Using Lie symmetry analysis, we obtain an exact solution of the conformable heat equation.

    Citation: Youness Chatibi, El Hassan El Kinani, Abdelaziz Ouhadan. Lie symmetry analysis of conformable differential equations[J]. AIMS Mathematics, 2019, 4(4): 1133-1144. doi: 10.3934/math.2019.4.1133

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  • In this paper, we construct a proper extension of the classical prolongation formula of point transformations for conformable derivative. This technique is illustrated and employed to construct a symmetry group admitted by a conformable ordinary and partial differential equations. Using Lie symmetry analysis, we obtain an exact solution of the conformable heat equation.


    In recent years, fractional differential equations have received considerable attention owing to their applicability in different fields of sciences such as chemistry, biology, diffusion, control theory, rheology, viscoelasticity and so on [1,2,3,4]. Many authors has been shown that the fractional derivatives used in this theory lost some of the basic properties that usual derivatives have such as the product rule and the chain rule. Recently, a conformable derivative is introduced and it satisfies the basic properties of usual derivatives [5,6,7].

    To construct exact solutions for fractional differential equations is not an easy task. Therefore, several methods such as homotopy perturbation method [8], sub-ODE method [9,10], generalized tanh method [11], residual power series method [9,12] and so on [13,14,15,16,17,18] are developed to obtain solutions of some nonlinear fractional differential equations.

    Lie symmetry analysis or Lie group method has become of great interest in many aspects of the exact sciences [19,20,21,22], and has been extensively applied to construct exact solutions of ordinary and partial differential equations. However, this method has not been used much to investigate invariance properties of fractional partial differential equations. Gazizov et al. [23] have established the prolongation formula in the case of fractional differential equations in which the fractional derivative is considered in the sense of Caputo and Riemann-Liouville. Their fundamental work has become a key to investigate symmetry properties of some fractional differential equations and a crucial tool for other works, see for example [9,12,20,21,22,23,24,25,26,27].

    In this work, we develop Lie symmetry method to establish the prolongation formula admitted by a conformable differential equations. The obtained prolongation formula is given in case of the dependent variables that are supposed to be differentiable functions. This method is used to obtain some solutions of conformable heat equation.

    This paper is organized as follows: In Section 2, we recall some basic properties of conformable derivatives. Section 3 is devoted to determine the prolongation formula in the case of a conformable ordinary differential equation. While in Section 4, we establish the prolongation formula of a conformable partial differential equation, the method is illustrated by conformable heat equation. We concluded the paper in the last section by some comments.

    Definition 2.1. We consider f:RR is a function, and α(0,1].

    The left conformable derivative of order α is defined by

    aTαtf(t)=limh0f(t+h(ta)1α)f(t)h. (2.1)

    The right conformable derivative of order α is

    tTαbf(t)=limh0f(th(bt)1α)f(t)h. (2.2)

    If a=0 and f is α-differentiable in some (0,x), i.e., limt0+f(α)(t) exists, so we define

    f(α)(0)=limt0+f(α)(t).

    We can write f(α)(t) for 0Tαtf(t) to denote the conformable derivative of f of order α. In addition, if the conformable derivative of f of order α exists, then we simply say f is α-differentiable. Some useful formulas and properties of conformable derivative are summarized in [5,6] and here we present some of its basic properties.

    Proposition 1. Let be f and g two α-differentiable functions at any point t>a such that g0, and λ,μR, we have

    aTαt(λf+μg)=λaTαtf+μaTαtg, (2.3)
    aTαt(fg)=faTαtg+gaTαtf, (2.4)
    aTαt(fg)=gaTαtffaTαtgg2. (2.5)

    Note that the above properties (2.4) and (2.5) are not preserved with other known fractional derivatives [4]. Furthermore, if α,β in (0,1), and f is a 2-times differentiable, we have non commutative and non stability properties of conformable derivatives.

    aTαt(aTβtf)aTβt(aTαtf),andaTαt(aTβtf)aTα+βtf. (2.6)

    Proposition 2. If f and g two functions such that g is differentiable in all t, and f is differentiable in all g(t). we have

    aTαt(fg)(t)=f(g(t))aTαtg(t). (2.7)

    In this section, we consider the one parameter group of point transformations with y=y(t) is a differentiable function,

    ˉt=φ(t,y,ϵ)=t+ϵξ(t,y)+o(ϵ2), (3.1)
    ˉy=ϕ(t,y,ϵ)=y+ϵη(t,y)+o(ϵ2). (3.2)

    Where

    ξ(t,y)=φϵ|ϵ=0andη(t,y)=ϕϵ|ϵ=0. (3.3)

    Along with (3.1) and (3.2), we consider the corresponding infinitesimal operator

    X=ξ(t,y)t+η(t,y)y. (3.4)

    With the condition of invariance of the equation t=0, we obtain

    ξ(t,y(t))|t=0=0. (3.5)

    Theorem 3.1. The infinitesimal transformation of the conformable derivative definition, we have

    0Tαˉtˉy(ˉt)=0Tαty(t)+ϵη(α)+o(ϵ2), (3.6)

    where

    η(α)=t1αηt+(ηy+1αtξξt)(0Tαty(t))tα1ξy(0Tαty(t))2. (3.7)

    Proof. According to the conformable derivative definition, we have

    0Tαˉtˉy(ˉt)=limh0ˉy(ˉt+hˉt1α)ˉy(ˉt)h (3.8)
    =ˉt1αlimh0ˉy(ˉt+hˉt1α)ˉy(ˉt)hˉt1α (3.9)
    =ˉt1αdˉydˉt. (3.10)

    In addition, from Eqs. (3.1) and (3.2), we get

    dˉydˉt=y+ϵ(ηt+ηyy)+o(ϵ2)1+ϵ(ξt+ξyy)+o(ϵ2), (3.11)

    and,

    ˉt1α=(t+ϵξ(t,y)+o(ϵ2))1α (3.12)
    =t1α(1+ϵtξ)1α+o(ϵ2) (3.13)
    =t1α(1+(1α)ϵtξ)+o(ϵ2). (3.14)

    Furthermore,

    11+ϵ(ξt+ξyy)+o(ϵ2)=1ϵ(ξt+ξyy)+o(ϵ2). (3.15)

    Substituting expressions (3.11), (3.14) and (3.15) in Eq. (3.10), we get

    0Tαˉtˉy(ˉt)=(t1α(1+(1α)ϵtξ)+o(ϵ2))(y+ϵ(ηt+ηyy)+o(ϵ2))×(1ϵ(ξt+ξyy)+o(ϵ2))=(t1αy+ϵt1α(ηt+(ηy+1αtξ)y)+o(ϵ2))×(1ϵ(ξt+ξyy)+o(ϵ2)) (3.16)
    =t1αy+ϵ[t1α(ηt+(ηy+1αtξ)y)t1αy(ξt+ξyy)]+o(ϵ2). (3.17)

    Consequently,

    0Tαˉtˉy(ˉt)=0Tαty(t)+ϵη(α)+o(ϵ2), (3.18)

    where

    η(α)=t1αηt+(ηy+1αtξξt)(0Tαty(t))tα1ξy(0Tαty(t))2. (3.19)

    Example 1. Consider the conformable Riccati equation

    y(α)(t)+y2(t)=0, (3.20)

    where y(α)(t) denotes the conformable derivative of y. In this example, we will compute the Lie symmetries of the above equation. Its corresponding infinitesimal generator is given by

    X=ξt+ηy, (3.21)

    and the α-th order prolongation of X is

    X(α)=ξt+ηy+η(α)y(α), (3.22)

    where η(α) is given by the formula (3.7). If we put Δ=y(α)(t)+y2(t), so the invariance criterion is given

    X(α)[Δ]|Δ=0=0. (3.23)

    Then,

    2ηy+η(α)|Δ=0=0. (3.24)

    Substituting for η(α) its expression, the above criterion leads to

    t1αηt+2ηy(ηy+1αtξξt)y2tα1ξyy4=0. (3.25)

    As ξ(t,y)=ξ(t) and η(t,y)=0, we obtain the equation

    ξ(t)1αtξ(t)=0. (3.26)

    Hence,

    ξ(t)=Ct1α,C=cte. (3.27)

    Consequently, the conformable Riccati equation (3.20) is invariant under the one-parameter group of transformations generated by the infinitesimal operator

    X=t1αt. (3.28)

    Let us assume that a conformable partial differential equation of independent variables (x,t) and dependent variable u is invariant under a one-parameter (ϵ) continuous point transformations

    ˉt=t+ϵξ(t,x,y)+o(ϵ2), (4.1)
    ˉx=x+ϵτ(t,x,y)+o(ϵ2), (4.2)
    ˉy=y+ϵη(t,x,y)+o(ϵ2). (4.3)

    Where

    dˉtdϵ|ϵ=0=ξ(t,x,y),dˉxdϵ|ϵ=0=τ(t,x,y),dˉydϵ|a=0=η(t,x,y). (4.4)

    Then, we have the following result.

    Theorem 4.1. The infinitesimal transformation of the conformable derivative is given by

    0Tαˉtˉy=0Tαty+ϵη(α)t+o(ϵ2), (4.5)

    where

    η(α)t=0Tαtη(tα1(0Tαtξ)1αtξ)(0Tαty)yx.(0Tαtτ), (4.6)

    Proof. Equations (4.1) and (4.2) may be inverted (locally) to give t and x in terms of ˉt and ˉx, provided that the Jacobian is nonzero, that is,

    J=|Dt[ˉt]Dt[ˉx]Dx[ˉt]Dx[ˉx]|0, (4.7)

    where y=y(t,x) and

    Dt=t+yty+yttyt+ytxyx+... (4.8)
    Dx=x+yxy+yxxyx+yxtyt+..., (4.9)

    denote the total derivatives that treat the dependent variable y and its derivatives according to the independent variables. if Eq. (4.7) is satisfied, then the equation (4.3) can be rewritten as:

    ˉy=ˉy(ˉt,ˉx). (4.10)

    Applying the chain rule to (4.10), we obtain

    [Dt[ˉy]Dx[ˉy]]=[Dt[ˉt]Dt[ˉx]Dx[ˉt]Dx[ˉx]][ˉyˉtˉyˉx], (4.11)

    and therefore, by Cramer's rule, we obtain

    ˉyˉt=1J|Dt[ˉy]Dt[ˉx]Dx[ˉy]Dx[ˉx]|,ˉyˉx=1J|Dt[ˉt]Dt[ˉy]Dx[ˉt]Dx[ˉy]|. (4.12)

    Dropping all terms with of order ϵ2, we get

    J=1+ϵ((ξt+ytξy)+(τx+yxτy))+o(ϵ2). (4.13)

    Thus,

    1J=1ϵ((ξt+ytξy)+(τx+yxτy))+o(ϵ2). (4.14)

    Then,

    0Tαˉtˉy=ˉt1αˉyˉt (4.15)
    =(t1α(1+(1α)ϵtξ)+o(ϵ2))(1ϵ((ξt+ytξy)+(τx+yxτy))+o(ϵ2))×[(yt+ϵ(ηt+ytηy)+o(ϵ2))(1+ϵ(τx+yxτy)+o(ϵ2))(yx+ϵ(ηx+yxηy)+o(ϵ2))(ϵ(τt+ytτy)+o(ϵ2))] (4.16)
    =(t1α(1+(1α)ϵtξ)+o(ϵ2))[yt+ϵ(ηt+(ηyξtytξy)yt(τt+ytτy)yx)+o(ϵ2)]=0Tαty+ϵ[t1αηt+t1α(ηy+1αtξξt)ytt1α(τt+ytτy)yxt1αξyy2t]+o(ϵ2). (4.17)

    Hence,

    0Tαˉtˉy=0Tαty+ϵη(α)t+o(ϵ2), (4.18)

    where

    η(α)t=0Tαtη(tα1(0Tαtξ)1αtξ)(0Tαty)yx.(0Tαtτ), (4.19)

    with

    0Tαtτ=t1αDtτ=t1α(τt+ytτy)and0Tαtη=t1αDtη=t1α(ηt+ytηy).

    Example 2. We consider the time conformable heat equation defined by

    y(α)t=yxx. (4.20)

    In the particular case α=1, the above equation is invariant under the following group transformations

    ˉx=x+ϵ,ˉt=t,ˉy=y. (4.21)
    ˉx=x,ˉt=t+ϵ,ˉy=y. (4.22)
    ˉx=x,ˉt=t,ˉy=eϵy. (4.23)
    ˉx=eϵx,ˉt=e2ϵt,ˉy=y. (4.24)
    ˉx=x+2ϵt,ˉt=t,ˉy=ye(ϵx+ϵ2t). (4.25)
    ˉx=x14ϵt,ˉt=t14ϵt,ˉy=y14ϵtexp(ϵx214ϵt). (4.26)
    ˉx=x,ˉt=t,ˉy=y+ϵβ(x,t). (4.27)

    Where β(t,x) is an arbitrary solution of the heat equation. Now, we show that transformations (4.21) to (4.27) do not all preserve the conformable heat equation (4.20) where α0:

    In the case of transformation (4.22), we have

    ˉy(α)ˉt=ˉt1αˉyˉt=(t+ϵ)1αyˉt=(t+ϵ)1αyt=(1+ϵt)1αy(α)t=(1+ϵt)1αyxx, (4.28)

    and ˉyˉxˉx=yxx, consequently,

    ˉy(α)ˉtˉyˉxˉx. (4.29)

    In a similar way, we find also that transformations (4.24), (4.25) and (4.26) do not preserve the conformable heat equation. However, the transformations (4.21) satisfies

    ˉyˉxˉx=ˉxˉyˉx=ˉxyx=yxx=y(α)t=ˉy(α)ˉt, (4.30)

    then, it preserves equation (4.20). Similarly, the transformations (4.23) and (4.27) will also leave the equation (4.20) invariant.

    Now, Let us assume that the conformable heat equation (4.20) is invariant under a one-parameter (ϵ) continuous point transformations

    ˉt=t+ϵξ(t,x,y)+o(ϵ2), (4.31)
    ˉx=x+ϵτ(t,x,y)+o(ϵ2), (4.32)
    ˉy=y+ϵη(t,x,y)+o(ϵ2), (4.33)
    ˉy(α)ˉt=y(α)t+ϵη(α)t+o(ϵ2), (4.34)
    ˉyˉx=yx+ϵη(1)x+o(ϵ2), (4.35)
    ˉyˉxˉx=yxx+ϵη(2)x+o(ϵ2). (4.36)

    Where η(1)x and η(2)x are given by

    η(1)x=ηx+(ηyτx)yxξxytτyy2xξyyxyt, (4.37)
    η(2)x=ηxx+(2ηxyτxx)yxξxxyt+(ηyy2τxy)y2x2ξxyyxytτyyy3xξyyy2xyt+(ηy2τx)yxx2ξxyxt3τyyxxyxξyyxxyt2ξyyxtyx. (4.38)

    Suppose

    X=ξ(t,x,y)t+τ(t,x,y)x+η(t,x,y)y, (4.39)

    be a symmetry operator for the Eq. (4.20). According to the invariance criterion, Eq. (4.20) admits the group transformations (4.31)–(4.33) if the prolonged X(α) annihilates (4.20) on its solution, namely

    X(α)(Δy)|(Δy=0)=0, (4.40)

    where Δy=y(α)tyxx, and

    X=X+η(1)xyx+η(2)xyxx+η(α)ty(α)t. (4.41)

    Then, the invariance criterion is written as

    t1αηt+(ηy+1αtξξt)yxxtα1ξy(yxx)2t1α(τt+tα1yxxτy)yxη(2)x=0. (4.42)

    Solving the above equation, we derive the determining system

    t1ατt+τxx2ηxy=0, (4.43)
    2τxyηyy=0, (4.44)
    τyy=0, (4.45)
    ξt+1αtξ+2τx+tα1ξxx=0, (4.46)
    tα1ξy+ξytα1=0, (4.47)
    2ξxytα1+2τy=0, (4.48)
    ξyytα1=0, (4.49)
    2ξx=0, (4.50)
    2ξy=0, (4.51)
    t1αηtηxx=0. (4.52)

    Solution of the determining system, gives

    ξ(t,x,y)=c6t1α+tα(4c1αtα+c3), (4.53)
    τ(t,x,y)=(4c1αtα+c3)x+(2c2αtα+c4), (4.54)
    η(t,x,y)=(c1x2+c2x+(2c1αtα+c5))y+c7γ(t,x). (4.55)

    where c1,...,c6 are arbitrary constants and γ(t,x) is an arbitrary solution of the conformable heat equation. Thus, the Lie symmetry algebra admitted by equation (4.20) is spanned by the operators

    X1=x,X2=t1αt,X3=yy, (4.56)
    X4=2tαt+xx,X5=2tααxxyy, (4.57)
    X6=4t1+αα2t+4xtααx(x2+2tαα)yy. (4.58)

    and the infinite dimensional subalgebra

    X=γ(t,x)y, (4.59)

    where γ is an arbitrary solution of the conformable heat equation. To obtain the group transformation generated by each infinitesimal symmetry Xk, we solve the system of first order ordinary differential equations,

    dˉtda=ξ(ˉt,ˉx,ˉy),dˉxda=τ(ˉt,ˉx,ˉy),dˉyda=η(ˉt,ˉx,ˉy),

    subject to the initial conditions

    ˉt(ϵ=0)=t,ˉx(ϵ=0)=x,ˉy(ϵ=0)=y.

    The one-parameter groups Gi generated by Xi are given as follows

    G1:(t,x,y)(t,x+ϵ,y), (4.60)
    G2:(t,x,y)((tα+ϵα)1/α,x,y), (4.61)
    G3:(t,x,y)(t,x,eϵy), (4.62)
    G4:(t,x,y)(e2ϵ/αt,eϵx,y), (4.63)
    G5:(t,x,y)(t,x+2ϵ(tα/α),ye(ϵx+ϵ2(tα/α))), (4.64)
    G6:(t,x,y)((tα14ϵ(tα/α))1/α,x14ϵ(tα/α),y14ϵ(tα/α)exp(ϵx214ϵ(tα/α))), (4.65)
    G:(t,x,y)(t,x,y+ϵγ(t,x)). (4.66)

    Consequently, since Gi is a symmetry, if y=f(t,x) is a solution of the conformable heat equation, so are

    y1=f(t,xϵ), (4.67)
    y2=f((tαϵα)1/α,x), (4.68)
    y3=eϵf(t,x), (4.69)
    y4=f(e2ϵ/αt,eϵx), (4.70)
    y5=f(t,x2ϵ(tα/α))e(ϵxϵ2(tα/α)), (4.71)
    y6=f((tα1+4ϵ(tα/α))1/α,x1+4ϵ(tα/α))1+4ϵ(tα/α)exp(ϵx21+4ϵ(tα/α)), (4.72)
    y=f(t,x)+ϵγ(t,x). (4.73)

    If we let y=1 be a constant solution of the conformable heat equation, we conclude that the function

    G6(ϵ).1=11+4ϵ(tα/α)exp(ϵx21+4ϵ(tα/α)), (4.74)

    is also solution. Furthermore, we can derive other exact solutions of conformable heat equation if we continue this iteration process.

    In this work, we have shown that the Lie symmetry analysis could be extended to conformable differential equations as it has been done for fractional differential equations based on the Riemann-Liouville or the Caputo approaches. This extension might be more investigated to employ various developed properties of invariance to derive exact solutions of other linear and nonlinear conformable differential equations. In addition, this work will bring new opportunities in studying some non autonomous system of partial differential equations with p-dependent variables and q-independent variables. Finally, it will be important to develop the Lie symmetry analysis for differential equations involving different fractional derivatives such as Caputo-Fabrizio derivative and Atangana-Baleanu derivative.

    The authors would like to thank anonymous reviewers for their valuable comments and suggestions that improved this paper.

    All authors declare no conflicts of interest in this paper.



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