On Lie Symmetry Analysis of Certain Coupled Fractional Ordinary Differential Equations

K. Sethukumarasamy; P. Vijayaraju; P. Prakash

doi:10.2991/jnmp.k.210315.001

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Volume 28, Issue 2, June 2021, Pages 219 - 241

On Lie Symmetry Analysis of Certain Coupled Fractional Ordinary Differential Equations

Authors

K. Sethukumarasamy¹^{, *}, P. Vijayaraju¹, P. Prakash²^,

¹Department of Mathematics, Anna University, Chennai – 600 025, India

²Department of Mathematics, Amrita School of Engineering, Amrita Vishwa Vidyapeetham, Coimbatore – 641 112, India

^*Corresponding author. Email: mathsethu@gmail.com

Corresponding Author

K. Sethukumarasamy

Received 11 November 2020, Accepted 13 March 2021, Available Online 31 March 2021.

DOI: 10.2991/jnmp.k.210315.001 How to use a DOI?
Keywords: System of fractional ODEs; Lie group formalism; Laplace transform technique; Mittag-Leffler function; Exact solution
Abstract: In this article, we explain how to extend the Lie symmetry analysis method for n-coupled system of fractional ordinary differential equations in the sense of Riemann-Liouville fractional derivative. Also, we systematically investigated how to derive Lie point symmetries of scalar and coupled fractional ordinary differential equations namely (i) fractional Thomas-Fermi equation, (ii) Bagley-Torvik equation, (iii) two-coupled system of fractional quartic oscillator, (iv) fractional type coupled equation of motion and (v) fractional Lotka-Volterra ABC system.The dimensions of the symmetry algebras for the Bagley-Torvik equation and its various cases are greater than 2 and for this reason we construct optimal system of one-dimensional subalgebras. In addition, the exact solutions of the above mentioned fractional ordinary differential equations are explicitly derived wherever possible using the obtained symmetries.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Fractional calculus has been effectively used in recent years to study many complex nonlinear phenomena. A natural phenomenon may depend not only on the instantaneous time but also on the past history of time. Such a phenomenon can be successfully modeled by the theory of differential equations involving derivatives and integrals of arbitrary order. With the virtue of nonlocal property of fractional derivatives, the fractional differential equations (FDEs) have been used to describe such scenarios precisely in the area of science and engineering such as viscoelasticity [3], hydrodynamics [64], quantum mechanics [36] and so on [16,24–26,32,42,63].

The derivation of the exact solution of the differential equation is important because the geometrical and physical meaning of the problem can be easily obtained. In literature, no well-defined analytic methods exist for deriving the exact solutions of fractional ordinary differential equations (FODEs). However, the derivation of the exact solution of FDEs is not an easy task since the properties of a fractional derivative are harder than the classical derivative. Recently, several research groups have been developed to derive the exact and numerical solutions of FDEs such as invariant subspace method [11,21,39,51–53,57,58,71], variational iteration method [44], homotopy perturbation method [45], operational matrix method [55], collocation method [40] and so on [14,23,68,69].

Among those methods, the Lie symmetry analysis method is an algorithmic approach that provides an efficient tool to construct an exact solution of FDEs in a systematic way. Initially, this method was proposed by Norwegian mathematician Sophus Lie during the 19th century and was further developed by Ovsianikov [49] and others [5,10,27,31,34,38,46–48,56,62]. The Lie symmetry analysis method is to find continuous transformations of one or more parameters leaving the differential equation invariant in the new coordinate system wherein the resulting differential equation is easier to solve. Gazizov et al. [18] extended the Lie symmetry analysis of differential equations to FDEs. Recently, many mathematicians and physicists have investigated and developed the theory of Lie symmetry analysis for FODEs and fractional partial differential equations (FPDEs) [4,6,7,9,15,18–20,29,30,37,54,59–61,70].

We would like to mention that only a very few applications of Lie point symmetries of scalar and coupled FODEs have been investigated. To the best of our knowledge, no one has extended the Lie symmetry analysis to n-coupled FODEs. The main aim of this article is to demonstrate how the Lie symmetry approach provides an efficient tool to derive exact solutions of scalar and coupled nonlinear FODEs. More precisely, Lie symmetries of fractional Thomas-Fermi equation, Bagley-Torvik equation, two-coupled system of fractional quartic oscillator, fractional type coupled equation of motion and fractional Lotka-Volterra ABC system are derived. Using the obtained symmetries, we explicitly derived the exact solution of the above-mentioned FODEs wherever possible. In addition, the dimensions of the symmetry algebras for the Bagley-Torvik equation and its various cases are greater than 2 and for this reason, we construct the optimal system of one-dimensional subalgebras.

The article is organized as follows: For the sake of completeness, in section 2, we recall certain basic definitions pertaining to the fractional calculus. The Lie symmetry analysis of the coupled system of FODEs is presented. The applicability and effectiveness of this method have been illustrated through the above mentioned FODEs in section 3. In section 4, we summarize our results as concluding remarks of this paper.

2. PRELIMINARIES

In this section, we present some basic concepts and results of the fractional calculus. We also present brief details of symmetry analysis for scalar and coupled system of FODEs.

Definition 2.1.

[50] Let x(t) ∈ L¹[a, b] and α ∈ ℝ⁺. Then the Riemann-Liouville (R-L) fractional differential operator of order α > 0 is defined by

aDtαx(t)={1Γ(n-α)dndtn∫at(t-ξ)n-α-1x(ξ)dξ,ifα∈(n-1,n),n∈𝕅x(n)(t),ifα=n∈𝕅 (2.1)

for t > a.

Note that L¹[a, b] denotes the space of absolutely integrable functions on [a, b].

Note 1. [32,50] The R-L fractional derivative of order α for x(t) = t^β is as follows:

0Dtαtβ=Γ(β+1)Γ(β-α+1)tβ-α,β>-1,α≥0.

Note 2. The Leibniz formula for the R-L fractional derivative is of the following form:

aDtα(x1(t)x2(t))=∑k=0∞(αk)aDtα-kx1(t)Dtkx2(t),α>0, (2.2)

where (αk)=Γ(1+α)Γ(α-k+1)Γ(k+1).

Note 3. [50] Let α ∈ (n − 1, n], n ∈ ℕ. Then the Laplace transformation of R-L derivative (2.1) is given by

L{0Dtαx(t)}=sαx(s)-∑k=0n-1skDα-k-1x(t)|t=0, 𝔕(s)>0, (2.3)

where x(s)=L{x(t)}=∫0∞e-stx(t)dt.

Definition 2.2.

[43] The generalized Mittag-Leffler function with three parameters is defined as

Eρ,μγ(z)=∑k=0∞(γ)kzkΓ(ρk+μ)k!,ρ,μ,γ∈𝔺,𝔕(ρ)>0,𝔕(μ)>0, (2.4)

where (γ)k=Γ(γ+k)Γ(γ)and (γ)₀ = 1 for ℛ(γ) > 0.

Note 4. The Laplace transformation of the Mittag-Leffler functions are given by [6]

(i)
L{tμ-1Eρ,μγ(±atρ)}={sργ-μ(sρ∓a)γ,ifμ≠ργ,𝔕(s)>|a|1ρ,1(sρ∓a)γ,ifμ=ργ,𝔕(s)>|a|1ρ.
(ii)
L{tμ-1Eρ,μ(∓atρ)}=[sρ-μsρ±a;t[.
(iii)
L{tρ-δ∑r=0∞(-a)rt(ρ-μ)rEρ,ρ+(ρ-μ)r-δ+1r+1(-btρ)}=[sδ-1sρ+asμ+b;t].

2.1. Symmetry Analysis for FODE

Consider the following generalized FODE of the form

aDtα+kx=G(t,x,aDtαx,aDtα+1x,…,aDtα+k-1x),α∈𝕉+,t>0,k∈𝕅∪{0}, (2.5)

where t and x are independent and dependent variables respectively. Assume that equation (2.5) is invariant under the following one parameter (∊) Lie group of continuous point transformations [6,18,19,29,54]

t¯↦t+∊τ(t,x)+O(∊2),x¯↦x+∊ζ(t,x)+O(∊2),x¯(k)↦x(k)+∊ζ(k)+O(∊2),k∈𝕅,aDt¯αx¯↦aDtαx+∊ζ(α)+O(∊2),⋮aDt¯α+kx¯↦aDtα+kx+∊ζ(α+k)+O(∊2),k∈𝕅. (2.6)

We apply the transformations (2.6) in (2.5) and omit the higher powers of ∊. Then by comparing the coefficients of ∊ on both the sides of the resulting equation we obtain

[ζ(α+k)-τ∂G∂t-ζ∂G∂x-∑r=0kζ(α+r-1)∂G∂aDtα+r-1x-∑r=0kζ(r)∂G∂x(r)]|H=0=0,

where H=aDtα+kx-G, k ∈ ℕ ∪ {0}, which is the invariant equation for (2.5). Therefore, the infinitesimal generator becomes

X=τ(t,x)∂∂t+ζ(t,x)∂∂x,

where

τ(t,x)=dt¯d∊|∊=0,ζ(t,x)=dx¯d∊|∊=0.

Therefore, the prolongation formula for (2.5) can be written as

Pr(α+k)X(H)=[ζ(α+k)-XG-∑r=0kζ(α+r-1)∂G∂aDtα+r-1x-∑r=0kζ(r)∂G∂x(r)]|H=0=0,

where H=aDtα+kx-G, k ∈ ℕ ∪ {0} and the infinitesimals are calculated as follows [6,18,19,29,54]

ζ(k)=D(ζ(k-1))-x(k)D(τ)=Dk(ζ-τx(1))+τx(k+1), k∈𝕅, (2.7)

ζ(α+k)=aDtα+k(ζ-τx(1))+τaDtα+k-1x=aDtα+k(ζ)+∑n=0∞(α+kn)(n-(α+k)n+1)aDtα+k-nxDtn+1(τ), α>0. (2.8)

Note 5. The lower limit of the R-L fractional derivative (2.1) is fixed. Therefore, it should be invariant at t = a, under the given transformation (2.6). Thus, we obtain the invariant condition as

τ(a,x)=a. (2.9)

Definition 2.3.

A function F(x, t) is an invariant of FODE (2.5) if F(x, t) is an invariant surface, that is XF = 0 which implies the following

(τ∂F∂t+ζ∂F∂x)=0.

Note 6. [28] The Lie bracket [X₁, X₂] of operators

X1=τ1∂∂t+ζ1∂∂x and X2=τ2∂∂t+ζ2∂∂x

is given by

[X1,X2]=X1(X2)-X2(X1)=(X1(τ2)-X2(τ1))∂∂t+(X1(ζ2)-X2(ζ1))∂∂x. (2.10)

Definition 2.4.

[28] A Lie algebra is a real linear space ℒ together with a binary operation [·, ·]: ℒ × ℒ → ℒ satisfying the following properties:

(i)
[X, Y] = − [Y, X],
(ii)
[aX + bY, Z] = a [X, Z] + b [Y, Z],
(iii)
[[X, Y], Z] + [[Y, Z], X] + [[Z, X], Y] = 0,

for X, Y, Z ∈ ℒ and a, b ∈ ℝ. The binary operation [·, ·] is known as Lie bracket defined in equation (2.10).

2.2. Symmetry Analysis for Coupled System of FODEs

Let us first present the Lie symmetry analysis for the two-coupled system of FODEs in the sense of R-L fractional derivative. Then we shall extend this method for the n-coupled system of FODEs.

2.2.1. Symmetry analysis for two-coupled system of FODEs

Consider the following two-coupled system of FODE in the sense of R-L fractional derivative

aDtαx1(t)=f1(t,x1,x2),aDtαx2(t)=f2(t,x1,x2), α>0, (2.11)

where aDtαxi(t)=dαxidtα for i = 1, 2 and f₁, f₂ are arbitrary functions. Here we assume that system (2.11) is invariant under one-parameter (∊) Lie group of continuous point transformations

t¯↦t+∊τ(t,x1,x2)+O(∊2),x¯1↦bluex1+∊ζx1(t,x1,x2)+O(∊2),x¯2↦x2+∊ζx2(t,x1,x2)+O(∊2),aDt¯αx¯1↦aDtαx1+∊ζx1(α)(t,x1,x2)+O(∊2),aDt¯αx¯2↦aDtαx2+∊ζx2(α)(t,x1,x2)+O(∊2), (2.12)

where ζx1(α) and ζx2(α) are αth extended infinitesimals that are derived in the following theorem.

Theorem 2.1.

The αth (α ∈ ℝ⁺) extended infinitesimals for the two-coupled system of FODEs in the sense of R-L fractional derivative are given by

ζx1(α)=aDtα(ζ-τx1(1))+τaDtα+1(x1),ζx2(α)=aDtα(ζ-τx2(1))+τaDtα+1(x2). (2.13)

Proof. The Leibniz rule for fractional derivative is given by

aDtα(ϕ(t)ψ(t))=∑n=0∞(αn)aDtα-nϕ(t)ψ(n)(t), α>0, (2.14)

where the binomial coefficient is given by (αn)=Γ(1+α)Γ(α-n+1)Γ(n+1). By substituting ϕ(t) = 1 and ψ(t)=x¯1(t¯) in Leibniz rule given by equation (2.14) we obtain

aDtα(x¯1(t¯))=∑n=0∞(αn)(t¯-a)n-αΓ(n-α+1)x¯1(n)(t¯). (2.15)

Similarly, we obtain

aDtα(x¯2(t¯))=∑n=0∞(αn)(t¯-a)n-αΓ(n-α+1)x¯2(n)(t¯). (2.16)

By the definition of extended αth infinitesimals, we have

ζx1(α)=dd∊[aDtα(x¯1(t¯))]∊=0, (2.17)

ζx2(α)=dd∊[aDtα(x¯2(t¯))]∊=0. (2.18)

Substituting equation (2.15) in (2.17), we obtain

ζx1(α)=dd∊[∑n=0∞(αn)(t¯-a)n-αΓ(n-α+1)x¯2(n)(t¯)]∊=0. (2.19)

After the use of infinitesimal transformation given by equation (2.12) and simplification, the above equation (2.19) becomes

ζx1(α)=∑n=0∞(αn)(t-a)n-αζ(n)Γ(n-α+1)+∑n=0∞τ(n-α)(t-a)n-α-1x1(n)(t)Γ(n-α+1). (2.20)

Substituting equation (2.7) in the above equation (2.20), we obtain

ζx1(α)=∑n=0∞(αn)(t-a)n-αDn(ζ-τx1(1)(t))Γ(n-α+1)+∑n=0∞τ(t-a)n-αx1(n+1)(t)Γ(n-α+1) +∑n=0∞(αn)τ(n-α)(t-a)n-α-1x1(n)(t)Γ(n-α+1). (2.21)

In the first term of the above equation (2.21) we apply (2.15). Then replacing n by n − 1 in the second term and substituting n = 0 in the third term of the above equation (2.21) we obtain

ζx1(α)=aDtα(ζ-τx1(1))+∑n=1∞(αn-1)τ(t-a)n-α-1x1(n)(t)Γ(n-α)-τα(t-a)-α-1x1(t)Γ(1-α) +∑n=0∞(αn)τ(n-α)(t-a)n-α-1x1(n)(t)Γ(n-α+1). (2.22)

The above equation (2.22) can be further simplified by using the relation (αn-1)+(αn)=(α+1n) to get

ζx1(α)=aDtα(ζ-τx1(1))+∑n=0∞(α+1n)(t-a)n-α-1τx1(n)Γ(n-α). (2.23)

Applying (2.15) in (2.23) we obtain,

ζx1(α)=aDtα(ζ-τx1(1))+τaDtα+1(x1). (2.24)

In a similar manner, we obtain

ζx2(α)=aDtα(ζ-τx2(1))+τaDtα+1(x2). (2.25)

Remark 1.

Applying the generalized Leibniz rule (2.2) in the extended infinitesimals given by equations (2.24) and (2.25) we obtain

ζx1(α)=aDtα(ζx1)-αDt(τ)aDtα(x1)+∑n=1∞(αn)(n-αn+1)aDtα-n(x1)Dtn+1(τ),ζx2(α)=aDtα(ζx2)-αDt(τ)aDtα(x2)+∑n=1∞(αn)(n-αn+1)aDtα-n(x2)Dtn+1(τ).

2.2.2. Symmetry analysis for n-coupled system of FODEs

In this subsection, we consider the n-coupled system of FODEs in the sense of R-L fractional derivative, having the form

aDtαx1=f1(t,x1,x2,…,xn),aDtαx2=f2(t,x1,x2,…,xn),⋮aDtαxn=fn(t,x1,x2,…,xn), (2.26)

where aDtαxi=dαxidtα, for i = 1, 2, …, n, α > 0 and f₁, f₂, …, f_n are arbitrary functions. Assuming that system (2.26) is invariant under one-parameter (∊) Lie group of continuous point transformations, we obtain

t¯↦t+∊τ(t,x1,x2,…,xn)+O(∊2),x¯1↦x1+∊ζx1(t,x1,x1,…,xn)+O(∊2),x¯2↦x2+∊ζx2(t,x1,x2,…,xn)+O(∊2),⋮x¯n↦xn+∊ζxn(t,x1,x2,…,xn)+O(∊2),aDt¯αx¯1↦aDtαx1+∊ζx1(α)(t,x1,x2,…,xn)+O(∊2),aDt¯αx¯2↦aDtαx2+∊ζx2(α)(t,x1,x2,…,xn)+O(∊2),⋮aDt¯αx¯n↦aDtαxn+∊ζxn(α)(t,x1,x2,…,xn)+O(∊2), (2.27)

where ζxi(α), i = 1, 2,..., n are infinitesimals that can be obtained in the following form

ζxi(α)=aDtα(ζxi)-αDt(τ)aDtα(xi)+∑n=1∞(αn)(n-αn+1)aDtα-n(xi)Dtn+1(τ), (2.28)

i = 1, 2,..., n. Making use of transformation (2.27) in (2.26) and equating the coefficients ∊ and omitting the terms of higher powers of ∊, we obtain the invariant equations for (2.26) as follows

(ζxi(α)-τ∂fi∂t-ζx1∂fi∂x1-⋯-ζxn∂fi∂xn)|dαx1dtα=f1,dαx2dtα=f2,⋯,dαxndtα=fn=0, i=1,2,…,n.

Therefore, the infinitesimal generator X reads

X=τ(t,x1,x2,…,xn)∂∂t+ζx1(t,x1,x2,…,xn)∂∂x1+ζx2(t,x1,x2,…,xn)∂∂x2+⋯+ζxn(t,x1,x2,…,xn)∂∂xn, (2.29)

where

τ(t,x1,x2,…,xn)=dt¯d∊|∊=0,ζxi(t,x1,x2,…,xn)=dx¯id∊|∊=0, i=1,2,…,n.

Therefore, the obtained prolongation formula for (2.26) is as follows

Pr(α)X(H)|H=0=0, where H=aDtαxi-fi, i=1,2,…,n,

which can be written as

ζxi(α)-Xfi=0, i=1,2,…,n,

where the infinitesimals ζxi(α) are given in (2.28).

3. SYMMETRIES AND EXACT SOLUTIONS FOR SCALAR AND COUPLED FODEs

3.1. Fractional Thomas-Fermi Equation

Consider the fractional Thomas-Fermi equation [12,17] of the form

0Dtα+1x=t-α2x32, α∈(0,1),t>0. (3.1)

Note that, when α = 1, the above equation is the well-known classical Thomas-Fermi equation that describes the charge distribution of a neutral atom which is a function of radius t. The analytical and numerical solutions of the classical Thomas-Fermi equation were discussed in [1,72].

3.1.1. Lie point symmetries

We assume that the Thomas-Fermi equation (3.1) is invariant under one-parameter Lie group of infinitesimal point transformations given in (2.6). Hence, we obtain the following invariant equation

ζ(α+1)-32x12t-α2ζ(t,x)+α2x32t-α2-1τ(t,x)=0. (3.2)

Substituting the expression for the infinitesimal ζ^(α+1) given in (2.8) into the above equation (3.2), we obtain

0Dtα+1(ζ)+∑n=0∞(α+1n)(n-α-1n+1)0Dtα+1-n(x)Dtn+1(τ) -32x12t-α2ζ(t,x)+α2x32t-α2-1τ(t,x)=0. (3.3)

The above equation (3.3) is not solvable for the arbitrary infinitesimals τ(t, x) and ζ(t, x) in general. In order to find the infinitesimals in equation (3.3), we assume

τ=τ(t), ζ=a1(t)x+a2(t),

where a₁(t) and a₂(t) are the unknown functions to be determined. Using the above expressions in (3.3), we obtain

a1(t)0Dtα+1x+0Dtα+1a2(t)-(α+1)t-α2x32Dt(τ)-32x12t-α2(a1(t)x+a2(t)) +α2x32t-α2-1τ+∑n=1∞(α+1n)[dna1(t)dtn+(n-α-1n+1)Dtn+1(τ)]0Dtα+1-n(x)=0. (3.4)

Making use of fractional Thomas-Fermi equation (3.1) in (3.4), we obtain

∑n=1∞(α+1n)[dna1(t)dtn+(n-α-1n+1)Dtn+1(τ)]0Dtα+1-n(x) +[-12t-α2a1(t)-(α+1)t-α2Dt(τ)+α2t-α2-1τ]x32-32t-α2x12a2(t)+0Dtα+1a2(t)=0 (3.5)

which reduces to the following overdetermined system of equations

dna1(t)dtn+(n-α-1n+1)Dtn+1(τ)=0, (3.6)

-12a1(t)t-α2-(α+1)t-α2Dt(τ)+α2t-α2-1τ=0, (3.7)

a2(t)=0. (3.8)

Solving the above equations (3.6)–(3.8) subject to τ(0) = 0, we obtain the explicit form of infinitesimals as

τ(t)=c1t, ζ(t,x)=-c1(α+2)x, c1∈𝕉.

Thus, the fractional Thomas-Fermi equation (3.1) is invariant under the following one-parameter Lie group transformation

t¯=t+∊c1t,x¯=x-∊c1(α+2)x

with the infinitesimal operator X = c₁X₁, where X1=t∂∂t-(α+2)∂∂x.

3.1.2. Construction of exact solution

In this subsection, we explain how to construct an exact solution for equation (3.1). The obtained infinitesimals of (3.1) are

τ(t)=c1t, ζ(t,x)=-c1(α+2)x.

Let us assume that F(x, t) is an invariant of fractional Thomas-Fermi equation (3.1), that is

X1F=t∂F∂t-(α+2)x∂F∂x=0,

which gives the characteristic equation as

dtt=dx-(α+2)x.

On solving the above equation, we obtain the invariant function

F(x,t)=k=xt(α+2), k∈𝕉.

Hence this yields an invariant solution as

x(t)=kt-(α+2). (3.9)

Substituting the above invariant solution in equation (3.1), we have k=(Γ(-(α+1))Γ(-2(α+1)))2. Hence, we obtain an exact solution of Thomas-Fermi equation (3.1) as

x(t)=(Γ(-(α+1))Γ(-2(α+1)))2t-(α+2), α∈(0,1). (3.10)

The gamma coefficient (Γ(-(α+1))Γ(-2(α+1)))in the above equation (3.30) becomes zero when α=12 and hence the equation (3.1) has a trivial solution x(t) = 0. The exact solution (3.30) of fractional Thomas-Fermi equation (3.1) for different values of α is shown in Figure 1.

3.2. Bagley-Torvik Equation

Let us consider the Bagley-Torvik equation [3]

Ax¨(t)+B 0Dt32x(t)+Cx(t)=f, t>0, (3.11)

where A ≠ 0, B, C are arbitrary constants and f is a function of t. It describes the motion of a rigid plate immersed in a Newtonian fluid [3]. The analytical and approximate solutions of the Bagley-Torvik equation were discussed through various methods [14,50,66].

3.2.1. Lie point symmetries

Here, we assume that the Bagley-Torvik equation (3.11) is invariant under the one-parameter Lie group of continuous point transformations (2.6). Thus, the obtained invariant equation is of the form

Aζ(2)+Bζ(32)+Cζ=τft. (3.12)

Substituting α=32 in (2.8), we obtain the expression for ζ(32) which in turn can be used in the above equation (3.12) to obtain

A[ζtt+(2ζtx-τtt)x˙+(ζxx-2τtx)x˙2-τxxx˙3+(ζx-2τt-3x˙τx)x¨]+B[0Dt(32)(ζ)+∑n=0∞(32n)(n-32n+1)Dtn+1τ 0Dt32-n(x)]-τft+Cζ=0. (3.13)

The above equation (3.13) is unsolvable for the arbitrary infinitesimals τ(t, x) and ζ(t, x). In order to find the infinitesimals in equation (3.13), we assume

τ=τ(t),ζ=a1(t)x+a2(t),

where a₁(t) and a₂(t) are the unknown arbitrary functions to be determined. Substituting the above expressions along with

0Dt32(a1(t)x)=∑n=0∞(32n)dna1(t)dtn0Dt32-nx

and substituting equation (3.11) in (3.13) and then rearranging term by term, we have

-12Aτtx¨(t)+(2Ada1dt-Aτtt)x˙(t)+(Ad2a1dt2+cτt)x(t) +B∑n=1∞(32n)[dna1dtn+(n-32n+1)Dtn+1(τ)]0D32-nx+Ca2(t)-τft +Ad2a2dt2+(a1(t)-32τt)f(t)+B 0Dt32a2(t)=0

which reduces to the following overdetermined system

Ca2(t)-τft+Ad2a2dt2+(a1(t)-32τt)f(t)+B 0Dt32a2(t)=0, (3.14)

dndtna1(t)+(n-32n+1)Dtn+1τ=0, (3.15)

Ad2a1dt2+32Cτt=0, (3.16)

2Ada1dt-Aτtt=0, (3.17)

Aτt=0. (3.18)

On solving the last three equations with A ≠ 0 and using τ (0) = 0, we obtain

τ(t)=0, a1(t)=k1, and ζ(t,x)=k1x+a2(t), k1,k2∈𝕉. (3.19)

Substituting (3.19) in (3.14), we obtain

Aa¨2(t)+B0Dt32a2(t)+k1f(t)+Ca2(t)=0. (3.20)

Making use of Laplace transformation on both sides of the equation (3.20) and regrouping the terms, we obtain the following equation

a¯2(s)=(AB)a2(0)ss32+(AB)s2+(CB)+(AB)a˙2(0)s32+(AB)s2+(CB)+D12a2(0)s32+(AB)s2+(CB)+sD12a2(0)s32+(AB)s2+(CB)-(1B)k1f¯(s)s32+(AB)s2+(CB) (3.21)

whose inverse Laplace transformation along convolution is

a2(t)=k2(AB)t-12∑r=0∞(-AB)rt-12rE32,12(1-r)r+1(-CBt32)+k3(AB)t12∑r=0∞(-AB)rt-12rE32,12(3-r)r+1(-CBt32)+k4t12∑r=0∞(-AB)rt-12rE32,12(3-r)r+1(-CBt32)+k5t-12∑r=0∞(-AB)rt-12rE32,12(1-r)r+1(-CBt32)-k1(1B)∫0tf(t-ξ)ξ12∑r=0∞(-AB)ξ-12rE32,12(1-r)r+1(-CBξ32)dξ, (3.22)

where the constants k₂ = a₂(0), k3=a˙2(0), k4=D12a2(0) and k5=D-12a2(0). Thus, the Bagley-Torvik equation (3.11) is invariant under

t¯=t,x¯=x+∊(k1x+a2(t)),

where a₂(t) is given in (3.22). The infinitesimal operator takes the following form

X=∑i=15kiXi, (3.23)

where

X1=[x-(1B)∫0tf(t-ξ)ξ12∑r=0∞(-AB)ξ-12rE32,12(1-r)r+1(-CBξ32)dξ]∂∂x,X2=[(AB)t-12∑r=0∞(-AB)rt-12rE32,12(1-r)r+1(-CBt32)]∂∂x,X3=[(AB)t12∑r=0∞(-AB)rt-12rE32,12(3-r)r+1(-CBt32)]∂∂x,X4=[t12∑r=0∞(-AB)rt-12rE32,12(3-r)r+1(-CBt32)]∂∂x,X5=[t-12∑r=0∞(-AB)rt-12rE32,12(1-r)r+1(-CBt32)]∂∂x.

Note that when A ≠ 0, B ≠ 0, C ≠ 0 and f (t) ≠ 0, the infinitesimal generator cannot be written as a linear combination of the space coordinate and the time coordinate. For this reason, the invariant and exact solutions of (3.11) cannot be obtained.

3.2.2. One-dimensional optimal system

In this subsection, we construct a one-dimensional optimal system for the symmetry algebra generated by X₁, X₂, X₃, X₄ and X₅. In order to obtain the optimal system, we follow the algorithm provided by Olver [48] and Ibragimov [28]. The commutator table for the symmetry generators X_i, i = 1, 2,...,5 is given in Table 1.

*[X_i, X_j]*	X₁	X₂	X₃	X₄	X₅
X₁	0	−X₂	−X₃	−X₄	−X₅
X₂	X₂	0	0	0	0
X₃	X₃	0	0	0	0
X₄	X₄	0	0	0	0
X₅	X₅	0	0	0	0

Table 1

Commutator table for infinitesimal generators of equation (3.11)

From the table we observe that the symmetry generators of Bagley-Torvik equation (3.11) form a closed Lie algebra ℒ of dimension five. Now we determine optimal system of one-dimensional subalgebras for (3.11). The adjoint representation of the infinitesimal generators is obtained using the following series

Ad(exp(∊Xi))Xj=Xj-∊[Xi,Xj]+12!∊2[Xi,[Xi,Xj]]-⋯ (3.24)

Hence we obtain the adjoint representation of form given in Table 2.

Ad(exp(∊X_i))X_j	X₁	X₂	X₃	X₄	X₅
X₁	X₁	e^∊X₂	e^∊X₃	e^∊X₄	e^∊X₅
X₂	X₁ − ∊X₂	X₂	X₃	X₄	X₅
X₃	X₁ − ∊X₃	X₂	X₃	X₄	X₅
X₄	X₁ − ∊X₄	X₂	X₃	X₄	X₅
X₅	X₁ − ∊X₅	X₂	X₃	X₄	X₅

Table 2

Adjoint representation for infinitesimal generators of equation (3.11)

Theorem 3.1.

The one-dimensional optimal system of Lie algebra ℒ for the Bagley-Torvik equation (3.11) is X₁.

Proof. Define the maps 𝒡i∊:𝒧→𝒧as ℱi∊(X)=Ad(exp(∊Xi))X, for X ∈ ℒ and i = 1, 2,..., 5. Let X ∈ ℒ be arbitrary. Then X=∑i=15aiXi. The composition of maps on X takes the form

𝒡1∊∘𝒡2∊∘𝒡3∊∘𝒡4∊∘𝒡5∊(X)=a1X1+∑i=25(ai-a1∊)e∊Xi (3.25)

If a₁ ≠ 0, then by taking ∊=aia1 for each i = 1, 2,..., 5, the arbitrary vector X is the scaling of X₁. This completes the proof.

3.2.3. Special cases

The special cases for Bagley-Torvik equation (3.11) are discussed below.

Case 1: Let B = 0 and f(t) = 0, t > 0. Then the equation (3.11) reads

Ax¨(t)+Cx(t)=0 (3.26)

which can be written as

x¨(t)+kx(t)=0, (3.27)

where k=CA. For k > 1, the above equation (3.27) models the motion of a simple harmonic oscillator [67]. Let (3.27) be invariant under the Lie group of transformation (2.6). Then the invariant equation is

ζtt+(2ζtx-τtt+3kτxx(t))x˙+(ζxx-2τtx)x˙2-τxxx˙3-kζxx(t)+2kτtx(t)+kζ=0. (3.28)

Proceeding in the above similar manner, the above equation (3.26) is invariant under

t¯=t+∊τ(t,x),x¯=x+∊ζ(t,x),

where

τ(t,x)=[k3cos(kt)+k4sin(kt)]x+k5sin(2kt)2k-k6cos(2kt)2k+k7,ζ(t,x)=[-k3ksin(kt)+k4kcos(kt)]x2+12[k5cos(2kt)+k6sin(2kt)]x+k8x+k1cos(kt)+k2sin(kt),

k_i, i = 1, 2, 3,..., 8 are constants of integration and k=CA, A ≠ 0. Hence the infinitesimal operator reads

X=∑i=18kiXi,

where

12X7-kX6

The commutator table for the infinitesimal generators X_i, for i = 1, 2,..., 8, is given in Table 3. By taking k = 1, the linear combinations X₂ − X₃, X₂ + X₃, X₁ + X₄, X₁ − X₄, X₇, X₈, 2X₅ and −2X₆ coincide with the operators obtained for one-dimensional harmonic oscillator equation in [67]. Also, as discussed in [67] the combinations X₂ − X₃, X₁ + X₄ and X₇ form a compact subalgebra. The Lie symmetries of the second order ODEs are discussed in [2,8,27,49,67].

*[X_i, X_j]*	X₁	X₂	X₃	X₄	X₅	X₆	X₇	X₈
X₁	0	0	12X7-kX6	k(X5+32X8)	12X1	12X2	kX2	X₁
X₂	0	0	kX5-32X8	12X7+kX6	-12X2	12X1	-kX1	X₂
X₃	kX6-12X7	32X8-kX5	0	0	12X3	12X4	kX4	−X₃
X₄	-k(X5+32X8)	-(12X7+kX6)	0	0	-12X4	12X3	-kX3	−X₄
X₅	-12X1	12X2	-12X3	12X4	0	12kX7	2kX6	0
X₆	-12X2	-12X1	-12X4	-12X3	-12kX7	0	-2kX5	0
X₇	-kX2	kX1	-kX4	kX3	-2kX6	2kX5	0	0
X₈	−X₁	−X₂	X₃	X₄	0	0	0	0

Table 3

Commutator table for infinitesimal generators of equation (3.27)

Construction of exact solution

Consider the infinitesimal generator X = X₇ + X₁ + X₂. Let us assume that F(x, t) is an invariant of equation (3.27), that is

XF=∂F∂t+[cos(kt)+sin(kt)]∂F∂x=0,

which gives the characteristic equation as

dt1=dxcos(kt)+sin(kt).

On solving the above equation, we obtain the invariant function

F(x,t)=c1=x-1k(sin(kt)-cos(kt)), c1∈𝕉.

Hence this yields an invariant solution as

x(t)=1k(sin(kt)-cos(kt))+c1. (3.29)

Substituting the above invariant solution (3.29) in equation (3.27), we have c₁ = 0. Hence, we obtain an exact solution of equation (3.27) as

x(t)=1k(sin(kt)-cos(kt)). (3.30)

Case 2: For C = 0 and f(t) = 0, t > 0, the equation (3.11) takes the form

Ax¨(t)+B0Dt32x(t)=0. (3.31)

The above equation (3.31) is invariant under

t¯=t,x¯=x+∊(k1x+a2(t)),

where

a2(t)=k2E12,1(-BAt12)+k3tE12,2(-BAt12)+(BA)k4tE12,2(-BAt12)+(BA)k5tE12,1(-BAt12),

k₂ = a₂(0), k3=a˙2(0), k4=D12a2(0) and k5=D-12a2(0). Hence the infinitesimal operator reads

X=∑i=15kiXi,

where

X1=x∂∂x,X2=E12,1(-BAt12)∂∂x,X3=tE12,2(-BAt12)∂∂x,X4=(BA)tE12,2(-BAt12)∂∂xandX5=(BA)E12,1(-BAt12)∂∂x.

The Lie brackets for the above case are [X₁, X_j] = −X_j for j = 2, 3, 4, 5, [X_i, X_j] = 0 for i, j = 2, 3, 4, 5..., and [X₁, X₁] = 0. In this case the commutator table and adjoint representation are same as that of Tables 1 and 2 respectively.

Note 7. Since the adjoint representation of the infinitesimal operators are same as that of Bagley-Torvik equation (3.11), the one-dimensional optimal system of Lie algebra is X₁.

Case 3: Let C = 0, f(t)=34[At-12+Bπ]. Then (3.11) becomes

Ax¨(t)+B0Dt32x(t)=34[At-12+Bπ] (3.32)

which is invariant under

t¯=t,x¯=x+∊(k1x+a2(t)),

where

a2(t)=k2E12,1(-BAt12)+k3tE12,2(-BAt12)+(BA)k4tE12,2(-BAt12)+(BA)k5tE12,1(-BAt12)+(34)k1πt32E12,52(-BAt12)+(3B4A)k1πt2E12,3(-BAt12),

where k₂ = a₂(0), k3=a˙2(0), k4=D12a2(0) and k5=D-12a2(0). Hence the infinitesimal operator reads

X=∑i=15kiXi,

where

X1=[x+(34)πt32E12,52(-BAt12)+(3B4A)πt2E12,3(-BAt12)]∂∂x,X2=E12,1(-BAt-12)∂∂x,X3=tE12,2(-BAt-12)∂∂x,X4=(BA)tE12,2(-BAt-12)∂∂xandX5=(BA)E12,1(-BAt-12)∂∂x.

The commutator table and the adjoint representation of the equation (3.32) take the form same as that of Tables 1 and 2 respectively and hence the one-dimensional optimal system of Lie algebra is X₁.

Note 8. We would like to point out that in the above case 2 and case 3, the infinitesimal generator cannot be written as a linear combination of the space coordinate and the time coordinate. For this reason, the invariant and exact solutions of (3.31) and (3.32) cannot be obtained.

3.3. Two-Coupled System of Fractional Quartic Oscillator

Consider the two-coupled system of fractional quartic oscillator as follows

0Dtα+1x1=4k3x13+2k5x1x22,0Dtα+1x2=4k4x23+2k5x12x2, α∈(0,1), t>0. (3.33)

Note that, when α = 1, k₃ = 1, k₄ = 8 and k₅ = 6, the above system (3.33) is known as classical quartic oscillator. The integrability of the classical system was discussed in [35]. It is noteworthy to mention that when α = 1, k₃ = −B, k₄ = −1, k₅ = −A, the above system (3.33) becomes the equations of motion for quartic potential and its detailed discussion can be found in [33].

3.3.1. Lie point symmetries

If the system given by equation (3.33) is invariant under one-parameter Lie group of transformation as given in (2.27), then the invariant equations are as follows

ζx1(α+1)=12k3x12ζx1+2k5x22ζx1+4k5x1x2ζx2, (3.34)

ζx2(α+1)=12k4x22ζx2+2k5x12ζx2+4k5x1x2ζx1. (3.35)

The above invariant system (3.34)–(3.35) is not solvable for arbitrary infinitesimals τ(t, x) and ζ(t, x). Thus, we assume the infinitesimals of the form ζx1=a(t)x1+b(t)x2+c(t) and ζx2=p(t)x1+q(t)x2+r(t). Hence, the equation (3.34) can be written as

ζx1(α+1)=aDtα+1(ζx1)-(α+1)Dt(τ)aDtα+1(x1)+∑n=1∞(α+1n)(n-α-1n+1)aDtα+1-n(x1)Dtn+1(τ). (3.36)

Substituting the expression for the infinitesimal ζx1(α+1) in the above equation, we obtain

0Dtα+1(a(t)x1+b(t)x2+c(t))-(α+1)Dt(τ)0Dtα+1(a(t)x1+b(t)x2+c(t))+∑n=1∞(α+1n)(n-α-1n+1)0Dtα+1-n(x1)Dtn+1(τ)=2k1(a(t)x1+b(t)x2+c(t))+12k3x12(a(t)x1+b(t)x2+c(t))+2k5x22(a(t)x1+b(t)x2+c(t))+4k5x1x2(p(t)x1+q(t)x2+r(t)). (3.37)

We recall that

0Dtα+1a(t)x1=∑n=0∞(α+1n)dndtna(t)0Dtα+1-nx1, (3.38)

0Dtα+1b(t)x2=∑n=0∞(α+1n)dndtnb(t)0Dtα+1-nx2. (3.39)

Using equations (3.38) and (3.39) in equation (3.37), we obtain

a(t)0Dtα+1x1+∑n=1∞(α+1n)dndtna(t)0Dtα+1-nx1+b(t)0Dtα+1x2+∑n=1∞(α+1n)dndtnb(t)0Dtα+1-nx2+0Dtα+1c(t)-2k1(α+1)Dt(τ)x1-4k3(α+1)Dt(τ)x13-2k5(α+1)Dt(τ)x1x22+∑n=1∞(α+1n)(n-α-1n+1)0Dtα+1-n(x1)Dtn+1(τ)=2k1a(t)x1+2k1b(t)x2+2k1c(t)+12k3a(t)x13+12k3b(t)x12x2+12k3c(t)x12+2k5a(t)x22x1+2k5b(t)x22+2k5c(t)x22+4k5p(t)x12x2+4k5q(t)x1x22+4k5r(t)x1x2. (3.40)

Making use of fractional quartic oscillator equation (3.33) in the above equation (3.40), we obtain

-[8k3a(t)+4k3(α+1)Dt(τ)]x13+[2k2b(t)-2k1b(t)]x2 +[4k4b(t)-2k5b(t)]x23+[2k5b(t)-12k3b(t)-4k5p(t)]x12x2 -[2k1(α+1)Dt(τ)]x1-[2k5(α+1)Dt(τ)+4k5q(t)]x1x22 -12k3c(t)x12-2k5c(t)x22-4k5r(t)x1x2 +∑n=1∞(α+1n)[dndtna(t)+(n-α-1n+1)Dtn+1(τ)]0Dtα+1-nx1 +Dtα+1c(t)-2k1c(t)+∑n=1∞(α+1n)dndtnb(t)0Dtα+1-nx2=0. (3.41)

In a similar manner, equation (3.35) can be reduced to

[2k1p(t)-2k2p(t)]x1+[4k3p(t)-2k5p(t)]x13+[2k5p(t)-12k4p(t)-4k5b(t)]x1x22 +[4k4q(t)-4k4(α+1)Dt(τ)-12k4q(t)]x23-2k2(α+1)Dt(τ)x2 -[2k5(α+1)Dt(τ)+4k5a(t)]x12x1-12k4r(t)x22-2k5r(t)x12+0Dtα+1r(t) -2k2r(t)-4k5c(t)x1x2+∑n=1∞(α+1n)dndtnp(t)0Dtα+1-nx1 +∑n=1∞(α+1n)[dndtnq(t)+(n-α-1n+1)Dtn+1(τ)]0Dtα+1-nx2=0. (3.42)

Equations (3.41) and (3.42) reduces to the following overdetermined system of equations

dndtna(t)+(n-α-1n+1)Dtn+1(τ)=0,n∈𝕅, (3.43)

dndtnq(t)+(n-α-1n+1)Dtn+1(τ)=0,n∈𝕅, (3.44)

k5b(t)-6k3b(t)-2k5p(t)=0, (3.45)

k5p(t)-6k4p(t)-2k5b(t)=0, (3.46)

k5(α+1)Dt(τ)+2k5q(t)=0, (3.47)

k5(α+1)Dt(τ)+2k5a(t)=0, (3.48)

2k3a(t)+k3(α+1)Dt(τ)=0, (3.49)

2k4q(t)+k4(α+1)Dt(τ)=0, (3.50)

b(t)=c(t)=0, (3.51)

p(t)=r(t)=0. (3.52)

Equation (3.43) subjected to the condition τ(0) = 0 gives

τ(t)=c1t, a(t)=-(α+1)c12,

where c₁ is an arbitrary constant. Hence we obtain infinitesimal

ζx1=-(α+1)c1x12.

Similarly, from equation (3.44) we obtain

ζx2=-(α+1)c1x22.

Thus, the two-coupled system of fractional quartic oscillator (3.33) is invariant under

t¯=t+∊c1t,x¯1=x1-∊(α+1)c1x12,x¯2=x2-∊(α+1)c1x22

with infinitesimal generator as

X=c1t∂∂t-(α+1)c12x1∂∂x1-(α+1)c12x2∂∂x2. (3.53)

3.3.2. Construction of exact solution

Let F(t, x) be an invariant of equation (3.33). Then

XF=t∂F∂t-(α+1)2x1∂F∂x1-(α+1)2x2∂F∂x2.

The characteristic equation of the above equation becomes

dtt=dx1-(α+1)2x1=dx2-(α+1)2x2.

First we consider

dtt=dx1-(α+1)2x1,

which gives the invariant solution x1(t)=kt-(α+1)2, where k is the constant of integration and is given by

k=±[14k3(4k3k4-2k3k54k3k4-k52)Γ(12(1-α))Γ(-12(1+3α))]12.

Hence, we obtain the exact solution of the coupled system of fractional quartic oscillator equations (3.33) as

x1(t)=±[14k3(4k3k4-2k3k54k3k4-k52)Γ(12(1-α))Γ(-12(1+3α))]12t-(α+1)2

and by proceeding in the similar manner, we obtain

x2(t)=±[(2k3-k58k4k3-2k52)Γ(12(1-α))Γ(-12(1+3α))]12t-(α+1)2, k3,k4,k5∈𝕉, α∈(0,1).

For α=13, the system (3.33) has a trivial solution x_i = 0 for i = 1, 2 because of singularity of Gamma function at nonpositive integers. The graphical representations of solutions of fractional system of two-coupled quartic oscillator equation (3.33) for various values of α with k₃ = 1, k₄ = 8 and k₅ = −5 are shown in Figure 2.

3.4. Fractional Type Coupled Equation of Motion

Consider the following fractional type coupled equation of motion having the form

0Dtα+1x1=-2Bx1x2-3Cx12,0Dtα+1x2=-3x22-Bx12, α∈(0,1), (3.54)

where B, C are constants. Note that, when α = 1, the system (3.54) is referred to as equations of motion for a cubic potential and its integrability is discussed in [33].

3.4.1. Lie point symmetries

Assume that the fractional type coupled equation of motion (3.54) is invariant under one-parameter Lie group of transformation (2.27). Then, we obtain the invariant system as

ζx1(α+1)=-2Bx1ζx2-2Bx2ζx1-6Cx1ζx1,ζx2(α+1)=-6x2ζx2-2Bx1ζx1. (3.55)

The above system (3.55) is not solvable for τ and ζxi, i = 1, 2, 3 in general. Then by using the following assumptions for ζx1, ζx2 and ζx3,

ζx1=a1(t)x1+a2(t)x2+a3(t),ζx2=b1(t)x1+b2(t)x2+b3(t)

and also by using the series representation given in (2.28), we obtain the following overdetermined system

dndtna1(t)+(n-αn+1)Dtn+1(τ)=0, n∈𝕅,dndtnb2(t)+(n-αn+1)Dtn+1(τ)=0, n∈𝕅,a1(t)+(α+1)Dt(τ)=0,b2(t)+(α+1)Dt(τ)=0,-b2(t)+(α+1)Dt(τ)+2a1(t)=0,a2(t)=a3(t)=0,b1(t)=b3(t)=0. (3.56)

Solving the above system (3.56) with τ(0) = 0, we obtain the infinitesimals as

τ=ct, ζx1=-(α+1)cx1, ζx2=-(α+1)cx2.

Thus, the fractional type coupled equation of motion (3.54) is invariant under one-parameter Lie group of transformations

t¯=t+∊ct,x¯1=x1-∊(α+1)cx1,x¯2=x2-∊(α+1)cx2

with infinitesimal generator

X=ct∂∂t-(α+1)cx1∂∂x1-(α+1)cx2∂∂x2,

which can be written as X = cX₁, where

X1=t∂∂t-(α+1)(x1∂∂x1+x2∂∂x2).

3.4.2. Construction of exact solution

The invariant function F(x, t) of (3.54) reads

X1F=t∂F∂t-(α+1)(x1∂F∂x1+x2∂F∂x2)=0,

which gives as follows

dtt=dx1-(α+1)x1=dx2-(α+1)x2.

Thus, we obtain an exact solution of the fractional type coupled equation of motion (3.54) as follows

x1(t)=[9C±81C2-(16B3-24B2)Γ(-α)Γ(-(2α+1))4B2]t-(α+1),x2(t)=-12B[27C2±3C81C2-(16B3-24B2)Γ(-α)Γ(-(2α+1))4B2+Γ(-α)Γ(-(2α+1))]t-(α+1), (3.57)

where α ∈ (0, 1) and 81C²Γ(−(2α + 1)) ≥ (16B³ − 24B²)Γ(−α) must hold for the existence of real solutions to the system (3.54). In the Figure 3, the geometrical representations of solutions of fractional type coupled equation of motion for various values of α are given, in which the constants are assumed to be B = 2 and C = 8.

3.5. Fractional Lotka-Volterra ABC System

Consider the following fractional form of Lotka-Volterra ABC system [65]

0Dtαx1=x1(Cx2+x3),0Dtαx2=x2(x1+Ax3),0Dtαx3=x3(Bx1+x2), α∈(0,1), (3.58)

where A, B, C are constants, x₁, x₂ and x₃ are functions of t. The approximate analytic solution of fractional two-dimensional Lotka-Volterra system was investigated in [13]. Note that, when α = 1, the system (3.58) is referred to as the classical Lotka-Volterra system or the Lotka-Volterra ABC system. The integrability of (3.58) with α = 1 was discussed in [22,35,41].

3.5.1. Lie point symmetries

Assuming that the system (3.58) is invariant under Lie group transformations given in equation (2.27), we get

ζx1(α)=Cζx2x1+Cζx1x2+ζx3x1+ζx1x3,ζx2(α)=ζx1x2+ζx2x1+Aζx3x2+Aζx2x3,ζx3(α)=Bζx1x3+Bζx3x1+ζx2x3+ζx3x2.

Let us assume that the infinitesimals ζx1, ζx2 and ζx3 are of the form

ζx1=a1(t)x1+a2(t)x2+a3(t)x3+a4(t),ζx2=b1(t)x1+b2(t)x2+b3(t)x3+b4(t),ζx3=c1(t)x1+c2(t)x2+c3(t)x3+c4(t).

Proceeding in the above similar manner, we obtain

τ=ct, ζx1=-αcx1, ζx2=-αcx2, ζx3=-αcx3.

Hence, the fractional Lotka-Volterra ABC system (3.58) is invariant under

t¯=t+c∊t,x¯1=x1-∊αcx1,x¯2=x2-∊αcx2,x¯3=x3-∊αcx3.

Hence, the infinitesimal generator is

X=ct∂∂t-αcx1∂∂x1-αcx2∂∂x2-αcx3∂∂x3.

3.5.2. Construction of exact solution

The invariant function F of system (3.58) satisfies the equation XF = 0. Hence its characteristic equation becomes

dtt=dx1-αx1=dx2-αx2=dx3-αx3.

On solving the above characteristic equation, we obtain an exact solution of fractional Lotka-Volterra ABC system (3.58) as

x1(t)=[(AC-A+1ABC+1)Γ(1-α)Γ(1-2α)]t-α,x2(t)=[(AB-B+1ABC+1)Γ(1-α)Γ(1-2α)]t-α,x3(t)=[(BC-C+1ABC+1)Γ(1-α)Γ(1-2α)]t-α, (3.59)

where α ∈ (0, 1). Note that for α=12, the solutions of the system (3.58) takes the form x₁(t) = x₂(t) = x₃(t) = 0 which is trivial. In the Figure 4, for A = 2, B = 1 and C = −1 the geometrical representations of solutions of the fractional Lotka-Volterra system (3.58) for various values of α are given.

Note 9. Observe that the obtained invariant solutions of the form x(t) ∝ t^−α holds only for the R-L fractional derivative because

dαdtα(t-α)=Γ(1-α)Γ(1-2α)t-2α, α∈(0,1), t>0.

These type of solutions are not available in the sense of Caputo fractional derivative [32,52].

4. CONCLUDING REMARKS AND DISCUSSION

In this article, we have shown how we can extend the Lie symmetry analysis method for n-coupled system of FODEs with R-L fractional derivative. Also, we systematically presented how to derive the Lie point symmetries of scalar and coupled FODEs namely (i) fractional Thomas-Fermi equation (3.1), (ii) Bagley-Torvik equation (3.12), (iii) two-coupled system of fractional quartic oscillator (3.33), (iv) fractional type coupled equation of motion (3.54) and (v) fractional Lotka-Volterra ABC system (3.58). Furthermore, we explicitly derived the exact solutions of the above mentioned FODEs wherever possible using the obtained Lie point symmetries. The numerical solutions of fractional Thomas-Fermi equation with Caputo derivative was discussed in [12]. In the presented work, we derived the invariant solution of Thomas-Fermi equation (3.1) in the sense of R-L fractional derivative of the form x(t) ∝ t⁻⁽^α⁺²⁾. Since the dimensions of the symmetry algebras for Bagley-Torvik equation and its various cases are greater than 2, we constructed the optimal system of one-dimensional subalgebras. We know that the maximal dimension of symmetry algebra for second-order classical ODE is eight. For the second-order Bagley-Torvik equation (3.11), the dimension of symmetry algebra is five due to the presence of R-L fractional derivative of order 32. We also systematically found the optimal system of Lie algebra for (3.11) with various cases.

We not only extended the Lie symmetry analysis method for n-coupled system of FODEs with R-L fractional derivative but also established the efficiency of the method by solving the two-coupled system of fractional quartic oscillator (3.33), fractional type coupled equation of motion (3.54) and fractional Lotka-Volterra ABC system (3.58). For the system (3.33) the invariant solution xi∝t-(α+1)2 for i = 1, 2. The fractional type coupled equation of motion (3.54) admits the invariant solution of the form x_i ∝ t⁻⁽^α⁺¹⁾ for i = 1, 2. Similarly, we observe that the invariant solution of the fractional Lotka-Volterra ABC system (3.58) is of singular type and is of the form x_i ∝ t⁻^α for i = 1, 2, 3. These type of obtained exact solutions are possible in R-L fractional derivative but not in Caputo sense [32,52]. The exact solutions of the given scalar and coupled FODEs were graphically shown for various values of α.

CONFLICTS OF INTEREST

The authors declare they have no conflicts of interest.

ACKNOWLEDGMENT

The authors wish to thank the editors and all the anonymous reviewers for their fruitful comments and suggestions for the significant improvement of the manuscript.

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 28 - 2
Pages: 219 - 241
Publication Date: 2021/03/31
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TY  - JOUR
AU  - K. Sethukumarasamy
AU  - P. Vijayaraju
AU  - P. Prakash
PY  - 2021
DA  - 2021/03/31
TI  - On Lie Symmetry Analysis of Certain Coupled Fractional Ordinary Differential Equations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 219
EP  - 241
VL  - 28
IS  - 2
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.k.210315.001
DO  - 10.2991/jnmp.k.210315.001
ID  - Sethukumarasamy2021
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