Conservation laws, analytical solutions and stability

EL-Kalaawy et al. Advances in Difference Equations (2019) 2019:445 https://doi.org/10.1186/s13662-019-2375-y

R E S E A R C H Open Access

Conservation laws, analytical solutions andstability analysis for the time-fractionalSchamel–Zakharov–Kuznetsov–BurgersequationO.H. EL-Kalaawy1*, S.M. Moawad1, M.M. Tharwat1 and Rasha B. Al-Denari1

*Correspondence:[email protected] of Mathematics andComputer Science, Faculty ofScience, Beni-Suef University,Beni-Suef, Egypt

AbstractIn this paper, we consider the (3 + 1)-dimensional time-fractionalSchamel–Zakharov–Kuznetsov–Burgers (SZKB) equation. With the help of theRiemann–Liouville derivatives, the Lie point symmetries of the (3 + 1)-dimensionaltime-fractional SZKB equation are derived. By applying the Lie point symmetrymethod as well as Erdélyi–Kober fractional operator, we get the similarity reductionsof the time-fractional SZKB equation. Conservation laws of the time-fractional SZKBare constructed. Moreover, we obtain its power series solutions with the convergenceanalysis. In addition, the analytical solution is obtained by modified trial equationmethod. Finally, stability is analyzed graphically in different planes.

Keywords: Lie point symmetries; Conservation laws; Time-fractionalSchamel–Zakharov–Kuznetsov–Burgers equation; Riemann–Liouville derivatives;Similarity reduction; Explicit power series; Modified trial equation

1 IntroductionPartial differential equations (PDEs) are frequently used to describe most of the phe-nomena that arise in engineering fields, mathematical physics, plasma physics, solid statephysics, quantum mechanics, fluid mechanics, ecology, optical fibers, biology, chemicalkinematics, geochemistry, meteorology, electricity and so on. Therefore, investigating an-alytical solutions (traveling wave solutions or soliton solutions) is very interesting. As aresult, many new techniques have been successfully proposed, developed and extendedby groups of researchers to find exact or analytical solutions for PDEs, such as the (G′/G)-expansion method [1–3], the Kudryashov method [4, 5], the functional variable method[5, 6], the first integral method [7], the exp(–φ(ξ ))-expansion method [8], the sine–cosinefunction method and Bernoulli’s equation approach [5, 9], the trial solution method [5,10], Hirota’s bilinear method [3] etc.

Fractional calculus was developed as one of the best tools for studying various modelsin each discipline [11, 12]. Using fractional models to describe the special characteristicphenomenon of fractional order, it can well reveal the essence of the nature of phenom-ena and their behavior. Fractional calculus is a generalization of integer calculus, integer

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calculus is a special case of fractional calculus. It has universal significance to study thefractional calculus system. Compared with the integer order model in describing the ac-tual physical process, the physical meaning of the fractional order model is clearer and theexpression is more concise.

On the other hand, in recent years, fractional ordinary differential equations (FODEs)and fractional partial differential equations (FPDEs) have been widely used to describemany various physical effects and many complex nonlinear phenomena. This is due totheir accurate description of nonlinear phenomena in fluid mechanics, viscoelasticity,electrical chemistry, quantum biology, physics, engineering mechanics [13–19] and otherscientific fields. So the study of FODEs and FPDEs has attracted much attention. In otherwords, by using the theory of derivatives and integrals of fractional order, many physicalphenomena can be accurately modeled.

The exact solutions of most of the FPDEs cannot be found easily, so analytical and nu-merical methods [20, 21] must be used. The solutions of the FPDEs are investigated bymany authors using powerful analytical methods. Several numerical methods such as thehomotopy perturbation method [22, 23], the Adomian decomposition method [24], thevariational iteration method [25], the differential transform method [26], the fractionalRiccati expansion method [27], the fractional sub-equation method [28–34] have beensuggested for solving FDEs. However, solutions obtained through all these methods areof a local nature and it is important to explore other techniques in order to find exactanalytical solutions of FDEs.

Lie point symmetry plays a very important role in various fields of science, especiallyin integrable systems where infinitely many symmetries exist. So Lie symmetry analy-sis is considered to be one of the efficient approaches for obtaining analytical solutionsof nonlinear partial differential equations (NLPDEs). A large number of studies are de-voted to the theory of Lie point symmetry and their applications to DEs. The symmetryanalysis of FDEs and the fractional derivatives are proposed by Gazizov and his collab-orators [35]. They proposed a prolongation formula for two basic fractional derivatives:Riemann–Liouville and Caputo. This method has been used to study many of the FDEs[36–38]. Furthermore Lie point symmetry is used to construct conservation laws whichplay an important role in the study of nonlinear physical phenomena [12]. Conservationlaws are a mathematical formulation which statement that the total amount of a certainphysical quantity stays the same during the evolution of a physical system [12]. Conserva-tion laws are also used in the development of numerical methods to establish the existenceand uniqueness of solution. There are many methods of constructing conservation laws fordifferential equations (DEs) [39–45]. The well-known Noether theorem [46] establishes aconnection between Lie point symmetries and conservation laws of DEs, provided that theequations are Euler–Lagrange equations. Ibragimov [47] suggested a new conservationlaw theorem. In [39], conservation laws for time-fractional sub-diffusion and diffusion-wave equations were obtained based on the new conservation laws theorem [47]. Alsothere are researches that discussed conservation laws for time FPDEs [12, 48, 49].

In this work, we focus on the time-fractional SZKB equation of the form

∂αu∂tα

+ a√

uux + buxxx + c(uxyy + uxzz) + duxx = 0, (1.1)

where ∂αt is the fractional derivative of order α, with 0 < α < 1 and u(x, y, z, t) is the poten-

tial function of space x, y, z and time t. If α = 1, Eq. (1.1) is reduced to the classical SZKB


equation, which describes the nonlinear plasma-dust ion acoustic waves DIAWs in a mag-netized dusty plasma and it is derived using the standard reductive perturbation techniquein small amplitude. The coefficients of dispersion b, non-linearity a, mixed derivative c anddissipation d are given in [50, 51]. Note that Sahoo and Ray [52] have studied the (3 + 1)-dimensional time-fractional mKdV-ZK equation without the Schamel and Burgers terms.The main purpose of this paper is to obtain the Lie point symmetries, conservation lawsand analytical solutions of the time-fractional SZKB equation.

The paper is organized as follows: The introduction is presented in Sect. 1. In Sect. 2,some definitions and description of Lie symmetry analysis for fractional partial differentialequations (FPDEs) are briefly presented. Lie point symmetries and similarity reduction ofthe Eq. (1.1) are obtained In Sect. 3. In Sect. 4, the conservation laws of the Eq. (1.1) areobtained. In Sect. 5, which is based on the power series, the analytical solution of theEq. (1.1) is constructed with convergence analysis. We construct the analytical solution ofEq. (1.1) by using a modified trial equation method In Sect. 6. Finally, the discussions andconclusions of this paper are presented in Sect. 7.

2 PreliminariesHere in this section, we focus on some of the concepts that revolve around the subject ofour article

Definition 1 Let α > 0. The operator Iα defined by

Iαf (t) =1

Γ (α)

∫ t

0(t – s)α–1f (s) ds, (2.1)

is called the Riemann–Liouville (R-L) fractional integral operator of order α, and Γ (·)denotes the gamma function.

Definition 2 Let α > 0. The operator Dαt defined by

Dαt f (t) =

⎧⎨⎩

1Γ (n–α)

dn

dtn∫ t

0 (t – s)n–α–1f (s) ds if n – 1 < α < n, n ∈ N ,dnf (t)

dtn if α = n, n ∈ N ,(2.2)

is called the R-L fractional partial derivative [16, 53–56].

2.1 Description of Lie point symmetriesLet us consider the symmetry analysis for a FPDE of the form

Dαt u(x, y, z, t) = G(x, y, z, t, u, ux, ut , uy, uz, uxx, uxy, . . .), 0 < α < 1. (2.3)

Now, let Eq. (2.3) be invariant under the following one-parameter Lie group of point trans-formations acting on both the dependent and the independent variables:

x = x + εξ (x, y, z, t) + O(ε2),

y = y + εζ (x, y, z, t) + O(ε2),

z = z + εν(x, y, z, t) + O(ε2),


t = t + ετ (x, y, z, t) + O(ε2),

u = u + εη(x, y, z, t) + O(ε2),

Dαt u = Dα

t u + εη0α(x, y, z, t) + O

(ε2), (2.4)

∂u∂ x

=∂u∂x

+ εηx(x, y, z, t) + O(ε2),

∂2u∂ x2 =

∂2u∂x2 + εηxx(x, y, z, t) + O

(ε2),

∂3u∂ x3 =

∂3u∂x3 + εηxxx(x, y, z, t) + O

(ε2),

∂3u∂ x ∂ y2 =

∂3u∂x ∂y2 + εηxyy(x, y, z, t) + O

(ε2),

∂3u∂ x ∂ z2 =

∂3u∂x ∂z2 + εηxzz(x, y, z, t) + O

(ε2),

where ε � 1 is the Lie group parameter and ξ , ζ , ν , τ , η are the infinitesimals of the trans-formations for dependent and independent variables, respectively. The explicit expres-sions of ηx, ηxx, ηxxx, ηxyy, ηxzz are given by

ηx = Dx(η) – uxDx(ξ ) – uyDx(ζ ) – uzDx(ν) – utDx(τ ),

ηxx = Dx(ηx) – uxxDx(ξ ) – uxyDx(ζ ) – uxzDx(ν) – uxtDx(τ ),

ηxxx = Dx(ηxx) – uxxxDx(ξ ) – uxxyDx(ζ ) – uxxzDx(ν) – uxxtDx(τ ),

ηxyy = Dx(ηyy) – uxxyDx(ξ ) – uxyyDx(ζ ) – uxyzDx(ν) – uxytDx(τ ),

ηxzz = Dx(ηzz) – uxxzDx(ξ ) – uxyzDx(ζ ) – uxzzDx(ν) – uxztDx(τ ),

(2.5)

where Dx, Dy, Dz , and Dt are the total derivatives with respect to x, y, z, and t, respectively,that are defined for x1 = x, x2 = y, x3 = z as

Dxj =∂

∂xj + uj∂

∂u+ ujk

∂

∂uk+ · · · , j, k = 1, 2, 3, . . . ,

where uj = ∂u∂xj , ujk = ∂2u

∂xj∂xk and so on.The corresponding Lie algebra of symmetries consists of a set of vector fields of the form

V = ξ∂

∂x+ ζ

∂

∂y+ ν

∂

∂z+ τ

∂

∂t+ η

∂

∂u.

The invariance condition of Eq. (2.3) under the infinitesimal transformations is given as

Pr(n) V (�)∣∣�=0 = 0, n = 1, 2, 3, . . . ,

where

� := Dαt u(x, y, z, t) – G(x, y, z, t, u, ux, ut , uy, uz, uxx, uxy, . . .).


Also, the invariance condition gives

τ (x, y, z, t, u)|t=0 = 0. (2.6)

The αth extended infinitesimal related to RL fractional time derivative with Eq. (2.6) canbe represented as follows:

η0α = Dα

t (η) + ξDαt (ux) – Dα

t (ξux) + ζDαt (uy) – Dα

t (ζuy) + νDαt (uz)

– Dαt (νuz) + Dα

t(Dt(τ )u

)– Dα+1

t (τu) + τDα+1t (u), (2.7)

where Dαt is the total fractional derivative operator and by using the generalized Leibnitz

rule

Dαt(f (t)g(t)

)=

∞∑n=0

(α

n

)Dα–n

t f (t)Dnt g(t),

(α

n

)=

(–1)n–1αΓ (n – α)Γ (1 – α)Γ (n + 1)

.

By applying the Leibnitz rule, Eq. (2.7) becomes

η0α = Dα

t (η) – αDαt (τ )

∂αu∂tα

–∞∑

n=1

(α

n

)Dn

t (ξ )Dα–nt ux –

∞∑n=1

(α

n

)Dn

t (ζ )Dα–nt uy

–∞∑

n=1

(α

n

)Dn

t (ν)Dα–nt uz –

∞∑n=1

(α

n + 1

)Dn+1

t (ξ )Dα–nt u. (2.8)

Now the chain rule for the compound function is defined as follows:

dnφ(h(t))dtn =

n∑k=0

k∑r=0

(kr

)1k!

[–h(t)

]r dn

dtn

[–h(t)k–r] × dkφ(h)

dhk .

By applying this rule and the generalized Leibnitz rule with f (t) = 1, we have

Dαt (η) =

∂αη

∂tα+ ηu

∂αu∂tα

– u∂αηu

∂tα+

∞∑n=1

(α

n

)∂nηu

∂tn Dα–nt (u) + μ,

where

μ =∞∑

n=2

n∑m=2

m∑k=2

k–1∑r=0

(α

n

)(nm

)(kr

)1k!

× tn–α

Γ (n + 1 – α)[–u]r ∂m

∂tm

[uk–r] ∂n–m+kη

∂tn–m∂uk .

Therefore, Eq. (2.8) yields

η0α =

∂αη

∂tα+

(ηu – αDα

t (τ ))∂αu∂tα

– u∂αηu

∂tα+ μ

+∞∑

n=1

[(α

n

)∂αηu

∂tα–

(α

n + 1

)Dn+1

t (τ )]

Dα–nt (u) +

∞∑n=1

(α)Dnt (ξ )Dα–n

t ux

–∞∑

n=1

(α

n

)Dn

t (ζ )Dα–nt uy –

∞∑n=1

(α

n

)Dn

t (ν)Dα–nt uz. (2.9)


Definition 3 The function u = θ (x, y, z, t) is an invariant solution of Eq. (2.3) associatedwith the vector field W , such that

1. u = θ (x, y, z, t) is an invariant surface of Eq. (2.3), this means

Vθ = 0 ⇔(

ξ∂

∂x+ ζ

∂

∂y+ ν

∂

∂z+ τ

∂

∂t+ η

∂

∂u

)θ = 0,

2. u = θ (x, y, z, t) satisfies Eq. (2.3).

3 Lie point symmetries and similarity reduction for Eq. (1.1)In this section, we implemented Lie group method for Eq. (1.1) and then, used these sym-metries to reduce Eq. (1.1) to be a FODE as shown in the next two subsections

3.1 Lie point symmetries for Eq. (1.1)Let us consider Eq. (1.1) as an invariant under Eq. (2.4), we get

∂αu∂ tα

+ a√

uux + buxxx + c(uxyy + uxzz) + duxx = 0, (3.1)

such that u = u(x, y, z, t) satisfies Eq. (1.1), then using the point transformations Eq. (2.4)in Eq. (3.1), we get the invariant equation

η0α + a

√uηx +

a2√

uηux + bηxxx + c

(ηxyy + ηxzz) + dηxx = 0, (3.2)

By substituting Eq. (2.5) and Eq. (2.9) into Eq. (3.2), we acquire

∂αη

∂tα+

(ηu – αDα

t (τ ))∂αu∂tα

– u∂αηu

∂tα+ μ

+∞∑

n=1

[(α

n

)∂αηu

∂tα–

(α

n + 1

)Dn+1

t (τ )]

Dα–nt (u) +

∞∑n=1

(α)Dnt (ξ )Dα–n

t ux

–∞∑

n=1

(α

n

)Dn

t (ζ )Dα–nt uy –

∞∑n=1

(α

n

)Dn

t (ν)Dα–nt uz

+ a√

u(ηx + (ηu – ξx)ux – ξuu2

x – (ζx + ζuux)uy – (νx + νxux)uz – (τx + τuux)ut)

+a

2√

uηux + b

(ηxxx + (ηxxu – ξxxx)ux + (3ηxu – 3ξxx)uxx + (ηu – 3ξx)uxxx

+ (3ηxuu – 3ξxxu)u2x + (ηuuu – ξxuu)u3

x – ξuuuu4x + (3ηuu – 9ξxu)uxuxx

– 3ξuuu2xuxx – 3ξuu2

xx – 4ξuuxuxxx – 3(ζx + ζuux)uxxy – 3(νx + νxux)uxxz

– 3(τx + τuux)uxxt – 3(ζxx + 2ζxuux + ζuuu2

x + ζuuxx)uxy

– 3(νxx + 2νxuux + νuuu2

x + νuuxx)uxz – 3

(τxx + 2τxuux + τuuu2

x + τuuxx)uxt

–(ζxxx + 3ζxxuux + 3ζxuuu2

x + ζuuuu3x + 3ζuuuxuxx + 3ζxuuxx + ζuuxxx

)uy

–(νxxx + 3νxxuux + 3νxuuu2

x + νuuuu3x + 3νuuuxuxx + 3νxuuxx + νuuxxx

)uz

–(τxxx + 3τxxuux + 3τxuuu2

x + τuuuu3x + 3τuuuxuxx + 3τxuuxx + τuuxxx

)ut

)

+ c(ηxyy + (ηuyy – ξxyy)ux + (2ηuy – ζyy – 2ξxy)uxy


+ (2ηxuy + 2ηuuyux – ζxyy – ζuyyux)uy + (ηu – 2ζy – ξx – ζuux)uxyy

+ 2(ηuu – 2ζuy)uyuxy + (ηxu + ηuuux – 2ζxy – 2ζuyux)uyy – 3ζuuu2yuxy

+ (3ηuux + ηuuuux – 2ζxuy – 2ζuuyux)u2y – 3ζuuxyuyy – (ζuux + ζuuuux)u3

y

– 3ζuuyuxyy – 3(ζxu + ζuuux)uyuyy – (2ξy + 2ξuuy + ξx + ξuux)uxxy

– 2(ξuyux + ξxuuy + ξuuuxuy + ξuuxy)uxy – (2νy + 2νuuy + νx + νuux)uxyz

– 2(νuyux + νxuuy + νuuuxuy + νuuxy)uzy(2τy + 2τuuy + τx + τuux)uxyt

– 2(τuyux + τxuuy + τuuuxuy + τuuxy)uyt –(ξyyux + 2ξuyuy + ξuuu2

y + ξuuyy)uxx

–(νyy + 2νuyuy + νuuu2

y + νuuyy)uxz –

(τyy + 2τuyuy + τuuu2

y + τuuyy)uxt

–(ξxyy + ξuyyux + 2ξxyuuy + 2ξuuyuxuy + 2ξuyuxy + ξxuuu2

y + ξuuuuxu2y + 2ξuuuxuxy

+ ξxuuyy + ξxuuxuyy + ξuuxyy)ux –

(νxyy + νuyyux + 2νxyuuy + 2νuuyuxuy + 2νuyuxy

+ νxuuu2y + νuuuuxu2

y + 2νuuuxuxy + νxuuyy + νxuuxuyy + νuuxyy)uz

–(τxyy + τuyyux + 2τxyuuy + 2τuuyuxuy + 2τuyuxy + τxuuu2

y + τuuuuxu2y + 2τuuuxuxy

+ τxuuyy + τxuuxuyy + τuuxyy)ut + ηxzz + (ηuzz – ξxzz)ux + (2ηuy – ζyy – 2ξxy)uxz

+ (2ηxuz + 2ηuuzux – νxzz – νuzzux)uz + (ηu – 2νz – ξx – νuux)uxzz

+ 2(ηuu – 2νuz)uzuxz + (ηxu + ηuuux – 2νxz – 2νuzux)uzz – 3νuuu2z uxz

+ (3ηuux + ηuuuux – 2νxuz – 2νuuzux)u2z – 3νuuxzuzz – (νuux + νuuuux)u3

z – 3νuuzuxzz

– 3(νxu + νuuux)uzuzz – (2ξz + 2ξuuz + ξx + ξuux)uxxz

– 2(ξuzux + ξxuuz + ξuuuxuz + ξuuxz)uxz – (2ζz + 2ζuuz + ζx + ζuux)uxyz

– 2(ζuzux + ζxuuz + ζuuuxuz + ζuuxz)uzy – (2τz + 2τuuz + τx + τuux)uxzt

– 2(τuzux + τxuuz + τuuuxuz + τuuxz)uzt –(ξzzux + 2ξuzuz + ξuuu2

z + ξuuzz)uxx

–(ζzz + 2ζuzuy + ζuuu2

z + ζuuzz)uxy –

(τzz + 2τuzuy + τuuu2

z + τuuzz)uxt

–(ξxzz + ξuzzux + 2ξxzuuz + 2ξuuzuxuz + 2ξuzuxz + ξxuuu2

z + ξuuuuxu2z + 2ξuuuxuxz

+ ξxuuzz + ξxuuxuzz + ξuuxzz)ux –

(ζxzz + ζuzzux + 2ζxzuuz + 2ζuuzuxuz + 2ζuzuxz

+ ζxuuu2z + ζuuuuxu2

z + 2ζuuuxuxz + ζxuuzz + ζxuuxuzz + ζuuxzz)uy

–(τxzz + τuzzux + 2τxzuuz + 2τuuzuxuz + 2τuzuxz + τxuuu2

z

+ τuuuuxu2z + 2τuuuxuxz + τxuuzz + τxuuxuzz + τuuxzz

)ut)

+ d(ηxx + (2ηxu – ξxx)ux + (ηu – 2ξx)uxx + (ηuu – 2ξxu)u2

x – ξuuu3x – 3ξuuxuxx

– 2(ζx + ζuux)uxy – 2(νx + νuux)uxz – 2(τx + τuux)uxt

–(ζxx + 2ζxuux + ζuuu2

x + ζuuxx)uy –

(νxx + 2νxuux + νuuu2

x + νuuxx)uz

–(τxx + 2τxuux + τuuu2

x + τuuxx)ut

)= 0, (3.3)


now, for Eq. (3.3) by making the whole powers of derivatives of u to zero, we get the fol-lowing system of equations:

(α

n

)∂αηu

∂tα–

(α

n + 1

)Dn+1

t (τ ) = 0, n = 1, 2, 3, . . . ,

a2√

uη + a

√u(ηu – ξx) + b(3ηxxu – ξxxx) + c(ηyyu – ξxyy + ηzzu – ξxzz) + d(2ξxu – ξxx) = 0,

a√

uηx + bηxxx + c(ηxyy + ηxzz) + dηxx + μ = 0,

τu = ξu = νu = τx = ζu = νu = 0,

τuu = ξuu = νuu = ζuu = νuu = 0,

Dnt (ξ ) = 0, n = 1, 2, 3, . . . ,

ηu – αDt(τ ) = 0,

ηu – 2ζy – ξx = 0,

ηu – 3ξx = 0,

ηu – 2νz – ξx = 0.

On solving the previous system of equations, we get the following infinitesimals:

ξ = c1x + c2, τ =3α

c1t + c3, ζ = c1y + c4, ν = c1z + c5, η = –4c1u, (3.4)

where ci, i = 1, 2, 3, 4, 5 are arbitrary constants. So, the associated vector fields are given by

V1 =∂

∂x, V2 =

∂

∂t, V3 =

∂

∂y, V4 =

∂

∂z,

V5 = x∂

∂x+ y

∂

∂y+ z

∂

∂z+

3α

t∂

∂t– 4u

∂

∂u,

(3.5)

and then, all of the infinitesimal generators of Eq. (1.1) can be expressed as follows:

V = c1V1 + c2V2 + c3V3 + c4V4 + c5V5.

3.2 The similarity reduction for Eq. (1.1)In order to reduce Eq. (1.1) to become a FODE, we used the infinitesimal generator V5

defined in Eq. (3.5) to form the following characteristic equation:

dxx

=dyy

=dzz

=dt3α

t=

du–4u

.

Solving the above equation yields the following similarity function:

u = t– 4α3 f (ξ ), ξ = (x + y + z)t– α

3 . (3.6)

By means of the similarity transformation u = t– 4α3 f (ξ ), the time-fractional SZKB Eq. (1.1)

can be reduced to a nonlinear FODE with a new independent variable ξ = (x + y + z)t– α3 .

Consequently, one can get the following theorem.


Theorem 1 The transformation equation (3.6) reduces the time-fractional SZKB Eq. (1.1)to the following nonlinear FODE:

(P1– 7α

3 ,α3α

f)(ξ ) + a

√f fξ + (b + 2c)fξξξ + dt

α3 fξξ = 0, (3.7)

with Erdélyi–Kober (EK) fractional differential operator (Pτ ,αβ f )(ξ ), which is defined as

(Pτ ,α

β f)(ξ ) =

n–1∏j=0

(τ + j –

1β

ddτ

)(K τ+α,n–α

β f)(ξ ), n =

⎧⎨⎩

|α| + 1, n /∈ N ,

α, n ∈ N ,(3.8)

where

(K τ+α,n–α

β f)(ξ ) =

⎧⎨⎩

1Γ (α)

∫ ∞1 (u – 1)α–1u–(τ+α)f (ξu

1β ) du, α > 0,

f (ξ ), α = 0,(3.9)

is the EK fractional integral operator.

Proof Let n – 1 < α < 1, n = 1, 2, 3, . . . , we obtain the similarity transformation based onRiemann–Liouville fractional derivatives as follows:

∂αu∂tα

=∂n

∂tn

[1

Γ (n – α)

∫ t

1(t – s)n–α–1s

–43 αf

((x + y + z)s– α

3)

ds]

. (3.10)

Let v = ts , one can get ds = – t

v2 , thus Eq. (3.10) can be written as

∂αu∂tα

=∂n

∂tn

[tn– 7

3 α

Γ (n – α)

∫ ∞

1(v – 1)n–α–1v–(n+1– 7

3 α)f(ξv

α3)

dv]

, (3.11)

by using the definition of EK fractional differential operator, then Eq. (3.11) becomes

∂αu∂tα

=∂n

∂tn

[tn– 7

3 α(K1– α

3 ,n–α

3α

f)(ξ )

], (3.12)

and now it is appropriate to simplify the right hand side of Eq. (3.11). Consider ξ = χ t– α3 ,

ϕ ∈ (0,∞) we get

t∂

∂tϕ(ξ ) = tχ

(–

α

3

)t– α

3 –1ϕ′(ξ ) = –α

3ξ

ddξ

ϕ(ξ ).

From this,

∂n

∂tn

[tn– 7

3 α(K1– α

3 ,n–α

3α

f)(ξ )

]=

∂n–1

∂tn–1

[∂

∂t(tn– 7

3 α(K1– α

3 ,n–α

3α

f)(ξ )

)]

=∂n–1

∂tn–1

[tn– 7

3 α–1(

n –73α –

α

3ξ

∂

∂ξ

)(K1– α

3 ,n–α

3α

f)(ξ )

],


and thus repeating we have

∂n

∂tn

[tn– 7

3 α(K1– α

3 ,n–α

3α

f)(ξ )

]=

∂n–1

∂tn–1

[∂

∂t(tn– 7

3 α(K1– α

3 ,n–α

3α

f)(ξ )

)]

=∂n–1

∂tn–1

[tn– 7

3 α–1(

n –73α –

α

3ξ

ddξ

)(K1– α

3 ,n–α

3α

f)(ξ )

]

...

= t– 73 α

n–1∏j=0

[(1 –

73α + j –

α

3ξ

ddς

)(K1– α

3 ,n–α

3α

f)(ξ )

].

On using the definition of the EK fractional differential operator Eq. (3.9), we get

∂n

∂tn

[tn– 7

3 α(K1– α

3 ,n–α

3α

f)(ξ )

]= t– 7

3 α(P1– 7

3 α,α3α

f)(ξ ). (3.13)

By substituting Eq. (3.13) into Eq. (3.12), we have

∂αu∂tα

= t– 73 α

(P1– 7

3 α,α3α

f)(ξ ). (3.14)

Thus, Eq. (3.13) can be reduced into a fractional ordinary differential equation

(P1– 7

3 α,α3α

f)(ξ ) + a

√f fξ + (b + 2c)fξξξ + dt

α3 fξξ = 0. (3.15)

The proof of the theorem is completed. �

4 Conservation laws for Eq. (1.1)In this section, the conservation laws of the time-fractional SZKB equation (1.1) are de-rived, based on the formal lagrangian and Lie point symmetries as described in the fol-lowing explanation:

Consider a vector C = (Ct , Cx, Cy, Cz) admitting the following conservation equation:

[Dt

(Ct) + Dx

(Cx) + Dy

(Cy) + Dz

(Cz)]

Eq. (1.1) = 0, (4.1)

where Ct = Ct(x, y, z, t, u, . . .), Cx = Cx(x, y, z, t, u, . . .), Cy = Cy(x, y, z, t, u, . . .), and Cz =Cz(x, y, z, t, u, . . .) are called the conserved vectors for Eq. (1.1). According to the new con-servation theorem due to Ibragimov [47], the formal Lagrangian for Eq. (1.1) can be givenby

L = ω(x, y, z, t)[

∂αu∂tα

+ a√

uux + buxxx + c(uxyy + uxzz) + duxx

]= 0, (4.2)

here ω(x, y, z, t) is a new dependent variable. Depending on the definition of the La-grangian, we get an action integral as follows:

∫ t

0

∫Ω1

∫Ω2

∫Ω3

L(x, y, z, t, u,ω, Dα

t , ux, uxxx, uxyy, uxzz, uxx)

dx dy dz dt.


The Euler–Lagrange operator is defined as

δ

δu=

∂

∂u+

(Dα

t)∗ ∂

∂Dαt u

– Dx∂

∂ux+ D2

x∂

∂uxx– D3

x∂

∂uxxx– DxD2

y∂

∂uxyy– DxD2

z∂

∂uxzz,

where (Dαt )∗ denotes to the adjoint operator of Dα

t , and the adjoint equation to the non-linear by means of the Euler–Lagrange equation given by

δLδu

= 0.

The adjoint operator (Dαt )∗ for R-L is defined by

(Dα

t)∗ = (–1)nIn–α

T(Dn

t) ≡ C

t DαT ,

where In–αT is the right-sided operator of fractional integration of order n–α that is defined

by

In–αT f (t, x) =

1Γ (n – α)

∫ T

t(τ – t)n–α–1f (τ , x) dτ .

Considering the case of one dependent variable u(x, y, z, t) with four independent variablesx, y, z, t, we get

X + Dt(τ )I + Dx(ξ )I + Dy(ζ )I + Dz(ν)I = Wδ

δu+ Dt

(Ct) + Dx

(Cx) + Dy

(Cy) + Dz

(Cz),

where X is defined by

X = τ∂

∂t+ ξ

∂

∂x+ ζ

∂

∂y+ ν

∂

∂z+ η

∂

∂u+ η0

α

∂

∂Dαt u

+ ηx ∂

∂ux+ ηxx ∂

∂uxx+ ηxxx ∂

∂uxxx+ ηxyy ∂

∂uxyy+ ηxzz ∂

∂uxzz,

and the Lie characteristic function is defined by

W = η – τut – ξux – ζuy – νuz, and then

W1 = –ux, W2 = –ut , W3 = –uy, Wt = –uz.(4.3)

For the R-L time-fractional derivative, the density component Ct of conservation law isdefined by

Ct = τL +n–1∑k=0

(–1)k0Dα–1–k

t (Wm)Dkt

∂L∂0Dα

t u– (–1)nJ

(Wm, Dn

t∂L

∂0Dαt u

), (4.4)

where the operator J(·) is defined by

J(f , g) =1

Γ (n – α)

∫ t

0

∫ T

t

f (τ , x, y, z)g(μ, x, y, z)(μ – τ )α+1–n dμdτ ,


and the other (flux) components are defined as

Ci = ξ iL + Wm

[∂L∂um

i– Dj

(∂L∂um

ij

)+ DjDk

(∂L

∂umijk

– · · ·)]

+ Dj(Wm)[

∂L∂um

ij– Dk

(∂L

∂umijk

)+ · · ·

]+ DjDk(Wm)

(∂L

∂umijk

– · · ·)

+ · · · , (4.5)

where ξ 1 = ξ , ξ 2 = ζ , ξ 3 = ν and m = 1, 2, . . . , 5.Now by using Eq. (4.3) with the help of Eqs. (4.4) and (4.5), we obtain the components

of the conservation laws for the time-fractional SZKB equation as follows.Case 1: W1 = –ux where ξ x = 1, ξ t = 0, ξ y = 0, ξ z = 0 and η = 0 we get the following

conserved vectors:

Ct1 = ω0Dα–1

t (–ux)∂L

∂0Dαt u

– J(

–ux, Dnt

∂L∂0Dα

t u

)= –ωDα

t (ux) – uxDαt (ω),

Cx1 = ω

[Dα

t u + a√

uux +c3

(uyxy + uyyx + uzxz + uzzx)]

– ux

[–dωx + bωxx +

c3

(ωyy + ωzz)]

+ bωxxωx +c3

(uxyωy + uxzωz),

Cy1 = –

c3

ux[ωxy + ωyx] +c3

(uxxωy + uxyωx) –c3ω(uxyx + uyxx),

Cz1 = –

c3

ux[ωxz + ωzx] +c3

(uxxωz + uxzωx) –c3ω(uxzx + uzxx).

Case 2: W2 = –ut where ξ x = 0, ξ t = 1, ξ y = 0, ξ z = 0 and η = 0 we acquire the followingconserved vectors:

Ct2 = ωLω0Dα–1

t (–ut)∂L

∂0Dαt u

– J(

–ut , Dnt

∂L∂0Dα

t u

)

= ω

[Dα

t u + a√

uux +c3

(uyxy + uyyx + uzxz + uzzx) + duxx

]– ωDα

t (ut) – utDαt (ω),

Cx2 = –ut

[aω

√u – dωx + bωxx +

c3

(ωyy + ωzz)]

– uxt[dωx – bωx] +c3

(uytωy + uztωz) – bωuxxt –c3ω(uyyt + uzzt),

Cy2 = –

c3

ut[ωxy + ωyx] +c3

(uxtωy + uytωx) –c3ω(uxyt + uyxt),

Cz2 = –

c3

ut[ωxz + ωzx] +c3

(uxtωz + uztωx) –c3ω(uxzt + uzxt).

Case 3: W3 = –uy where ξ x = 0, ξ t = 0, ξ y = 1, ξ z = 0 and η = 0 we obtain the followingconserved vectors:

Ct3 = ω0Dα–1

t (–uy)∂L

∂0Dαt u

– J(

–uy, Dnt

∂L∂0Dα

t u

)= –ωDα

t (uy) – uyDαt (ω),

Cx3 = –uy

[aω


c3

(ωyy + ωzz)]

– uxy[dωx – bωx] +c3

(uyyωy + uyzωz) – bωuxxy –c3ω(uyyy + uzzy),


Cy3 = ω

[Dα

t u + a√

uux +c3


]

–c3

uy[ωxy + ωyx] +c3

(uxyωy + uyyωx) –c3ω(uxyy + uyxy),

Cz3 = –

c3

uy[ωxz + ωzx] +c3

(uxyωz + uyzωx) –c3ω(uxzy + uzxy).

Case 4: W4 = –uz where ξ x = 0, ξ t = 0, ξ y = 0, ξ z = 0 and η = 0 we obtain the followingconserved vectors:

Ct4 = ω0Dα–1

t (–uz)∂L

∂0Dαt u

– J(

–uz, Dnt

∂L∂0Dα

t u

)= –ωDα

t (uz) – uzDαt (ω),

Cx4 = –uz

[aω


c3

(ωyy + ωzz)]

– uxy[dωx – bωx]

+c3

(uyyωy + uyzωz) – bωuxxy –c3ω(uyyy + uzzy),

Cy4 = ω

[Dα

t u + a√

uux +c3


]

–c3

uy[ωxy + ωyx] +c3

(uxyωy + uyyωx) –c3ω(uxyy + uyxy),

Cz4 = –

c3

uy[ωxz + ωzx] +c3

(uxyωz + uyzωx) –c3ω(uxzy + uzxy).

Remark We have verified that all the solutions satisfy the original equation.

5 Explicit power series and convergence analysis for Eq. (1.1)In this section, we derived the analytic solution for Eq. (1.1) via the power series method[57, 58] and proved the convergence of the power series solution as demonstrated in thenext two subsections.

5.1 Power series and analytical solutions for Eq. (1.1)Considering Eq. (1.1) and by using u(x, y, z, t) = ψ2(x, y, z, t), Eq. (1.1) becomes

∂αψ2

∂tα+ aψ

(ψ2)

x + b(ψ2)

xxx + c((

ψ2)xyy +

(ψ2)

xzz

)+ d

(ψ2)

xx = 0. (5.1)

Let us introduce the important transformation

ψ(x, y, z, t) = ψ(ξ ), ξ = px + qy + rz –ktα

Γ (1 + α), (5.2)

where p, q, r and k are constants to be determined later. The substitution of (5.2) into (5.1)leads to the following nonlinear ODE:

–k2(ψ2)′ +

a3(ψ3)′ +

12

p(bp2 + c

(q2 + r2))(ψ2)′′′ + dp2(ψ2)′′ = 0. (5.3)

By integrating Eq. (5.3) with respect to ξ , we get

–k2(ψ2) +

a3(ψ3) +

12

p(bp2 + c

(q2 + r2))(ψ2)′′ + dp2(ψ2)′ + λ = 0, (5.4)


where λ is the constant of integration. Let us assume that the solution of Eq. (5.4) has thefollowing form:

ψ(ξ ) =∞∑

n=0

anξn, (5.5)

where an are constants to be determined later. On substituting (5.5) into (5.4), we have thefollowing relation:

–k2

∞∑n=0

n∑m=0

(aman–m)ξn +a3

∞∑n=0

n∑m=0

n–m∑l=0

(alam–lan–m)ξn + p(bp2 + c

(q2 + r2))

×( ∞∑

n=0

n∑m=0

(n – m + 1)(n – m + 2)aman–m+2

+∞∑

n=0

n∑m=0

(n – m + 1)(m + 1)am+1an–m+1

)ξn

+ 2dp2∞∑

n=0

n∑m=0

((n – m + 1)aman–m+1

)ξn + λ = 0. (5.6)

From Eq. (5.6), by comparing the coefficients for n = 0 one can get

a2 = –2aa3

0 – 3ka20 + 12dp2a0a1 + 6(λ + bp2 + c(q2 + r2))a2

112p(bp2 + c(q2 + r2))a0

, (5.7)

where a0 and a1 are arbitrary constants where p and a0 = 0. Generally, for n ≥ 1 we canget

an+2 =1

6ρ(n + 1)(n + 2)a0

(3k

n∑m=0

(aman–m) – 2an∑

m=0

n–m∑l=0

(alam–lan–m) – 6ρ

×( n∑

m=0

(n – m + 1)(n – m + 2)aman–m+2 +n∑

m=0

(n + 1)(n – m + 1)am+1an–m+1

)

– 12dp2n∑

m=0

((n – m + 1)aman–m+1

)– 6λ

), (5.8)

where ρ = p(bp2 + c(q2 + r2)). Now, from (5.8) and (5.7) we can get all the coefficients an+2,n ≥ 1 of the power series (5.5) in which a0, p, q, r, and k are the arbitrary constants. Thus,the power series solution of Eq. (5.4) could be written as follows:

u(ξ ) =(

a0 + a1ξ + a2ξ2 +

∑∞n=1

an+2ξn+2

)2

= (a0 + a1ξ +(

–6λ – 3ka2

0 + 2aa30 + 6p(2dp + bp2 + c(q2 + r2))a0a1

12p(bp2 + c(q2 + r2))a0

)ξ 2

+∞∑

n=1

(1

6ρ(n + 1)(n + 2)a0

(3k

n∑m=0

(anan–m) – 2an∑

m=0

n–m∑l=0

(alam–lan–m) – 6ρ


×( n∑

m=0

(n – m + 1)(n – m + 2)aman–m+2 +n∑

m=0

(n + 1)(n – m + 1)am+1an–m+1

)

– 12dp2n∑

m=0

((n – m + 1)aman–m+1

)– 6λ

)ξn+2

)2

. (5.9)

Thus, we obtain the exact power series solution for Eq. (3.15) as follows:

u(x, y, z, t) =

(a0 + a1

(px + qy + rz –

ktα

Γ (1 + α

)

+(

–6λ – 3ka2

0 + 2aa30 + 6p(2dp + bp2 + c(q2 + r2))a0a1

12p(bp2 + c(q2 + r2))a0

)

×(

px + qy + rz –ktα

Γ (1 + α

)2

+∞∑

n=1

(1

6ρ(n + 1)(n + 2)a0

(3k

n∑m=0

(aman–m) – 2an∑

m=0

n–m∑l=0

(alam–lan–m)

– 6ρ

( n∑m=0

(n – m + 1)(n – m + 2)aman–m+2

+n∑

m=0

(n + 1)(n – m + 1)am+1an–m+1

)

– 12dp2n∑

m=0

((n – m + 1)aman–m+1

)– 6λ

))

×(

px + qy + rz –ktα

Γ (1 + α

)n+2)2

. (5.10)

We represented the solution defined by Eq. (5.10) using the 2D plot, see Fig. 1b, and the3D plots, see Fig. 1a and Fig. 2. The result in Eq. (5.10) shows that there is an analyticalpower series solution for Eq. (5.4). It is important to find new solutions, because either newexact solutions or numerical approximate solutions and analytical solutions may deepenour understanding of the physical phenomena.

5.2 Convergence analysisIn this subsection, to complete the analysis of Eq. (1.1), we study the convergence of thepower series solution Eq. (5.5) with the coefficients denoted by Eq. (5.7) and (5.8), whereEq. (5.8) can be enlarged to the following form:

|an+2| ≤ M

[ n∑m=0

|am||an–m| –n∑

m=0

n–m∑l=0

|al||am–l||an–m| –n∑

m=0

|am||an–m+2|

–n∑

m=0

|am+1||an–m+1| –n∑

m=0

|am||an–m+1|]

, (5.11)


Figure 1 (a) The effect of α on the 3D solitary wave solution of equation (5.10) with a fixed time t and theparameters k = 0.3, p = 0.2, q = 0.7, r = 0.3, a0 = 6, a1 = 1.7. (b) The corresponding 2D plot of (a), which showsthat the envelope of the wave decays far away from the peak quickly when the value of α increases meaningthat the solitary wave solution behaves as a dynamical system

Figure 2 The effect of α on the 3D solitary wave solution of equation (5.10) at different values of α = 0.1, 0.3with the suitable parameters k = 0.3, p = 0.2, q = 0.7, r = 0.3, a0 = 6, a1 = 1.7

where M = max{ 3kρ

, 2aρ

, 12dp2}, ρ = bp2 + c(q2 + r2). Now, consider another power series,

S(ξ ) =∞∑

n=0

snξn, (5.12)

with si = |ai|, i = 0, 1. Then we can get

sn+2 = M

[ n∑m=0

|sm||sn–m|

–n∑

m=0

n–m∑l=0

|sl||sm–l||sn–m| – |sm||sn–m+2| – |sm+1||sn–m+1| – |sm||sn–m+1|]

, (5.13)

where n = 0, 1, 2, . . . thus, it is easily noted that

|an+2| ≤ sn+2 this leads to |an| ≤ sn,

depending on this result, we could say that the series defined by Eq. (5.12) is a majorantseries of Eq. (5.5). Next, we have shown that the series S = S(ξ ) has positive radius of con-


vergence. Then, we write this series in the following form:

S(ξ ) = s0 + s1ξ +∞∑

n=2

snξn

= s0 + s1ξ + M

[ n∑m=0

|sm||sn–m|

–n∑

m=0

n–m∑l=0

|sl||sm–l||sn–m| – |sm||sn–m+2| – |sm+1||sn–m+1| – |sm||sn–m+1|]ξn+2

= s0 + s1ξ + M[S2(ξ )ξ 2 – S3(ξ )ξ 2 – 2(S – s0)(S – s0)

].

Consider an implicit functional system with respect to the independent variable ξ as fol-lows:

ϑ(ξ , S) = S – s0 – s1ξ – M[S2(ξ )ξ 2 – S3(ξ )ξ 2 – 2(S – s0)(S – s0)

], (5.14)

since ϑ is an analytic in a neighborhood of (o, s0), where ϑ(o, s0) = 0 and, ∂ϑ∂S (o, s0) = 0.

Theorem ([59]) Let f be a ℘ ′-mapping of an open set E ⊂ sn+m into Sn, such that f (a, b) = 0for some point (a, b) ∈ E. Assume that A = f ′(a, b) and Ax is invertible. Then the followingproperties hold in the open sets U ⊂ sn+m and W ⊂ Sm with (a, b) ∈ U and b ∈ W .

(i) For each y ∈ W , there exists a unique x such that (x, y) ∈ U and f (x, y) = 0.(ii) If x is defined to be g(y), then

g(b) = a,

f(g(y), y

)= 0 (y ∈ W ),

g ′(b) = –(Ax)–1Ay,

where g is a ℘ ′-mapping of W into Sn.(iii) The function g is implicitly defined by (ii).

One can see that S = S(ξ ) is analytical in a neighborhood of the point (o, s0) and has positiveradius. It shows that the power series Eq. (5.5) is convergent in a neighborhood of the point(o, s0).

6 The modified trial equation method for Eq. (1.1)In this section, we gave a brief description of the modified trial method that was proposedby Liu [60, 61]; some authors [62–64] improved this method. Now, let us consider the timeFPDEs, say in the four variables x, y, z and t. In the following we give the main steps of themodified trial method and we obtain some traveling wave solutions of Eq. (1.1):

Step 1: Consider the time FPDE defined by Eq. (2.3) and take the wave transformationas Eq. (5.2). Under this transformation, we were permitted to reduce Eq. (2.3) to ODE.


Step 2: Take the trial equation as follows:

ψ ′ =F(ψ)G(ψ)

=∑n

i=0 Aiψi

∑lj=0 Bjψ j

=A0 + a1ψ + A2ψ

2 + · · · + Anψn

B0 + B1ψ + B2ψ2 + · · · + Blψ l ,

ψ ′′ =F(ψ)(F ′(ψ)G(ψ) – F(ψ)G′(ψ))

G3(ψ),

(6.1)

where F and G are polynomials in ψ . On substituting (6.1) into (5.4) yields an equation ofΩ(ψ) of polynomial in ψ

Ω(ψ) = psψs + · · · + p1ψ + p0,

we found a relation between n and l according to the balance principle and determinedsome values of them.

Step 3: Setting the coefficients of Ω(ψ) to zero yields a system of algebraic equations:

pi = 0, i = 0, 1, . . . , s.

By solving the obtained system of algebraic equations, we can determine the values ofA0, . . . , An and B0, . . . , Bl .

Step 4: Rewrite Eq. (5.11) by the following integral form:

ξ – ξ0 =∫ F(ψ)

G(ψ)dψ , (6.2)

where ξ0 is the constant of integration. When we classify the roots of F(ψ) using the com-plete discrimination system, we obtain the analytical solutions of Eq. (1.1).

Now, applying the modified trial equation method for the (3 + 1)-dimensional time-fractional SZKB equation with the wave transformation defined by Eq. (5.2), by employing(6.1), (6.2), and using the balance principle yields n = l + 2. This resolution procedure wasapplied and we obtained results as follows.

Case 1: If we take l = 0, then n = 2, and

ψ ′ =A0 + A1ψ + A2ψ

2

B0,

ψ ′′ =(A1 + 2A2ψ)(A0 + A1ψ + A2ψ

2)B2

0,

(6.3)

thus, we have a system of algebraic equations from the coefficients of polynomial of u.Solving this system, we get

A0 = 0, A2 = –aB0

6dp, r = ±

√–

bp2 + cq2

c, k = –

4dp2A1

B0.

Substituting from the above coefficients into (6.2) and integrating, we get the solutions to(5.4), as follows:

ξ – ξ0 =∫ B0

A2ψ2 + a1ψdψ , (6.4)


this implies

ψ(ξ ) =3kB0

2p(6dP exp(± k4dp2B0

(ξ – ξ0)) – aB0), (6.5)

thus,

u(x, y, z, t) =(

3kB0

2p(6dP exp(± k4dp2B0

(px + qy + rz – ktαΓ (1+α) – ξ0)) – aB0)

)2

. (6.6)

For simplicity, take ξ0 = 0, A1 = A2 = 1 then the solution Eq. (6.6) is reduced to the followingsolution:

u(x, y, z, t) =(

1exp(γ (px + qy + rz – ktα

Γ (α+1) )) – 1

)2

, (6.7)

where γ = ±A1B0

. We represent the solution defined by Eq. (6.7) using the 2D plot, seeFig. 3b, and the 3D plots, see Fig. 3a and Fig. 4.

Figure 3 (a) The effect of α on the 3D double-layer solution of equation (6.7) with a fixed time t and theparameters k = 0.3, p = 0.2, q = 0.2. (b) The corresponding 2D plot of (a) with the same parameters

Figure 4 The 3D double-layer solution of equation (6.7) at different values of α = 0.5, 0.9 with the suitableparameters k = 0.3, p = 0.2, q = 0.2 which shows that when the value of α decreases the singularity becomesclearer


Case 2: If we take l = 1, then n = 3, and

ψ ′ =A0 + A1ψ + A2ψ

2 + A3ψ3

B0 + B1ψ,

ψ ′′ =((

A0 + A1ψ + A2ψ2 + A3ψ

3)((B0 + B1ψ)(

A1 + 2A2ψ + 3A3ψ2))

– B1(A0 + A1ψ + A2ψ

2 + A3ψ3))

/(B0 + B1ψ)3,

(6.8)

thus, we have a system of algebraic equations from the coefficients of polynomial of u.Solving this system, we get

A0 = 0, A2 =6dpA1B1 – aB2

06dpB0

, A3 = –aB1

6dp,

k =4dp2A1

B0, r = ±

√–

bp2 + cq2

c.

Substituting from the above coefficients into Eq. (6.8) and integrating, we get the solutionsto Eq. (5.4), as follows:

ξ – ξ0 =∫ B0 + B1u

A3ψ3 + A2ψ2 + a1ψdψ , (6.9)

this implies

ξ – ξ0 = –1

2A1

(B0 ln

(A1 + ψ(A2 + A3ψ)

)– 2B0 ln(ψ)

+1σ

(2(A2B0 – 2A1B1)tan–1

(A2 + 2A3ψ

σ

))), (6.10)

where σ =√

4A1A3 – 2A22. Substituting the values of Ai, i = 1, 2, 3, thus

ψ(ξ ) =(

3k2ap + exp( k(ξ0–ξ )

4dp2 )

), (6.11)

where ξ = px + qy + rz – ktαΓ (α+1) .

For simplicity, take ξ0 = 0 then the solution Eq. (6.11) is reduced to the following solu-tion:

u(x, y, z, t) =(

3k

2ap + exp(–k(px+qy+rz– ktα

Γ (α+1) )4dp2 )

)2

. (6.12)

We represent the solution defined by Eq. (6.12) using the 2D plot, see Fig. 5b, and the 3Dplots, see Fig. 5a and Fig. 6.

7 ConclusionIn this paper, we considered a (3 + 1)-dimensional time-fraction Schamel–Zakharov–Kuznetsov–Burgers equation, which described the nonlinear plasma-dust ion acoustic


Figure 5 (a) The effect of α on the 3D double-layer solution of equation (6.12) with a fixed time t and theparameters k = 0.3, p = 0.2, q = 0.2. (b) The corresponding 2D plot of (a) which shows that when the value ofα increases the interval of the stability of the solution also increases

Figure 6 The 3D double-layer solution of equation (6.12) at different values of α = 0.5, 0.9 with the suitableparameters k = 0.3, p = 0.2, q = 0.2

waves (DIAWs) in a magnetized dusty plasma. The Lie point symmetries were appliedsuccessfully to the study of the (3 + 1)-dimensional time-fraction Schamel–Zakharov–Kuznetsov–Burgers equation. Based on the Riemann–Liouville derivatives, we deducedthe corresponding vector fields, which helped us to construct the symmetry reductionsof the time-fractional SZKB equation. Furthermore, we obtained four kinds of conserva-tion laws with independent variables laying the foundation of Lie point symmetries. Also,we constructed a new set of analytical solutions via two powerful methods, which are theexplicit power series method and the modified trial equation method. Moreover, to intro-duce a better understanding of the dynamics of these solutions, we provided their graphicanalysis of Eq. (1.1). As shown in Fig. 3, It was noticed that the solution u(x, y, z, t) definedby Eq. (6.7) remained stable for –4.35 < x < 2.75 and has a singularity outside this interval.It can easily be observed in Figs. 4 and 6 that the solutions are in line with the values ofα, it means that the singularity becomes clearer when the value of α decreases. Thus, weexpect that the obtained results might serve as the explanation of the physical meaningof the time-fractional SZKB equation with more accuracy. However, we can extend thesymmetry analysis to the time-space fractional (3 + 1)-dimensional Schamel–Zakharov–Kuznetsov–Burgers equation where that work may be presented later elsewhere.

AcknowledgementsThe authors would like to thank the reviewers for insightful comments and constructive suggestions leading toimprovements into the current form of the paper. The authors expresses sincere thanks to the editor.


FundingThis research work is not supported by any funding agencies.

Competing interestsThe authors declare that no competing interests exist.

Authors’ contributionsThe authors declare that the work was realized in collaboration with the same responsibility. All authors read andapproved the final paper.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 12 July 2019 Accepted: 9 October 2019

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Conservation laws, analytical solutions and stability