New exact solutions for the conformable space-time fractional KdV, CDG, (2+1)-dimensional CBS and (2+1)-dimensional AKNS equations

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Journal of Taibah University for Science

ISSN: (Print) 1658-3655 (Online) Journal homepage: https://www.tandfonline.com/loi/tusc20

New exact solutions for the conformable

space-time fractional KdV, CDG, (2+1)-dimensional CBS

and (2+1)-dimensional AKNS equations

H. C. Yaslan & A. Girgin

To cite this article: H. C. Yaslan & A. Girgin (2019) New exact solutions for the conformable space-time fractional KdV, CDG, (2+1)-dimensional CBS and (2+1)-dimensional AKNS equations, Journal of Taibah University for Science, 13:1, 1-8, DOI: 10.1080/16583655.2018.1515303 To link to this article: https://doi.org/10.1080/16583655.2018.1515303

Published online: 30 Aug 2018.

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2019, VOL. 13, NO. 1, 1–8

https://doi.org/10.1080/16583655.2018.1515303

New exact solutions for the conformable space-time fractional KdV, CDG,

(2+1)-dimensional CBS and (2+1)-dimensional AKNS equations

H. C. Yaslan and A. Girgin

Department of Mathematics, Pamukkale University, Denizli, Turkey

ABSTRACT

In the present paper, G/G2expansion method is applied to the space-time fractional third order Korteweg-De Vries (KdV) equation, space-time fractional Caudrey-Dodd-Gibbon (CDG) equation, space-time fractional (2+1)-dimensional Calogero-Bogoyavlenskii-Schiff (CBS) equation and space-time fractional (2+1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) equation. Here, the fractional derivatives are described in conformable sense. The obtained traveling wave solutions are expressed by the hyperbolic, trigonometric, exponential and rational functions. The graphs for some of these solutions have been presented by choosing suitable values of parameters to visualize the mechanism of the given nonlinear fractional evolution equations.

ARTICLE HISTORY

Received 16 May 2018 Revised 31 July 2018 Accepted 10 August 2018

KEYWORDS

Space-time fractional third order KdV equation; space-time fractional CDG equation; space-time fractional (2+1)-dimensional CBS equation; space-time fractional (2+1)-dimensional AKNS equation; conformable derivative;G/G2_expansion

method

PACS NO

2.70.c; 02.60.Cb; 02.30.Jr.

1. Introduction

The nonlinear evolution equations are widely used as models to describe complex physical phenomena in various field of science, particularly in fluid mechan-ics, solid state physmechan-ics, plasma waves and chemical physics (see, for example, [1–4]). In this paper, we apply G/G2 expansion method (see, for example, [5]) to four space-time fractional nonlinear evolution equa-tions: space-time fractional third-order KdV equation, space-time fractional CDG equation, space-time frac-tional (2+1)-dimensional CBS equation and space-time fractional (2+1)-dimensional AKNS equation. Here, frac-tional derivatives are defined in conformable sense. In the literature, the solutions of these equations have been investigated by many authors using various meth-ods (see, for example, [6–29]).

KdV equation was first introduced by Boussinesq in 1877 and rediscovered by Diederik Korteweg and Gustav de Vries in 1895. It describes surface waves of long wavelength and small amplitude on shallow water and internal waves in a shallow density-stratified fluid. Natural transform and Homotopy perturbation meth-ods, Homotopy Perturbation Transform Method, Ric-cati Equation Approach, extended hyperbolic function method, projective Riccati equation method and the Exp-function method have been applied to the third

order KDV equation in [6–10]. Jacobi elliptic function expansion method has been applied to conformable space-fractional KdV equation in [11].

Physical understanding of the CDG equation has been investigated in [30] and its solutions have been studied in [12–15]. The sin-cosine method, the rational Exp-Function, sinh method, G/G-expansion method, Hirota’s bilinear method and exp-function method have been used to obtain solutions of the fifth order CDG equation in [12–14]. G/G-expansion method has been applied to conformable time fractional CDG equation in [15].

The CBS equation was first constructed by Bogoy-avlenskii and Schiff in different ways [31]. The modified simple equation method, the exp-function methods, Sine-Gordon expansion method, the simplest equation method, G/G-expansion method, a modified version of the Fan sub-equation method, improved G /G-expansion and extended tanh-function method, sym-metry method, Cole-Hopf transformation and the Hirota bilinear method have been implemented to com-pute solutions of the nonlinear (2+1)-dimensional CBS equation in [16–23], respectively.

The AKNS equation is one of the most important physical models (see, for example, [32]). In 1997, Lou and Hu have obtained the (2+1)-dimensional AKNS

CONTACT H. C. Yaslan hcerdik@pau.edu.tr. Department of Mathematics, Pamukkale University, 20070 Denizli, Turkey

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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2 H. C. YASLAN AND A. GIRGIN

equation from the inner parameter dependent sym-metry constraints of the KP equation [33]. Solutions of the AKNS equation have been investigated by many researchers. Hirota’s bilinear method, TANF (ξ/2)-expansion method, the ansatz method, the improved tanh method, the simplified form of the bilinear method to obtain some new exact solutions for high nonlinear form of (2+1)-dimensional AKNS equation have been presented in [24–28]. Bilinear Backlund transformation has been presented to obtain periodic wave solutions of (2+1)-dimensional AKNS equation in [29].

2. Description of the conformable fractional derivative and its properties

For a function f :(0, ∞) → R, the conformable frac-tional derivative of f of order 0< α < 1 is defined as (see, for example, [34])

T_tα_{f(t) = lim}

ε→0

f(t + εt1−α_{) − f(t)}

ε . (1)

Some important properties of the the conformable frac-tional derivative are as follows:

T_tα_{(af + bg)(t) = aT}_tα_{f(t) + bT}_tα_{g(t), ∀ a, b ∈ R, (2)} T_tα_(tμ_{) = μt}μ−α, (3)

T_tα_{(f(g(t)) = t}1−αg_(t)f_(g(t)). (4) 3. Analytic solutions to the conformable space-time fractional KdV equation

Conformable space-time fractional KdV equation is given as follows (see, for example, [8])

T_tα_{u + auT}xβu + bTxβTxβTxβu = 0, 0 < α ≤ 1, 0 < β ≤1, (5) where a= 0 and b are constants. Let us consider the following transformation

u(x, t) = U(ξ), ξ = ktα α + m

xβ

β , (6)

where k, m are constants. Substituting (6) into Equation (5) we obtain the following ordinary differen-tial equation (ODE)

kU_{+ amUU}_{+ bm}3_U_{= 0.} ₍₇₎

Integrating of Equation (7) with zero constant of inte-gration, we have

kU + amU2

2 + bm

3_U_{= 0.} ₍₈₎

Let us suppose that the solution of Equation (8) can be expressed in the following form:

U(ξ) = a0+ N i=1 ai _G(ξ) G(ξ)2 i + N i=1 bi _G(ξ) G(ξ)2 −i , (9)

where G(ξ) satisfies the following ODE _G G2 = μ + λ _G G2 2 , λ = 0, μ = 0, (10) where a0, ai, bi(i = 1, 2, . . . , N), λ and μ are constants to be determined. Equation (10) has different solutions as follows (see, for example, [5]):

Whenμλ > 0, G G2 = μ λ C cos(√μλξ) + D sin(√μλξ) D cos(√μλξ) − C sin(√μλξ). (11) Whenμλ < 0, G G2=− |μλ| λ C sinh(2√|μλ|ξ)+C cosh(2√|μλ|ξ)+D C sinh(2√|μλ|ξ)+C cosh(2√|μλ|ξ)−D. (12) Whenμ = 0, λ = 0, G G2 = − C λ(Cξ + D). (13)

Here C and D are nonzero constants. Substituting Equation (9) into Equation (8) and then by balancing the highest order derivative term and nonlinear term in result equation, the value of N can be determined as 2. The solution can be expressed as follows

U(ξ) = a0+ a1 _G G2 + a2 _G G2 2 + b1 _G G2 −1 + b2 _G G2 −2 . (14)

Substituting Equation (14) into Equation (8), collecting all the coefficients with the same power of G/G2, we can obtain a set of algebraic equations for the unknowns a0, a₁, b1, a2, b2,λ, μ, k, m: aa2 2m + 12ba2λ2m3= 0, 4a1bλ2m3+ 2aa1a2m = 0, aa2 1m + 16a2bλμm3+ 2aa0a2m + 2a2k = 0, 2a1k + 2aa0a1m + 2aa2b1m + 4a1bλm3μ = 0, 2a0k + aa20m + 4bb2λ2m3+ 4a2bm3μ2 + 2aa1b1m + 2aa2b2m = 0, 2b1k + 2aa0b1m + 2aa1b2m + 4bb1λm3μ = 0, ab2 1m + 16bb2λμm3+ 2aa0b2m + 2b2k = 0, 4bb1m3μ2+ 2ab1b2m = 0, ab2 2m + 12bb2m3μ2= 0.

Solving the algebraic equations in the Mathematica 10.0, we obtain the following set of solutions:

Case 1: a0=((−24bλm2μ)/a), a1=0, a2= ((−12bλ2 m2_{)/a), b}

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Whenμλ > 0, u(x, t) = −24bλm_a 2μ+−12bλ_a2m2 × _μ λ C cos(√μλξ) + D sin(√μλξ) D cos(√μλξ) − C sin(√μλξ) 2 +−12bμ_a 2m2 ⎛ ⎜ ⎜ ⎜ ⎝ μ λ C cos(√μλξ) +D sin(√μλξ) D cos(√μλξ) −C sin(√μλξ) ⎞ ⎟ ⎟ ⎟ ⎠ −2 . (15) Whenμλ < 0, u(x, t) =−24bλm_a 2μ +−12bλ_a2m2 × ⎛ ⎜ ⎜ ⎜ ⎝ |μλ| λ C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) + D C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) − D ⎞ ⎟ ⎟ ⎟ ⎠ 2 +−12bμ_a 2m2 ⎛ ⎜ ⎜ ⎜ ⎝ |μλ| λ C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) + D C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) − D ⎞ ⎟ ⎟ ⎟ ⎠ −2 . (16) Whenμ = 0, λ = 0, u(x, t) = −24bλm_a 2μ+−12bλ_a2m2 _C λ(Cξ + D) 2 +−12bμ_a2m2 _C λ(Cξ + D) ₋₂ . (17) Hereξ = 16bλm3μ(tα/α) + m(xβ/β). Case 2: a0= ((8bλm2μ)/a), a1= 0, a2= ((−12bλ2 m2_{)/a), b} 1= 0, b2=((−12bμ2m2)/a), k =−16bλm3μ : Whenμλ > 0, u(x, t) =8bλm_a2μ+−12bλ_a2m2 × _μ λ C cos(√μλξ) + D sin(√μλξ) D cos(√μλξ) − C sin(√μλξ) 2 +−12bμ_a 2m2 ⎛ ⎜ ⎜ ⎜ ⎝ μ λ C cos(√μλξ) +D sin(√μλξ) D cos(√μλξ) −C sin(√μλξ) ⎞ ⎟ ⎟ ⎟ ⎠ −2 . (18) Whenμλ < 0, u(x, t) =8bλm_a2μ+−12bλ_a2m2. |μλ| λ C sinh(2√|μλ|ξ) + C cosh(2√|μλ|ξ) + D C sinh(2√|μλ|ξ) + C cosh(2√|μλ|ξ) − D 2 +−12bμ_a 2m2. |μλ| λ C sinh(2√|μλ|ξ) + C cosh(2√|μλ|ξ)+D C sinh(2√|μλ|ξ) + C cosh(2√|μλ|ξ)−D −2 . (19) Whenμ = 0, λ = 0, u(x, t) = 8bλm_a 2μ +−12bλ_a2m2 _C λ(Cξ + D) 2 +−12bμ_a2m2 _C λ(Cξ + D) −2 . (20) Hereξ = −16bλm3μ(tα/α) + m(xβ/β).

Figure1shows 3D plot of the traveling wave solution

u(x, t) in Equation (15) for a = 6, b = 2, α = 0.5, β =

0.7,λ = 0.01, μ = 12, m = 0.05, D = 0.5, C = 1/3.

Figure 1.3D plot of the obtained traveling wave solutionu(x, t) of Equation (15) for a = 6, b = 2, α = 0.5, β = 0.7, λ = 0.01, μ = 12, m = 0.05, D = 0.5, C = 1/3, 0 ≤ x, t ≤ 50.

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4. Analytic solutions to the conformable space-time fractional CDG equation

Conformable space-time fractional CDG equation is given as follows (see, for example, [15])

T_tα_{u + 30uT}xβTxβTxβu + 30TxβuTxβTxβu + 180u2Txβu + TxβTxβTxβTxβTxβu = 0

0< α ≤ 1, 0 < β ≤ 1. (21) Using the transformations (6), Equation (21) reduces to the following ordinary differential equation

kU_{+ 30m}3_UU_{+ 30m}3_U_U

+ 180mU2_U_{+ m}5_U(5)_{= 0.} ₍₂₂₎

kU + 30m3_UU_{+ 60mU}3_{+ m}5_U(4)_{= 0.} ₍₂₃₎

Let us suppose that the solution of Equation (23) can be expressed in the form of Equation (9). Substituting Equation (9) into Equation (23) and then by balancing the highest order derivative term and nonlinear term in result equation, the value of N can be determined as 2. Therefore, Equation (9) reduces to

U(ξ) = a0+ a1 _G G2 + a2 _G G2 2 + b1 _G G2 −1 + b2 _G G2 −2 . (24)

Substituting Equation (24) into Equation (23), collect-ing all the terms with the same power of G/G2, we can obtain a set of algebraic equations for the unknowns a0, a1, b1,a2, b2,λ, μ, k,m,n:

60a3₂m + 180a2₂λ2m3+ 120a2λ4m5= 0,

180a1a22m + 240a1a2λ2m3+ 24a1λ4m5= 0,

180a2₁a2m + 60a21λ2m3+ 240μa22λm3+ 180a0a22m

+ 240μa2λ3m5+ 180a0a2λ2m3= 0,

60a3₁m + 300μa1a2λm3+ 360a0a1a2m + 40μa1λ3m5

+ 60a0a1λ2m3+ 180b1a22m + 180b1a2λ2m3= 0,

180a2₀a2m + 180a0a21m + 240a0a2λm3μ + 60a21λm3μ

+ 360b1a1a2m + 60b1a1λ2m3+ 60a22m3μ2

+ 180b2a22m + 136a2λ2m5μ2+ 240b2a2λ2m3

+ ka2= 0,

180a2₀a1m + 60a0a1λm3μ + 360a2b1a0m + 180b1a21m

+ 16a1λ2m5μ2+ 120b2a1λ2m3+ 60a2a1m3μ2

+ 360a2b2a1∗ m + ka1+ 300a2b1λm3μ = 0,

60a3₀m + 360a0a1b1m + 60b2a0λ2m3+ 60a2a0m3μ2

+ 360a2b2a0m + ka0+ 180b2a21m + 120a1b1λm3μ

+ 180a2b21m + 16b2λ3m5μ + 16a2λm5μ3

+ 480a2b2λm3μ = 0,

180a2₀b1m + 60a0b1λm3μ + 360a1b2a0m + 180a1b21m

+ 16b1λ2m5μ2+ 60b2b1λ2m3+ 120a2b1m3μ2

+ 360a2b2b1m + kb1+ 300a1b2λm3μ = 0,

180a2₀b2m + 180a0b21m + 240a0b2λm3μ + 60b21λm3μ

+ 360a1b1b2m + 60a1b1m3μ2+ 60b22λ2m3

+ 180a2b22m + 136b2λ2m5μ2

+ 240a2b2m3μ2+ kb2= 0,

60b3₁m + 300λb1b2m3μ + 360a0b1b2m + 40λb1m5μ3

+ 60a0b1m3μ2+ 180a1b22m + 180a1b2m3μ2= 0,

180b2₁b2m + 60b21m3μ2+ 240λb22m3μ + 180a0b22m

+ 240λb2m5μ3+ 180a0b2m3μ2= 0,

180b1b22m + 240b1b2m3μ2+ 24b1m5μ4= 0,

60b3₂m + 180b2₂m3μ2+ 120b2m5μ4= 0.

Solving the algebraic equations in the Mathematica 10.0, we obtain the following set of solutions:

Case 1: a0= 2 7 15λm2μ, a1= 0, a2= −λ2m2, b1= 0, b2= −m2μ2, k= −32(11λ2m5μ2+ √ 105λ2m5μ2) : Whenμλ > 0, u(x, t) = 2 7 15λm 2_{μ − λ}2_m2 × _μ λ C cos(√μλξ)+D sin(√μλξ) D cos(√μλξ)−C sin(√μλξ) 2 −m2_μ2 × _μ λ C cos(√μλξ) + D sin(√μλξ) D cos(√μλξ) − C sin(√μλξ) −2 . (25) Whenμλ < 0, u(x, t) = 2 7 15λm 2_{μ − λ}2_m2 × ⎛ ⎜ ⎜ ⎜ ⎝ |μλ| λ C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) + D C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) − D ⎞ ⎟ ⎟ ⎟ ⎠ 2 − m2_μ2 ⎛ ⎜ ⎜ ⎜ ⎝ |μλ| λ C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) + D C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) − D ⎞ ⎟ ⎟ ⎟ ⎠ −2 . (26)

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Figure 2.3D plot of the obtained traveling wave solutionu(x, t) of Equation (25) for α = 0.5, β = 0.75, λ = 0.05, μ = 0.2, m = 0.5, D = 1, C = 1, 0 ≤ x, t ≤ 50 0 ≤ x, t ≤ 20. Whenμ = 0, λ = 0, u(x, t) = 2 7 15λm 2_{μ − λ}2_m2 _C λ(Cξ + D) 2 − m2_μ2 _C λ(Cξ + D) ₋₂ . (27) Here ξ = −32(11λ2m5μ2+√105λ2m5μ2)(tα/α) + m(xβ_/β). Case 2: a₀_{= −2λm}2_{μ, a} 1= 0, a2= −λ2m2, b1= 0, b2= −m2μ2, k= −256λ2m5μ2: Whenμλ > 0, u(x, t) = −2λm2_{μ − λ}2_m2 × μ λ C cos(√μλξ)+D sin(√μλξ) D cos(√μλξ)−C sin(√μλξ) 2 −m2_μ2 × _μ λ C cos(√μλξ) + D sin(√μλξ) D cos(√μλξ) − C sin(√μλξ) −2 . (28) Whenμλ < 0, u(x, t) = −2λm2_{μ − λ}2_m2 × ⎛ ⎜ ⎜ ⎜ ⎝ |μλ| λ C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) + D C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) − D ⎞ ⎟ ⎟ ⎟ ⎠ 2 −m2_μ2 × ⎛ ⎜ ⎜ ⎜ ⎝ |μλ| λ C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) + D C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) − D ⎞ ⎟ ⎟ ⎟ ⎠ −2 . (29) Whenμ = 0, λ = 0, u(x, t) = −2λm2_{μ − λ}2_m2 _C λ(Cξ + D) 2 − m2_μ2 _C λ(Cξ + D) −2 . (30) Hereξ = −256λ2m5μ2(tα/α) + m(xβ/β).

u(x, t) in Equation (25) for α = 0.5, β = 0.75, λ = 0.05, μ = 0.2, m = 0.5, D = 1, C = 1.

5. Analytic solutions to the conformable space-time fractional (2 + 1)-dimensional CBS Equation

Conformable space-time fractional CBS equation is given in the following form: (see, for example, [21])

TxβTtαu + TxβTxβTxβTyθu + 4TxβuTxβTyθu + 2TxβTxβuTyθu = 0, 0< α ≤ 1, 0 < β ≤ 1, 0 < θ ≤ 1. (31) Using the following transformation

u(x, y, t) = U(ξ), ξ = kt_αα + mx_ββ + ny_θθ, (32) Equation (31) can be transformed into the following ordinary differential equation

kmU_{+ m}3_nU(4)_{+ 4m}2_nU_U_{+ 2m}2_nU_U_{= 0. (33)}

kmU_{+ m}3_nU_{+ 3m}2_n(U₎2_{= 0.} ₍₃₄₎

Let us suppose that the solution of Equation (34) can be expressed in the form of Equation (9). Substituting Equation (9) into Equation (34) and then by balancing

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Figure 3.3D plot of the obtained traveling wave solutionu(1, y, t) of Equation (37) for a0= 1, α = 0.5, β = 0.5, θ = 0.75, λ = −5,

μ = 1, m = 0.01, n = 0.5, D = 10, C = −10, 0 ≤ x, t ≤ 50.

the highest order derivative term and nonlinear term in result equation, the value of N can be determined as 1. Therefore, Equation (9) reduces to

U(ξ) = a0+ a1 _G G2 + b1 _G G2 −1 . (35)

Substituting Equation (35) into Equation (34), collecting all the terms with the same power of_GG2, we can obtain a set of algebraic equations for the unknowns a0, a1, b1, λ, μ, k, m, n:

3na2₁λ2m2+ 6na1λ3m3= 0,

6μna2₁λm2+ 8μna1λ2m3− 6b1na1λ2m2+ ka1λm=0,

3na2₁m2μ2− 12na1b1λm2μ + 2na1λm3μ2+ ka1mμ

+ 3nb2

1λ2m2− 2nb1λ2m3μ − kb1λm = 0,

6λnb2₁m2μ − 8λnb1m3μ2− 6a1nb1m2μ2−kb1mμ=0,

3nb2₁m2μ2− 6nb1m3μ3= 0.

Solving these algebraic equations in the Mathematica 10.0, we obtain the following set of solutions: a1=

−2λm, b1= 2mμ, k = 16λm2μn : Whenμλ > 0, u(x, y, t) = a0− 2λm μ λ C cos(√μλξ) + D sin(√μλξ) D cos(√μλξ) − C sin(√μλξ) + 2mμ ⎛ ⎜ ⎜ ⎜ ⎝ μ λ C cos(√μλξ) +D sin(√μλξ) D cos(√μλξ) −C sin(√μλξ) ⎞ ⎟ ⎟ ⎟ ⎠ −1 . (36) Whenμλ < 0, u(x, y, t) = a0− 2λm |μλ| λ C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) + D C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) − D + 2mμ ⎛ ⎜ ⎜ ⎜ ⎝ |μλ| λ C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ)+D C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ)−D ⎞ ⎟ ⎟ ⎟ ⎠ −1 . (37) Whenμ = 0, λ = 0, u(x, y, t) = a0+ 2λm _C λ(Cξ + D) − 2mμ _C λ(Cξ + D) −1 . (38) Hereξ = 16λm2μn(tα/α) + m(xβ/β) + n(yθ/θ). Figure3shows 3D plot of the traveling wave solu-tion u(x, 1, t) in Equation (37) for a0= 1, α = 0.5, β =

0.5, θ = 0.75, λ = −5, μ = 1, m = 0.01, n = 0.5, D = 10, C= −10.

6. Analytic solutions to the conformable space-time fractional (2 + 1)-dimensional AKNS equation

Finally, we consider conformable space-time fractional AKNS equation as follows (see, for example, [25])

4TxβTtαu + TxβTxβTxβTyθu + 8TxβTyθuTxβu + 4TyθuTxβTxβu − aTxβTxβu = 0,

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Figure 4.3D plot of the obtained traveling wave solutionu(x, 1, t) of Equation (45) for a0= 10, α = 0.25, β = 0.75, θ = 0.5,

λ = 2, μ = 0, m = 0.5, n = 5, D = 5, C = −6, 0 ≤ x, t ≤ 35.

where a is constant. Using the transformation (32) for Equation (39) we have the following ordinary differen-tial equation

4kmU+ m3nU(4)+ 8m2nUU

+ 4m2_nU_U_{− am}2_U_{= 0.} ₍₄₀₎

(4k − am)U+ m2_nU_{+ 6mn(U}₎2_{= 0.} ₍₄₁₎

Let us suppose that the solution of Equation (41) can be expressed in the form of Equation (9). Substituting Equation (9) into Equation (41) and then by balancing the highest order derivative term and nonlinear term in result equation, the value of N can be determined as 1. Therefore, Equation (9) reduces to

U(ξ) = a0+ a1 _G G2 + b1 _G G2 −1 . (42)

Substituting Equation (42) into Equation (41), collecting all the terms with the same power of_GG2, we can obtain a set of algebraic equations for the unknowns a0, a1, b1, λ, μ, k, m,n:

6na2₁λ2m + 6na1λ3m2= 0,

12μna2₁λm + 8μna1λ2m2− 12b1na1λ2m

− aa1λm + 4ka1λ = 0,

6na2₁mμ2− 24na1b1λmμ + 2na1λm2μ2

− aa1mμ + 4ka1μ + 6nb21λ2m − 2nb1λ2m2μ

+ ab1λm − 4kb1λ = 0,

12λnb2₁mμ − 8λnb1m2μ2− 12a1nb1mμ2

+ ab1mμ − 4kb1μ = 0,

6nb2₁mμ2− 6nb1m2μ3= 0.

Solving these algebraic equations in the Mathematica 10.0, we obtain the following set of solutions: a1=

−λm, b1= mμ, k = ((m(a + 16λmμn))/4) : Whenμλ > 0, u(x, y, t) = a0− λm × _μ λ C cos(√μλξ)+D sin(√μλξ) D cos(√μλξ)−C sin(√μλξ) +mμ × _μ λ C cos(√μλξ) + D sin(√μλξ) D cos(√μλξ) − C sin(√μλξ) −1 . (43) Whenμλ < 0, u(x, y, t) = a0+ λm × ⎛ ⎜ ⎜ ⎜ ⎝ |μλ| λ C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) + D C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) − D ⎞ ⎟ ⎟ ⎟ ⎠ −mμ ⎛ ⎜ ⎜ ⎜ ⎝ |μλ| λ C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) + D C sinh(2√|μλ|ξ) +C cosh(2√|μλ|ξ) − D ⎞ ⎟ ⎟ ⎟ ⎠ −1 . (44) Whenμ = 0, λ = 0, u(x, y, t) = a0+ m _C (Cξ + D) − mμ _C λ(Cξ + D) −1 . (45) Here ξ = ((m(a + 16λmμn))/4)(tα/α) + m(xβ/β) + n(yθ_/θ).

u(x, 1, t) in Equation (45) for a0= 10, α = 0.25, β =

0.75, θ = 0.5, λ = 2, μ = 0, m = 0.5, n = 5, D = 5,

(9)

7. Conclusion

In this article, G/G2 expansion method has been applied to obtain new exact solutions of the con-formable space-time fractional third-order KdV, CDG, (2+1)-dimensional CBS and (2+1)-dimensional AKNS equations. The exact solutions include hyperbolic, trigonometric, exponential and rational functions. Note that the obtained solutions are new form of the solu-tions and are not available in the literature. In this work, we have solved only four conformable nonlin-ear fractional differential equations. This method is use-ful in solving wide classes of conformable nonlinear fractional differential equations such as Sharma-Tasso-Olever, Zakharov Kuznetsov, Benjamin Bona Mahony, Boussinesq, Jimbo-Miwa, Burger equations.

Disclosure statement

No potential conflict of interest was reported by the authors. ORCID

A. Girgin http://orcid.org/0000-0002-2972-7583 References

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