Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Nonlinear Engineering

Modeling and Application

Editor-in-Chief: Nakahie Jazar, Gholamreza

Managing Editor: Skoryna, Juliusz


CiteScore 2017: 0.69

SCImago Journal Rank (SJR) 2017: 0.181
Source Normalized Impact per Paper (SNIP) 2017: 0.338

Open Access
Online
ISSN
2192-8029
See all formats and pricing
More options …

A note on Riccati-Bernoulli Sub-ODE method combined with complex transform method applied to fractional differential equations

Mahmoud A.E. Abdelrahman
Published Online: 2018-07-12 | DOI: https://doi.org/10.1515/nleng-2017-0145

Abstract

In this paper, the fractional derivatives in the sense of modified Riemann–Liouville and the Riccati-Bernoulli Sub-ODE method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear fractional Zoomeron equation and the (3 + 1) dimensional space-time fractional mKDV-ZK equation. These nonlinear fractional equations can be turned into another nonlinear ordinary differential equation by complex transform method. This method is efficient and powerful in solving wide classes of nonlinear fractional order equations. The Riccati-Bernoulli Sub-ODE method appears to be easier and more convenient by means of a symbolic computation system.

Keywords: Modified Riemann-Liouville derivative; Riccati-Bernoulli Sub-ODE method; exact solution; fractional Zoomeron equation; (3 + 1) dimensional space-time fractional mKDV-ZK equation

MSC 2010: 26A33; 34A08; 35A99; 35R11; 83C15; 65Z05

1 Introduction

Many physical phenomena such as mathematical biology, signal processing, optics, fluid mechanics, electromagnetic theory, etc., can be modeled using the fractional derivatives. Consequently, the investigation of exact solutions for FDEs turns out to be very useful in the study of scientific research. Moreover generalized forms of differential equations are described as fractional differential equations FDEs.

Recent past, an strong attention has been purposed by the researchers concerning the fractional partial differential equations FDEs. For an interesting overview and more applications of nonlinear FDEs, we refer to [1, 2, 3, 4].

However, even in most useful studies, there is no an efficient and general methods to solve them. Actually, various analytical and numerical methods to construct approximate and exact solutions of nonlinear FDEs have been put forward, such as the fractional sub-equation method [5, 6], the tanh-sech method [7], the (GG) expansion method [8, 19], the first integral method [10], the modified Kudryashov method [11], the exponential function method [12, 13] and others [14, 15].

The novelties of this paper are mainly exhibited in two aspects: First, we introduce a new method, which is not familiar, the so called Riccati-Bernoulli Sub-ODE method. We use this method to solve the nonlinear fractional Zoomeron equation and the (3 + 1) dimensional space-time fractional mKDV-ZK equation. Moreover, we show that the proposed method gives infinite sequence of solutions. Second, we obtain new types of exact analytical solutions. Moreover comparing our results with other results, one can see that our results are new and most extensive.

Actually, the proposed two fractional equations have many applications in various fields of theoretical physics, applied mathematics and engineering such as control theory viscoelasticity, modelling heat transfer, control, diffusion, signal and image processing, and many other physical and engineering processes. In more details, the (3 + 1) dimensional space-time fractional mKDV-ZK equation is derived for a plasma comprised of cool and hot electrons and a species of fluid ions, which have so many direct and indirect in engineering models. Furthermore the nonlinear fractional Zoomeron equation is a convenient model to display the novel phenomena associated with boomerons and trappons and further interesting engineering applications.

The Riccati-Bernoulli Sub-ODE technique has been used to solve some partial and fractional differential equations, see for example [16, 17, 18, 19, 20, 21]. These works show that this method is efficacious, robust and adequate for solving further equations.

The rest of the paper is arranged as follows: In Section 2, we recall some basic definitions and notions dealing with fractional calculus theory, which are used in the sequel in this article. In Sections 4 and 3, two examples, namely the nonlinear fractional Zoomeron equation and the (3 + 1) dimensional space-time fractional mKDV-ZK equation, are solved by the Riccati-Bernoulli Sub-ODE method. Conclusion will appear in Section 5.

2 Preliminaries and notation

We introduce some fundamental definitions and properties of fractional calculus theory, which turn to be very useful in order to complete this paper in a completely unified way. These are the Riemann–Liouville, the Grünwald-Letnikov, the Caputo and the modified Riemann-Caputo, Liouville derivative. The most commonly used definitions are the modified Riemann–Liouville and Caputo derivatives [22, 23]. Jumarie proposed a modified Riemann–Liouville derivative [24]. Firstly, we present some properties and definitions of the modified Riemann–Liouville derivative. Secondly, we give the description of the Riccati-Bernoulli Sub-ODE method.

Assume that f(t) denotes a continuous ℝ → ℝ function (but not necessarily first-order differentiable). The Jumarie’s modified Riemann–Liouville derivative is defined as

Dtαf(t)=1Γ(1α)ddt0t(tξ)α1(f(ξ)f(0))dξ,α<01Γ(1α)ddt0t(tξ)α(f(ξ)f(0))dξ,0<α<1(f(n)(t))(αn),nα<n+1,n1.,(2.1)

Important property of the fractional modified Riemann-Liouville derivative is [25]

Dtαtr=Γ(1+r)Γ(1+rα)trα.(2.2)

  • Step 1

    Any nonlinear fractional differential equation in two independent variables x and t can be expressed in following form:

    G(u,Dtαu,Dxαu,DtαDxαu,DxαDxαu,...)=0,(2.3)

    where 0 < α ≤ 1, Dtαu,Dxαu are modified Riemann-Liouville derivative of u and G is a polynomial in u(x, t) and its partial fractional derivatives.

  • Step 2

    Using the traveling wave transformation

    u(x,t)=U(ξ),ξ=kxαΓ(1+α)λtαΓ(1+α),(2.4)

    where k, λ are non zero constants and 1 < α≤ 1. By using the chain rule,

    Dtαu=σtdUdξDtαξ,Dxαu=σxdUdξDxαξ,(2.5)

    where σtandσx are called the sigma indexes, [26], without loss of generality, we can take σt=σx=L, where L is a constant.

    Superseding (2.4) with (2.2) and (2.5) into (2.3), the equation (2.3) transformed into the following ODE:

    H(U,U,U,U,.....)=0,(2.6)

    where prime denotes the derivation with respect to ξ.

  • Step 3

    Based on the Riccati-Bernoulli Sub-ODE method [16, 17, 18], we assume that equation (2.6) has the following solution:

    U=aU2n+bU+cUn,(2.7)

    where a, b, c and n are constants calculated later. From equation (2.7), we have

    U=ab(3n)U2n+a2(2n)U32n+nc2U2n1+bc(n+1)Un+(2ac+b2)U,(2.8)

    U=(ab(3n)(2n)U1n+a2(2n)(32n)U22n+n(2n1)c2U2n2+bcn(n+1)Un1+(2ac+b2))U.(2.9)

The exact solutions of equation (2.7), for an arbitrary constant μ are given as follow:

  1. For n = 1, the solution is

    U(ξ)=μe(a+b+c)ξ.(2.10)

  2. For n ≠ 1, b = 0 and c = 0, the solution is

    U(ξ)=a(n1)(ξ+μ)1n1.(2.11)

  3. For n ≠ 1, b ≠ 0 and c = 0, the solution is

    U(ξ)=ab+μeb(n1)ξ1n1.(2.12)

  4. For n ≠ 1, a ≠ 0 and b2 – 4ac < 0, the solution is

    U(ξ)=b2a+4acb22atan(1n)4acb22(ξ+μ)11n(2.13)

    and

    U(ξ)=b2a4acb22acot(1n)4acb22(ξ+μ)11n.(2.14)

  5. For n ≠ 1,a ≠ 0 and b2 – 4ac > 0, the solution is

    U(ξ)=b2ab24ac2acoth(1n)b24ac2(ξ+μ)11n(2.15)

    and

    U(ξ)=b2ab24ac2atanh(1n)b24ac2(ξ+μ)11n.(2.16)

  6. For n ≠ 1, a ≠ 0 and b2 – 4ac = 0, the solution is

    U(ξ)=1a(n1)(ξ+μ)b2a11n.(2.17)

2.1 Bäcklund transformation

When Um–1(ξ) and Um(ξ)(Um(ξ) = Um(Um–1(ξ))) are the solutions of equation (2.7), we have

dUm(ξ)dξ=dUm(ξ)dUm1(ξ)dUm1(ξ)dξ=dUm(ξ)dUm1(ξ)(aUm12n+bUm1+cUm1n),

namely

dUm(ξ)aUm2n+bUm+cUmn=dUm1(ξ)aUm12n+bUm1+cUm1n.(2.18)

Integrating equation (2.18) once with respect to ξ, we get a Bäcklund transformation of equation (2.7) as follows:

Um(ξ)=cK1+aK2Um1(ξ)1nbK1+aK2+aK1Um1(ξ)1n11n,(2.19)

where K1 and K2 are arbitrary constants. We use equation (2.19) to obtain infinite sequence of solutions for equation (2.7), as well for equation (2.3).

3 The nonlinear fractional Zoomeron equation

We are concerned with the nonlinear fractional Zoomeron equation([27]),

Dtt2αuxyuuxyuxx+2Dtαu2x=0,0<α1,(3.1)

where u(x, y, t) is the amplitude of the relevant wave mode.

Using the transformation

u(x,y,t)=U(ξ),(3.2)

ξ=lx+γywtαΓ(1+α),(3.3)

where l, γ and w are non zero constants and 0 < α ≤ 1.

Substituting (3.3) with (2.2) and (2.5) into (3.1), we have the ODE

lγw2UUγl3UU2lw(U2)=0.(3.4)

Integrating this equation twice, with the second constant of integration is vanishing, we obtain

lγ(w2l2)U2lwU3kU=0,(3.5)

where k is a nonzero constant of integration.

Substituting equations (2.8) into equation (3.5), we obtain

lγ(w2l2)ab(3m)U2m+a2(2m)U32m+mc2U2m1+bc(m+1)Um+(2ac+b2)U2lwU3kU=0.(3.6)

Setting m = 0, equation (3.6) is reduced to

lγ(w2l2)(3abU2+2a2U3+bc+(2ac+b2)U)2lwU3kU=0.(3.7)

Equating each coefficient of Ui (i = 0, 1, 2, 3) to zero, we have

lγ(w2l2)bc=0,(3.8)

lγ(w2l2)(2ac+b2)k=0,(3.9)

3lγ(w2l2)ab=0,(3.10)

2lγ(w2l2)a22lw=0.(3.11)

Solving equations (3.8)-(3.11), we get

b=0,(3.12)

ac=k2lγ(w2l2),(3.13)

a=±wγ(w2l2),(3.14)

Trigonometric function solutions

When klγ(w2l2)<0, substituting equations (3.12)-(3.14) and (3.3) into equations (2.13) and (2.14), we get the exact solutions for equation (3.1),

U1,2(x,y,t)=±k2wltank2lw(w2l2)(lx+γywtαΓ(1+α)+μ)(3.15)

and

U3,4(x,y,t)=±k2wlcotk2lw(w2l2)(lx+γywtαΓ(1+α)+μ),(3.16)

where l, k, γ, w, μ are arbitrary constants and 0 < α ≤ 1. Figure 1 illustrated the solution U2.

The solution U1(x, 0, t) in (3.15) for l=1.5, k=1, a=1, μ=0, w = 2 and –5 ≤ t, x ≤ 5.
Fig. 1

The solution U1(x, 0, t) in (3.15) for l=1.5, k=1, a=1, μ=0, w = 2 and –5 ≤ t, x ≤ 5.

Hyperbolic function solutions

When klγ(w2l2)>0, substituting equations (3.12)-(3.14) and (3.3) into equations (2.15) and (2.16), we obtain exact solutions for equation (3.1),

U5,6(x,y,t)=±k2wltanhk2lw(l2w2)(lx+γywtαΓ(1+α)+μ)(3.17)

and

U7,8(x,y,t)=±k2wltanhk2lw(l2w2)(lx+γywtαΓ(1+α)+μ),(3.18)

where l, k, γ, w, μ are arbitrary constants and 0 < α ≤ 1. Figure 2 illustrated the solution U5.

The solution U5(x, 0, t) in (3.17) for l=2, k=2, a=1, μ=1, w = 3.5 and –5 ≤ t, x ≤ 5.
Fig. 2

The solution U5(x, 0, t) in (3.17) for l=2, k=2, a=1, μ=1, w = 3.5 and –5 ≤ t, x ≤ 5.

Remark 3.1

Applying equation (2.19) to ui(x, t), i =1,2,…,8, we obtain an infinite sequence of solutions of equation (3.1). For illustration, by applying equation (2.19) to ui(x, t), i =1,2,…,8, once, we have new solutions of equation (3.1)

u1,2(x,t)=k2lw±A3k2wltank2lw(w2l2)(lx+γywtαΓ(1+α)+μ)A3±k2wltank2lw(w2l2)(lx+γywtαΓ(1+α)+μ),(3.19)

u3,4(x,t)=k2lw±A3k2wlcotk2lw(w2l2)(lx+γywtαΓ(1+α)+μ)A3±k2wlcotk2lw(w2l2)(lx+γywtαΓ(1+α)+μ),(3.20)

u5,6(x,t)=k2lw±A3k2wltanhk2lw(l2w2)(lx+γywtαΓ(1+α)+μ)A3±k2wltanhk2lw(l2w2)(lx+γywtαΓ(1+α)+μ),(3.21)

u7,8(x,t)=k2lw±A3k2wlcothk2lw(l2w2)(lx+γywtαΓ(1+α)+μ)A3±k2wlcothk2lw(l2w2)(lx+γywtαΓ(1+α)+μ),(3.22)

where A3, l, k, γ, w, μ are arbitrary constants and 0 < α ≤ 1.

4 The (3 + 1) dimensional space-time fractional mKDV-ZK equation

The second equation is the (3 + 1) dimensional space-time fractional mKDV-ZK equation which has the form ([28])

Dtαu+lu2Dxαu+Dx3αu+DxαDy2αu+DxαDz2αu=0,,t>0,0<α1,(4.1)

where l is an nonzero constant and 0 < α ≤ 1. The mKdV equation is used for representing physical and engineering phenomena such as to describe the ion-acoustic waves in a magnetized plasma, dipole blocking and study of coastal waves in ocean etc., see e.g. [29, 30, 31].

Using the transformation

u(x,y,z,t)=U(ξ),(4.2)

ξ=βxαΓ(1+α)+γyαΓ(1+α)+δzαΓ(1+α)wtαΓ(1+α),(4.3)

where β, γ, δ and w are non zero constants and 0 < α ≤ 1.

Substituting (4.3) with (2.2) and (2.5) into (4.1), we have the ODE

(β3+βγ2+βδ2)U+lβU2UwU=0,(4.4)

Integrating equation (4.4) once with respect to ξ with the zero constant of integration, we have

(β3+βγ2+βδ2)U+lβ3U3wU=0.(4.5)

Substituting equations (2.8) into equation (4.5), we obtain

(β3+βγ2+βδ2)ab(3m)U2m+a2(2m)U32m+mc2U2m1+bc(m+1)Um+(2ac+b2)U+lβ3U3wU=0.(4.6)

Putting m = 0, equation (4.6) becomes

(β3+βγ2+βδ2)(3abU2+2a2U3+bc+(2ac+b2)U)+lβ3U3wU=0.(4.7)

Putting each coefficient of Ui(i = 0, 1, 2, 3) to zero, we obtain

(β3+βγ2+βδ2)bc=0,(4.8)

(β3+βγ2+βδ2)(2ac+b2)w=0,(4.9)

3(β3+βγ2+βδ2)ab=0,(4.10)

2a2(β3+βγ2+βδ2)+lβ3=0.(4.11)

Solving equations (4.8)-(4.11), we get

b=0,(4.12)

ac=w2(β3+βγ2+βδ2),(4.13)

a=±l6(β2+γ2+δ2),(4.14)

Trigonometric function solutions

When wβ3+βγ2+βδ2<0, substituting equations (4.12)-(4.14) and (4.3) into equations (2.13) and (2.14), we obtain the exact solutions of equation (4.1),

U~1,2(x,y,z,t)=±3wβltanw2(β3+βγ2+βδ2)βxαΓ(1+α)+γyαΓ(1+α)+δzαΓ(1+α)wtαΓ(1+α)+μ(4.15)

and

U~3,4(x,y,z,t)=±3wβlcotw2(β3+βγ2+βδ2)βxαΓ(1+α)+γyαΓ(1+α)+δzαΓ(1+α)wtαΓ(1+α)+μ,(4.16)

where β, γ, δ, w and μ are non zero constants and 0 < α ≤ 1. Figure 3 illustrated the solution Ũ1.

The solution Ũ1(x, 0, 0, t) in (4.15) for β=-2.6, γ=-3.5, δ=3,l=-2, α=1, μ=0, w = -2 and –5 ≤ t, x ≤ 5.
Fig. 3

The solution Ũ1(x, 0, 0, t) in (4.15) for β=-2.6, γ=-3.5, δ=3,l=-2, α=1, μ=0, w = -2 and –5 ≤ t, x ≤ 5.

Hyperbolic function solutions

When wβ3+βγ2+βδ2>0, substituting equations (3.12)-(3.14) and (4.3) into equations (2.15) and (2.16), we obtain exact solutions of equation (4.1),

U~5,6(x,y,z,t)=±3wβltanhw2(β3+βγ2+βδ2)βxαΓ(1+α)+γyαΓ(1+α)+δzαΓ(1+α)wtαΓ(1+α)+μ(4.17)

and

U~7,8(x,y,z,t)=±3wβlcothw2(β3+βγ2+βδ2)βxαΓ(1+α)+γyαΓ(1+α)+δzαΓ(1+α)wtαΓ(1+α)+μ,(4.18)

where β, γ, δ, w and μ are non zero constants and 0 < α ≤ 1. Figure 4 illustrated the solution Ũ5.

The solution Ũ5(x, 0, 0, t) in (4.17) for β=2.6, γ=-3.5, δ=3,l=-2, α=0.5, μ=0, w = 3 and –5 ≤ t, x ≤ 5.
Fig. 4

The solution Ũ5(x, 0, 0, t) in (4.17) for β=2.6, γ=-3.5, δ=3,l=-2, α=0.5, μ=0, w = 3 and –5 ≤ t, x ≤ 5.

Remark 4.1

Similarly as shown in Remark (3.1), we can give an infinite solutions of equation (4.1).

Remark 4.2

  1. Comparing our results concerning equation (3.1) with the results in [27, 32], one can see that our results are new and most extensive.

  2. Comparing our results concerning equation (4.1) with the results in [31, 33], one can see that our results are new and most extensive.

  3. Comparing our solutions for equations (3.1) and (4.1) with [27, 31, 32, 33], it can be seen that by choosing suitable values for the parameters similar solutions can be verified.

  4. Actually the Riccati-Bernoulli Sub-ODE technique has a very important feature, that admits infinite sequence of solutions of equation, which is explained clearly in Section 2.1. In fact this feature has never given for any another method.

  5. Consequently, the method is efficacious, robust and adequate for solving other type of space-time fractional differential equations.

5 Conclusions

In this work, a Riccati-Bernoulli Sub-ODE technique has successfully been applied to exact solutions for the nonlinear fractional Zoomeron equation and the (3 + 1) dimensional space-time fractional mKDV-ZK equation with modified Riemann–Liouville derivative. Fractional complex transform is also used as the basic ingredient to obtain exact solutions for these nonlinear equations. As a result, some new exact solutions for them have successfully been obtained. The graphs of some solutions are depicted for suitable coefficients. Actually this method can be applied for many other nonlinear FDEs appearing in mathematical physics and natural sciences.

Acknowledgement

The author wants to thank the editor and reviewers for valuable comments.

References

  • [1]

    K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley: New York, 1993. Google Scholar

  • [2]

    I.Podlubny, Fractional Differential Equations. Academic Press: California, (1999). Google Scholar

  • [3]

    A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier: Amsterdam, (2006). Google Scholar

  • [4]

    I. Podlubny, Fractional Differential Equations. Academic Press: San Diego, (1999). Google Scholar

  • [5]

    S. Zhang and H-Q. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs. Physics Letters A, 375(2011), 1069-1073. Web of ScienceCrossrefGoogle Scholar

  • [6]

    B. Tong, Y. He, L. Wei and X. Zhang, A generalized fractional sub-equation method for fractional differential equations with variable coefficients. Physics Letters A, 376(2012), 2588-2590. CrossrefWeb of ScienceGoogle Scholar

  • [7]

    S. Saha Ray and S. Sahoo, A novel analytical method with fractional complex transform for new exact solutions of time-fractional fifth-order Sawada-Kotera equation, Reports on Math. Phys., 75(1) (2015), 63-72. Web of ScienceCrossrefGoogle Scholar

  • [8]

    N. Shang and B. Zheng, Exact Solutions for Three Fractional Partial Differential Equations by the GG Method, Int. J. of Appl. Math., 43(3) 2013, 1-6. Google Scholar

  • [9]

    B. Zheng, GG-expansion method for solving fractional partial differential equations in the theory of mathematical physics. Communications in Theoretical Physics, 58 (2012), 623-630. CrossrefGoogle Scholar

  • [10]

    B. Lu, The first integral method for some time fractional differential equations. Journal of Mathematical Analysis and Applications 395 (2012), 684-693. Web of ScienceCrossrefGoogle Scholar

  • [11]

    S.M. Ege and E. Misirli, The modified Kudryashov method for solving some fractional-order nonlinear equations, Advances in Difference Equations, 2014, 2014, 135. CrossrefWeb of ScienceGoogle Scholar

  • [12]

    S. Zhang, Q-A. Zong, D. Liu and Q. Gao, A generalized exp-function method for fractional riccati differential equations. Communications in Fractional Calculus 1(1) (2010), 48-51. Google Scholar

  • [13]

    A. Bekir, o. Guner and A.C. Cevikel, The fractional complex transform and exp-function methods for fractional differential equations, Abstr. and Appl. Anal., 2013, 2013 426-462. Google Scholar

  • [14]

    Z.Z. Ganjia, D.D. Ganjia and Y. Rostamiyan, Solitary wave solutions for a time-fraction generalized Hirota-Satsuma coupled KdV equation by an analytical technique, Applied Mathematical Modelling 33 (2009), 3107-3113. Web of ScienceCrossrefGoogle Scholar

  • [15]

    W. Liu and K. Chen, The functional variable method for finding exact solutions of some nonlinear time-fractional differential equations, Pramana J. Phys., 81(2013), 377-384. Web of ScienceCrossrefGoogle Scholar

  • [16]

    M.A.E. Abdelrahman and M.A. Sohaly, Solitary Waves for the Modified Korteweg-De Vries Equation in Deterministic Case and Random Case. J Phys Math. 8(1) (2017), []. CrossrefGoogle Scholar

  • [17]

    M.A.E. Abdelrahman and M.A. Sohaly, Solitary waves for the nonlinear Schrödinger problem with theprobability distribution function in the stochastic input case, Eur. Phys. J. Plus, (2017) 132: 339. CrossrefGoogle Scholar

  • [18]

    X.F. Yang, Z.C. Deng and Y. Wei, A Riccati-Bernoulli sub-ODE method for nonlinear partial differential equations and its application, Adv. Diff. Equa. 1 (2015) 117-133. Google Scholar

  • [19]

    B. Zheng, A new Bernoulli sub-ODE method for constructing traveling wave solutions for two nonlinear equations with any order, U. P. B. Sci. Bull., Series A, 73(3), (2011). Google Scholar

  • [20]

    M. Inc, A. I. Aliyu and A. Yusuf, Traveling wave solutions and conservation laws of some fifth-order nonlinear equations. Eur. Phys. J. Plus, 132 (2017) 224. CrossrefWeb of ScienceGoogle Scholar

  • [21]

    F. Xu and Q. Feng, A generalized sub-ODE method and applications for nonlinear evolution equations. J. Sci. Res. Report 2 (2013), 571-581. Google Scholar

  • [22]

    M. Caputo, Linear models of dissipation whose Q is almost frequency independent II, Geophys. J. Royal Astronom. Soc. 13 (1967), 529-539. CrossrefGoogle Scholar

  • [23]

    S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, (1993). Google Scholar

  • [24]

    G. Jumarie, Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions further results, Comput. Math. Appl. 51 (2006), 1367-1376. CrossrefGoogle Scholar

  • [25]

    G. Jumarie, Table of some basic fractional calculus formulae derived from a modified Riemann–Liouvillie derivative for nondifferentiable functions. Applied Mathematics Letters 22 (2009), 378-385. CrossrefGoogle Scholar

  • [26]

    J. H. He, S. K. Elegan and Z. B. Li, Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Phys. Lett. A 376 (2012), 257-259. CrossrefWeb of ScienceGoogle Scholar

  • [27]

    O. Guner, A. Bekir and H. Bilgil, A note on exp-function method combined with complex transform method applied to fractional differential equations, Adv. Nonlinear Anal., 4(3) (2015), 201-208. Web of ScienceGoogle Scholar

  • [28]

    R.L. Mace and M.A. Hellberg, The Korteweg-de Vries-Zakharov-Kuznetsov equation for electron-acoustic waves, Phys. Plasmas 8(6) (2001), 2649-2656. CrossrefGoogle Scholar

  • [29]

    F. Demontis, Exact solutions of the modified Korteweg–de Vries, Theoret. Math. Phys., 168 (1) (2011), 886-897. CrossrefGoogle Scholar

  • [30]

    I. Aslan, Exact solutions of a fractional-type differential–difference equation related to discrete MKdV equation, Commun. Theor. Phys. 61 (2014), 595-599. CrossrefGoogle Scholar

  • [31]

    S. Sahoo and S.S. Ray, Improved fractional sub-equation method for (3 + 1)-dimensional generalized fractional KdV-Zakharov-Kuznetsov equations, Comput. Math. Appl. 70 (2015), 158-166. Web of ScienceCrossrefGoogle Scholar

  • [32]

    Marwan Alquran and Kamel Al-khaled, Mathematical methods for a reliable treatment of the (2 + 1)-dimensional Zoomeron equation, Math. Sci. 6 (12) (2012). Google Scholar

  • [33]

    O. Guner, E. Aksoy, A. Bekir and A.C. Cevikel, Different methods for (3+1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation. Comput Math Appl. 71 (2016), 1259-1269. CrossrefWeb of ScienceGoogle Scholar

About the article

mahmoud.abdelrahman1983@gmail.com, Tel +2 050 2242388; Fax +20 050 2246781


Received: 2017-10-23

Revised: 2017-12-12

Accepted: 2018-01-13

Published Online: 2018-07-12

Published in Print: 2018-12-19


Citation Information: Nonlinear Engineering, Volume 7, Issue 4, Pages 279–285, ISSN (Online) 2192-8029, ISSN (Print) 2192-8010, DOI: https://doi.org/10.1515/nleng-2017-0145.

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.Get Permission

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Mahmoud A. E. Abdelrahman and M. A. Sohaly
Indian Journal of Physics, 2018

Comments (0)

Please log in or register to comment.
Log in