A New Approach to (3+1) Dimensional Boiti–Leon–Manna–Pempinelli Equation

  • 1 Faculty of Educational Sciences, Kyrenia, Turkey
  • 2 Department of Mathematics and Science Education, Faculty of Education, Turkey
Gülnur Yel and Tolga Aktürk
  • Corresponding author
  • Department of Mathematics and Science Education, Faculty of Education, Ordu University, Turkey
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Abstract

In this article, some new travelling wave solutions of the (3+1) dimensional Boiti–Leon–Manna–Pempinelli (BLMP) equation are obtained using the modified exponential function method. When the solution functions obtained are examined, it is seen that functions with periodic functions are obtained. Two and three dimensional graphs of the travelling wave solutions of the BLMP equation are drawn by selecting the appropriate parameters

1 Introduction

Nonlinear partial differential equations (NPDE) have an important role to describe natural phenomenon from biology to engineering. Especially in engineering, acoustic waves, water waves, electromagnetic waves have been model via NPDE. Physics and engineering applications have concentrated to the behavior of waves, for this reason solutions of such equations have attracted the attention of many scientists for many years. Hence, there are various analytical methods in the literature used by researchers to obtain solutions for such equations. Some of these are the multipliers method [1], the simplest equation method [2], the (G′/G)-expansion method [3, 4, 5, 6], the Sine-Gordon expansion method [7, 8, 9, 10, 11], the extended trial equation method [12, 13], the new function method [14, 15].

In this study, we used the Modified Exponential Function Method (MEFM) to the (3+1) dimensional Boiti-Leon-Manna-Pempinelli equation (BLMP) which is used to describe incompressible liquid in fluid mechanics. The equation is given as,

υyt+υzt+υxxxy+υxxxz3υx(υxy+υxz)3υxx(υy+υz)=0.

Eq.(1) is derived from (2+1)-dimensional Boiti–Leon–Manna–Pempinelli equation by Darvishi et al. [16]. They have submitted multisoliton solutions of both equations. There are several studies on the Boiti–Leon–Manna–Pempinelli equation in the literature. Authors have investigated the Lax pair of Eq.(1) by singular manifolds method [17]. Employing Hirota’s bilinear method different types of lump solitons of Eq.(1) have been submitted in [18]. They have disscused Nth-order soliton solutions, rational solutions, periodic wave solutions using Pffafian tecnique, the ansatz method and the Hirota-Riemann method respectively [19]. Further researchs can be seen [?, 2, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33].

This paper is rested as manner; we give steps of the modified exponential function method in section 2, then an application of the mention method is given in section 3. In last section 4, we give some conclusions on the obtained wave solutions.

2 The Manner of the Method

In this section, we give the manner of the MEFM [34, 35, 36].

Let’s consider the following general form of nonlinear partial differential equation;

P(u,ux,uy,uz,uxy,uyt,uxxz,uxxxz,)=0,
where u = u(x,y,z,t)is unknown solution function.

Step 1. Regarding travelling wave transformation as follows;

u(x,t)=U(ζ),ζ=x+y+zct,
where c is a non-zero real value, required derivative terms are substituted into Eq. (2). By this way, the following nonlinear ordinary differential equation is obtained,
N(U,U,U,)=0.

Step 2: We think the solution function Uin Eq. (4) as follows;

U(ζ)=i=0NAi[exp(Ω(ζ))]ij=0MBj[exp(Ω(ζ))]j=A0+A1exp(Ω)++ANexp(N(Ω))B0+B1exp(Ω)++BMexp(M(Ω)),
where Ai,Bj,(0 ≤ iN, 0 ≤ jM) are constants. Using the balancing principle, a relationship is set between the upper limits of Eq. (5), M and N values. (Balancing principle; it is obtained by equalizing the term containing the highest order derivative and the highest degree nonlinear term). AN ≠ 0, BM ≠ 0, and Ω = Ω (ζ) provides the following differential equation;
Ω(ζ)=exp(Ω(ζ))+μexp(Ω(ζ))+λ.

When the Eq. (6) is solved, the following solution families are obtained [37].

Family 1: When µ ≠ 0, λ2 − 4µ > 0,

Ω(ζ)=ln(λ24μ2μtanh(λ24μ2(ζ+E))λ2μ).

Family 2: When µ ≠ 0, λ2 − 4µ < 0,

Ω(ζ)=ln(λ2+4μ2μtan(λ2+4μ2(ζ+E))λ2μ).

Family 3: When µ = 0, λ ≠ 0, and λ2 − 4µ > 0,

Ω(ζ)=ln(λexp(λ(ζ+E))1).

Family 4: When µ ≠ 0, λ ≠ 0, and λ2 − 4µ = 0,

Ω(ζ)=ln(2λ(ζ+E)+4λ2(ζ+E)).

Family 5: When µ = 0, λ = 0, and λ2 − 4µ = 0,

Ω(ζ)=ln(ζ+E).
where A0,A1,...,AN,B0,B1,..., BM,E,λ, µ are constants.

Step 3: The Eq.(6) and the solution families are written into the Eq. (5) to obtain the algebraic equation system consisting of exp (Ω(ζup)).

The equation systems obtained are equalized to zero and A0,A1,...,AN,B0,B1,...,BM,E,λ, µ are obtained. These coefficients are written in Eq. (5) instead of Eq. (2) to provide the travelling wave solutions.

3 Applications

By using the wave transformation in equation (3) to equation (1), the following nonlinear differential equation is obtained;

cυ+υıv6υυ=0.

If the equation (12) is integrated,

cυ+υ3(υ)2=0.

It is described in a simpler way as follows by applying the transformation υ = ω to the nonlinear ordinary differential equation (13).

cω+ω3(ω)2=0.

When the balancing principle is applied to the equation (14), the following relation is found between M and N,

N=M+2.
N = 3 for M = 1 and values to Eq.(5), ω,ω can be written as follows,
ω(ζ)=ψφ=A0+A1eΩ(ζ)+A2e2Ω(ζ)+A3e3Ω(ζ)B0+B1eΩ(ζ),ω(ζ)=ψφψφφ2,ω(ζ)=ψφ3φ2ψφ(ψφ+ψφ)φ2+2(ψ)2ψφφ4.

Substituting Eq. (15) into Eq. (14), ω related solution functions are obtained. By integrating these solutions, the travelling wave solutions providing the equation (1) were obtained as follows.

CASE 1:

A0=2μB0,A1=2(λB0+μB1),A2=2(B0+λB1),A3=2B1,c=λ24μ.

Using the coefficients given above, the following solution families are obtained.

Family 1:

υ1,1(x,y,z,t)=λ34λμ+2λ24μμSinh[τ]λ22μ+2μCosh[τ],
where τ(x,y,z,t)=(EE+ζ)λ24μ.

Figure 1
Figure 1

The 3D and t = 1 for 2D of Eq. (17)

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00029

Family 2:

υ1,2(x,y,z,t)=λ3+4λμ+2μλ2+4μSin[ξ]λ22μ+2μCos[ξ],
where ξ(x,y,z,t)=(EE+ζ)λ2+4μ.

Figure 2
Figure 2

The 3D and t = 1 for 2D of Eq. (18)

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00029

Family 3:

υ1,3(x,y,z,t)=2λ1+eλ(EE+ζ),

Figure 3
Figure 3

The 3D and t = 1 for 2D of Eq.(19)

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00029

CASE 2:

A0=13(c6μ)B0,A1=2c+4μB013(c6μ)B1,A2=2(B0c+4μB1),A3=2B1,λ=c+4μ.

Equations (20) the following solution families are obtained from equations.

Family 1:

υ2,1(x,y,z,t)=3cc+4μc2(EE+ζ)+2cμ(EE+ζ)(1+Cos[κ])6cμSin[κ]3(c2μ2μCos[κ]),
where κ(x,y,z,t)=(EE+ζ)c.

Figure 4
Figure 4

The 3D andt = 1 for 2D of Eq. (21)

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00029

Family 2:

υ2,1(x,y,z,t)=3cc+4μc2(EE+ζ)+2cμ(EE+ζ)(1+Cos[κ])6cμSin[κ]3(c2μ2μCos[κ]),
where κ(x,y,z,t)=(EE+ζ)c.

Figure 5
Figure 5

The 3D and t = 1 for 2D of Eq. (22)

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00029

Family 3:

υ2,3(x,y,z,t)=(2c1+ec(EE+ζ)cζ3),

Figure 6
Figure 6

The 3D andt = 1 for 2D of Eq. (23)

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00029

CASE 3:

A0=16(3λ2+c)B0,A1=2λB0+16(3λ2+c)B1,A2=2(B0+λB1),A3=2B1,μ=14(λ2+c).

According to the coefficients, the following solution families are get.

Family 2:

υ3,2(x,y,z,t)=(3λ2+c(3+λ(EE+ζ))3λ+c+λ2λcTan[12c(EE+ζ)]),

Figure 7
Figure 7

The 3D andt = 1 for 2D of Eq. (25)

Citation: Applied Mathematics and Nonlinear Sciences 5, 1; 10.2478/amns.2020.1.00029

Remark

Since the above cases do not meet the requirements of the conditions, no suitable solution has been found for the families.

4 Conclusion

In this study, new travelling wave solutions of Boiti-Leon-Manna-Pempinelli (BLMP) equation have been successfully obtained by using modified exponential function method. When we compare our results with the solutions obtained for this equation in the literature, we see that all solutions are completely different. We have drawn two and three dimensional graphs of all travelling wave solutions by selecting the suitable constants. The solutions obtained can be said to be an effective method for obtaining analytical solutions of such nonlinear differential equations. The solutions found include trigonometric and hyperbolic functions. Such functions are also periodic functions. The advantage of such functions is that it allows us to comfortably comment on the physical behavior of the wave, regardless of the range of the graph of the resulting solution function. The hyperbolic functions and trigonometric functions are arisen in both mathematics and physics. For example, the hyperbolic cosine functions are shape of catenary, the hyperbolic tangent functions arise in calculate to magnetic moment and rapidity of special relativity, the hyperbolic secant functions arise in the profile of a laminar jet, the hyperbolic cotangent functions arise in the Langevin function for magnetic polarization [38].

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    Kumar, D., Hosseini, K., Samadani, F., The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitzéica type equations in nonlinear optics. Optik, 149, 439–446, 2017.

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    Yel, G., Baskonus, H. M., & Bulut, H., Novel archetypes of new coupled Konno–Oono equation by using sine–Gordon expansion method. Optical and Quantum Electronics, 49(9), 285, 2017.

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    Baskonus, H. M., & Bulut, H., New complex exact travelling wave solutions for the generalized-Zakharov equation with complex structures. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 6(2), 141–150, 2016.

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    Yongyı G., The exp(-(z))-expansion method for (3+1)-dimensional generalized Boiti-Leon-MannaPempinelli equation, IJRDO - Journal of Mathematics, 4(12), 2018.

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    Hongcai M., Yongbin B. and Aiping D., Exact three-wave solutions for the (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation, Advances in Difference Equations, 2013(??), 2013.

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    Baskonus, H. M., & Bulut, H., Exponential prototype structures for (2+ 1)-dimensional Boiti-Leon-Pempinelli systems in mathematical physics. Waves in Random and Complex Media, 26(2), 189–196, 2016.

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    Yokus, A., Sulaiman, T. A., Gulluoglu, M. T., & Bulut, H., Stability analysis, numerical and exact solutions of the (1+1)-dimensional NDMBBM equation. In ITM Web of Conferences (Vol. 22, p. 01064). EDP Sciences, 2018.

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    Pandey, P. K., A new computational algorithm for the solution of second order initial value problems in ordinary differential equations. Applied Mathematics and Nonlinear Sciences, 3(1), 167–174, 2018.

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    El-Shaboury, S. M., Ammar, M. K., & Yousef, W. M., Analytical solutions of the relative orbital motion in unperturbed and in J2-perturbed elliptic orbits. Applied Mathematics and Nonlinear Sciences, 2(2), 403–414, 2017.

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    Cattani C., Haar wavelet splines, Journal of Interdisciplinary Mathematicss, 4 (1), 35–47, 2001.

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    Heydari, M. H., Hooshmandasl, M. R., Ghaini, F. M., & Cattani, C., A computational method for solving stochastic Itô–Volterra integral equations based on stochastic operational matrix for generalized hat basis functions. Journal of Computational Physics, 270, 402–415, 2014.

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    Cattani, C., Connection coefficients of Shannon wavelets. Mathematical Modelling and Analysis, 11(2), 117–132, 2006.

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    Cattani, C., & Rushchitskii, Y. Y., Cubically nonlinear elastic waves: wave equations and methods of analysis. International applied mechanics, 39(10), 1115–1145, 2003.

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    Heydari, M. H., Hooshmandasl, M. R., Ghaini, F. M., & Cattani, C., Wavelets method for solving fractional optimal control problems. Applied Mathematics and Computation, 286, 139–154, 2016.

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    Cattani, C., Harmonic wavelet solutions of the Schrodinger equation. International Journal of Fluid Mechanics Research, 30(5), 2003.

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    Amkadni, M., Azzouzi, A., & Hammouch, Z., On the exact solutions of laminar MHD flow over a stretching flat plate. Communications in Nonlinear Science and Numerical Simulation, 13(2), 359–368, 2008.

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    Bulut, H., Aktürk, T., Yel, G., An Application of the Modified Expansion Method to Nonlinear Partial Differential Equation, Turk. J. Math. Comput. Sci., 10, 202–206, 2018.

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    Baskonus, H. M, Bulut, H. Analytical Studies on the (1+1)-dimensional Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation Defined by Seismic Sea Waves, Wavesin Random and Complex Media, 25(4), 576–586, 2015.

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    Baskonus, H. M., Bulut, H., Atangana, A., On the complex and hyperbolic structures of the longitudinal wave equation in a magneto-electro-elastic circular rod. Smart Materials and Structures, 25(3), 035022, 2016.

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  • [1]

    Gandarias, M. L., & Bruzón, M. S., Conservation laws for a Boussinesq equation. Applied Mathematics and Nonlinear Sciences, 2(2), 465–472, 2017.

  • [2]

    Moleleki, L. D., Motsepa, T., & Khalique, C. M., Solutions and conservation laws of a generalized second extended (3+ 1)-dimensional Jimbo-Miwa equation. Applied Mathematics and Nonlinear Sciences, 3(2), 459–474, 2018.

  • [3]

    Mingliang, W., Li, X. and Zhang. J., The (G′/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics, Physics Letters A372.4,417–423, 2008.

  • [4]

    Zhang, S., Tong, J. L., Wang, W., A generalized (G′ G)-expansion method for the mKdV equation with variable coefficients. Physics Letters A, 372(13), 2254–2257, 2008.

  • [5]

    Zhang, J., Wei, X., & Lu, Y., A generalized (G′ G)-expansion method and its applications. Physics Letters A, 372(20), 3653–3658, 2008.

  • [6]

    Khalique, C. M., & Mhlanga, I. E., Travelling waves and conservation laws of a (2+ 1)-dimensional coupling system with Kortewegde Vries equation. Applied Mathematics and Nonlinear Sciences, 3(1), 241–254, 2018.

  • [7]

    Kumar, D., Hosseini, K., Samadani, F., The sine-Gordon expansion method to look for the traveling wave solutions of the Tzitzéica type equations in nonlinear optics. Optik, 149, 439–446, 2017.

  • [8]

    Yel, G., Baskonus, H. M., & Bulut, H., Novel archetypes of new coupled Konno–Oono equation by using sine–Gordon expansion method. Optical and Quantum Electronics, 49(9), 285, 2017.

  • [9]

    Baskonus, H. M., & Bulut, H., New complex exact travelling wave solutions for the generalized-Zakharov equation with complex structures. An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 6(2), 141–150, 2016.

  • [10]

    Eskitaşçıoğglu, E. İ., Aktaş, M. B., & Baskonus, H. M., New Complex and Hyperbolic Forms for Ablowitz–Kaup–Newell–Segur Wave Equation with Fourth Order. Applied Mathematics and Nonlinear Sciences, 4(1), 105–112, 2019.

  • [11]

    Baskonus, H. M., Bulut, H., Sulaiman, T. A., New Complex Hyperbolic Structures to the Lonngren-Wave Equation by Using Sine-Gordon Expansion Method. Applied Mathematics and Nonlinear Sciences, 4(1), 141–150, 2019.

  • [12]

    Gurefe, Y., Misirli, E., Sonmezoglu, A., Ekici, M., Extended trial equation method to generalized nonlinear partial differential equations. Applied Mathematics and Computation, 219(10), 5253–5260, 2013.

  • [13]

    Ekici, M., Mirzazadeh, M., Sonmezoglu, A., Ullah, M. Z., Zhou, Q., Triki, H., Biswas, A. Optical solitons with anti-cubic nonlinearity by extended trial equation method. Optik, 136, 368–373, 2017.

  • [14]

    Bulut, H. Akturk, T. and Gurefe, Y., Traveling wave solutions of the (N+1)- dimensional sine-cosine-Gordon equation, AIP Conference Proceedings, Vol. 1637, 145–149, 2014.

  • [15]

    Bulut, H., Akturk, T. and Gurefe, Y., An application of the new function method to the generalized double sinh-Gordon equation., AIP Conference Proceedings, 1648(1):pp. 4, 2015.

  • [16]

    Darvishi, M., Najafi, M., Kavitha, L., & Venkatesh, M., Stair and Step Soliton Solutions of the Integrable (2+1) and (3+1)-Dimensional Boiti—Leon—Manna—Pempinelli Equations. Communications in Theoretical Physics, 58(6), 785, 2012.

  • [17]

    Mabrouk, S. M., & Rashed, A. S., Analysis of (3+ 1)-dimensional Boiti–Leon–Manna–Pempinelli equation via Lax pair investigation and group transformation method. Computers & Mathematics with Applications, 74(10), 2546–2556, 2017.

  • [18]

    Mohamed R. Ali and Wen-Xiu Ma, New Exact Solutions of Nonlinear (3 + 1)-Dimensional Boiti-Leon-Manna-Pempinelli Equation, Advances in Mathematical Physics, 1–7, Article ID 9801638, 2019.

  • [19]

    Jia, S. L., Gao, Y. T., Hu, L., Huang, Q. M., & Hu, W. Q., Soliton-like, periodic wave and rational solutions for a (3+1)-dimensional Boiti-Leon-Manna-Pempinelli equation in the incompressible fluid. Superlattices and Microstructures, 102, 273–283, 2017.

  • [20]

    Guner, O., New exact solution for (2+ 1) and (3+ 1) dimensional nonlinear partial differential equations. Aksaray University Journal of Science and Engineering, 2(2), 161–170, 2018.

  • [21]

    Yongyı G., The exp(-(z))-expansion method for (3+1)-dimensional generalized Boiti-Leon-MannaPempinelli equation, IJRDO - Journal of Mathematics, 4(12), 2018.

  • [22]

    Hongcai M., Yongbin B. and Aiping D., Exact three-wave solutions for the (3 + 1)-dimensional Boiti-Leon-Manna-Pempinelli equation, Advances in Difference Equations, 2013(??), 2013.

  • [23]

    Baskonus, H. M., & Bulut, H., Exponential prototype structures for (2+ 1)-dimensional Boiti-Leon-Pempinelli systems in mathematical physics. Waves in Random and Complex Media, 26(2), 189–196, 2016.

  • [24]

    Yokus, A., Sulaiman, T. A., Gulluoglu, M. T., & Bulut, H., Stability analysis, numerical and exact solutions of the (1+1)-dimensional NDMBBM equation. In ITM Web of Conferences (Vol. 22, p. 01064). EDP Sciences, 2018.

  • [25]

    Pandey, P. K., A new computational algorithm for the solution of second order initial value problems in ordinary differential equations. Applied Mathematics and Nonlinear Sciences, 3(1), 167–174, 2018.

  • [26]

    El-Shaboury, S. M., Ammar, M. K., & Yousef, W. M., Analytical solutions of the relative orbital motion in unperturbed and in J2-perturbed elliptic orbits. Applied Mathematics and Nonlinear Sciences, 2(2), 403–414, 2017.

  • [27]

    Cattani C., Haar wavelet splines, Journal of Interdisciplinary Mathematicss, 4 (1), 35–47, 2001.

  • [28]

    Heydari, M. H., Hooshmandasl, M. R., Ghaini, F. M., & Cattani, C., A computational method for solving stochastic Itô–Volterra integral equations based on stochastic operational matrix for generalized hat basis functions. Journal of Computational Physics, 270, 402–415, 2014.

  • [29]

    Cattani, C., Connection coefficients of Shannon wavelets. Mathematical Modelling and Analysis, 11(2), 117–132, 2006.

  • [30]

    Cattani, C., & Rushchitskii, Y. Y., Cubically nonlinear elastic waves: wave equations and methods of analysis. International applied mechanics, 39(10), 1115–1145, 2003.

  • [31]

    Heydari, M. H., Hooshmandasl, M. R., Ghaini, F. M., & Cattani, C., Wavelets method for solving fractional optimal control problems. Applied Mathematics and Computation, 286, 139–154, 2016.

  • [32]

    Cattani, C., Harmonic wavelet solutions of the Schrodinger equation. International Journal of Fluid Mechanics Research, 30(5), 2003.

  • [33]

    Amkadni, M., Azzouzi, A., & Hammouch, Z., On the exact solutions of laminar MHD flow over a stretching flat plate. Communications in Nonlinear Science and Numerical Simulation, 13(2), 359–368, 2008.

  • [34]

    Bulut, H., Aktürk, T., Yel, G., An Application of the Modified Expansion Method to Nonlinear Partial Differential Equation, Turk. J. Math. Comput. Sci., 10, 202–206, 2018.

  • [35]

    Baskonus, H. M, Bulut, H. Analytical Studies on the (1+1)-dimensional Nonlinear Dispersive Modified Benjamin-Bona-Mahony Equation Defined by Seismic Sea Waves, Wavesin Random and Complex Media, 25(4), 576–586, 2015.

  • [36]

    Baskonus, H. M., Bulut, H., Atangana, A., On the complex and hyperbolic structures of the longitudinal wave equation in a magneto-electro-elastic circular rod. Smart Materials and Structures, 25(3), 035022, 2016.

  • [37]

    He, J. H., Wu, X. H., Exp-function method for nonlinear wave equations, Chaos, Solitons & Fractals, 30(3), 700–708, 2006.

  • [38]

    Weisstein, E. W., CRC concise encyclopedia of mathematics. Chapman and Hall/CRC, 2002.

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