New Exact Solutions for Generalized (3+1) Shallow Water-Like (SWL) Equation

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Abstract

In this study, we use the improved Bernoulli sub-equation function method for exact solutions to the generalized (3+1) shallow water-like (SWL) equation. Some new solutions are successfully constructed. We carried out all the computations and the graphics plot in this paper by Wolfram Mathematica.

Abstract

In this study, we use the improved Bernoulli sub-equation function method for exact solutions to the generalized (3+1) shallow water-like (SWL) equation. Some new solutions are successfully constructed. We carried out all the computations and the graphics plot in this paper by Wolfram Mathematica.

1 Introduction

In various fields of physical sciences, nonlinear evolution equations (NLEEs) and their exact solutions are important for non-linear phenomena. In this paper, generalized (3 + 1) shallow water-like (SWL) equation [1, 2] which is one of these equations will be discussed and new solutions will be examined.

uxxxy+3uxxuy+3uxuxyuytuxz=0.

There are some studies on this equation. Rational solutions and lump solutions are obtained for equation(1) by Zhang et al. [1] and Grammian and Pfaffian solutions are obtained by Tang et al. [2]. Also, this equation solved by Tian and Gao [3] via the tanh method, by Zayed [4] via the (G′/G) expansion method. Lump-type solutions and their interaction solutions are generated by Sadat[5]. In this context, various papers were presented to the literature [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. The organization of this paper is as follows: firstly, we give the methodology of the improved Bernoulli sub-equation function method. Then we apply this method to the SWL equation for finding new exact solutions. At last, we give a conclusion.

2 Material ve Method

In this part, we use the improved Bernoulli sub-equation function method [24, 25, 26, 27, 28] for solutions eq. (1).

  1. Step 1Let’s consider the following partial differential equation;
    P(u,ux,uy,uz,ut,uxx,uxy,...)=0.
    and take the wave transformation
    u(x,y,z,t)=U(ξ),ξ=x+ky+mzwt,
    where k, m and w are nonzero constants.Substituting Eq. (2) into Eq. (3), we obtain the following nonlinear ordinary differential equation;
    N=(U,U,U,U,...)=0.
  2. Step 2Considering trial equation of solution in Eq. (4), it can be written as following;
    U(ξ)=i=0naiFiξj=0mbjFjξ
    According to the Bernoulli theory, we can consider the general form of Bernoulli differential equation for as following;
    F=αF+βFM,α,β0,MR0,1
    where F is Bernoulli differential polynomial. Substituting Eq. (5-b6) into Eq. (4), it converts an equations of polynomial σ(F) as following;
    σ(F)=ρsFs+...+ρ1F+ρ0=0
    According to the balance principle, we can determine the relationship between n, m and M.
  3. Step 3The coefficients of σ(F) all be zero will yield us an algebraic system of equations;
    ρi=0,i=0,...,s
    Solving this system of equation, we reach the values of a0, …, an and b0, …, bm. Step 4. When we solve nonlinear Bernoulli differential equation Eq. (6), we obtain the following two situations according to α and β;
    F(ξ)=[βα+Eeα(M1)ξ]11M,αβ
    F(ξ)=[(E1)+(E+1)tanh(α(1M)ξ21tanh(α(1M)ξ2]11M,α=β,ER.

3 Findings

In this section, application of the improved Bernoulli sub-equation function method to SWL equation is presented. Using the wave transformation on Eq. (1)

u(x,y,z,t)=U(ξ),ξ=x+ky+mzwt,

we get the following nonlinear ordinary differential equation:

kU(4)+6kUU+(kwm)U=0.

Integrating the equation in (12), we get

kU+3k[U]2+(kwm)U=0.

Finally, if we write V instead of U, the equation (13) becomes a second order nonlinear ordinary differential equation:

kV+3kV2+(kwm)V=0.

Balancing Eq. (14) by considering the highest derivative V and the highest power V2, we obtain

n+2=2M+m.

Choosing M = 2, m = 1, gives n = 3. Thus, the trial solution to Eq. (1) takes the following form:

U(ξ)=a0+a1F(ξ)+a2F2(ξ)+a3F3(ξ)b0+b1F(ξ).

where F = αF + βF2, α, β ≠ 0. Substituting Eq. (15), its second derivative and power along with F = αF + βF2, α, β ≠ 0, into Eq. (14), yields a polynomial in F. Solving the system of the algebraic equations, yields the values of the parameter involved. Substituting the obtained values of the parameters into Eq. (15), yields the solutions to Eq. (1). We can find following coefficients:

  1. Case 1
    a0=(1+k)wb03k,a1=2w3/2b0k1+k(1+k)wb13k,a2=2w2b0+2w3/2b1k1+k,a3=2w2b1,σ=wk1+k;
  2. Case 2
    a0=0,a1=0,a2=2iw3/2b1k1+k,a3=2w2b1,b0=0,σ=iwk1+k;
  3. Case 3
    a0=0,a1=mb13k,a2=4m3/2b1k3/2,a3=8m2b1k2,b0=0,w=2mk,σ=mk;
  4. Case 4
    a0=0,a1=(kw)b13k,a2=2iwmkwb1k,a3=2w2b1,b0=0,σ=imkwk;
    Choosing the suitable values of parameters, we performed the numerical simulations of the obtained solutions for (16, 17) case by plotting their 2D and 3D.
    Fig. 1
    Fig. 1

    The 3D and 2D surfaces of the solution for (16) for suitable values

    Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00031

    Fig. 2
    Fig. 2

    The 3D and 2D surfaces of the solution for (17) for suitable values

    Citation: Applied Mathematics and Nonlinear Sciences 4, 2; 10.2478/AMNS.2019.2.00031

4 Result and Discussion

In this article, new solutions are obtained for the SWL equation using the IBSEFM method. We have seen that the results we obtained are new solutions when we compare them with previous ones. The results may be useful to explain the physical effects of various nonlinear models in non-linear sciences. IBSEFM is a powerful and efficient mathematical tool that can be used to process various nonlinear mathematical models.

Communicated by Juan Luis García Guirao

References

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    Y.-N. Tang W.-X. Ma and W. Xu "Grammian and Pfaffian solutions as well as Pfaffianization for a (3+1)-dimensional generalized shallow water equation" Chinese Physics B vol. 21 no. 7 2012.

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    B. Tian and Y.-T. Gao "Beyond travelling waves: A new algorithmfor solving nonlinear evolution equations" Computer Physics Communications vol. 95 no. 2-3 pp. 139-142 1996.

    • Crossref
    • Export Citation
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    E. Zayed "travelling wave solutions for higher dimensional nonlinear evaluation equations using G’/G expansion method" Journal of Applied Mathematics and Informatics vol. 28 no. 1 2 pp. 383-395 2010.

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    R. Sadat M. Kassem and Wen-Xiu Ma "Abundant Lump-Type Solutions and Interaction Solutions for a Nonlinear (3+1) Dimensional Model" Advances in Mathematical Physics vol. 2018 Article ID 9178480 8 pages 2018. https://doi.org/10.1155/2018/9178480.

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    Akin L. 2017. "Some Weighted Martingale Inequalities On Rearrangement Invariant Quasi-Banach Function Spaces" MSU Journal of Science 5(2) 483-486.

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    Akin L. and Zeren Y.2017. Approximation To Generalized Taylor Derivatives By Integral Operator Families MSU Journal of Science 5(2) 421-423.

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    Baskonus H.M. and Bulut H. 2015. "An Effective Scheme for Solving Some Nonlinear Partial Differential Equation Arising In Nonlinear Physics" Open Physics 131 280-289.

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    Baskonus H.M. Sulaiman T.A. and Bulut H. 2017. "New Solitary Wave Solutions to the (2+1)-Dimensional Calogero-Bogoyavlenskii-Schi and the Kadomtsev-Petviashvili Hierarchy Equations" Indian Journal of Physics91101237-1243.

    • Crossref
    • Export Citation
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    Baskonus H.M. Bulut H. and Atas S.S. 2018. "Contour Surfaces in the (2+1)-dimensional Sine-Poisson Model" International Journal of Innovative Engineering Applications 2(2) 44-49.

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    Modanlı M. 2018. "Two numerical methods for fractional partial differential equation with nonlocal boundary value problem" Advances in Difference Equations 2018:333. https://doi.org/10.1186/s13662-018-1789-2

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    Modanlı M. and Akgül A. 2017. "Numerical solution of fractional telegraph differential equations by theta-method" Eur. Phys. J. Special Topics 226 3693-3703.

    • Crossref
    • Export Citation
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    Polat N. and Pişkin E. 2015. "Existence and asymptotic behavior of solution of Cauchy problem for the damped sixth-order Boussinesq equation" Acta Mathematicae Applicatae Sinica English Series 31(3) 735-746.

    • Crossref
    • Export Citation
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    Polat N. and Pişkin E. 2012. "Asymptotic behavior of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation" Applied Mathematics Letters 25 1871-1874.

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    Pişkin E. and Polat N. 2014. "Existence global nonexistence and asymptotic behavior of solutions for Cauchy problem of a multidimensional generalized damped Boussinesq-type equation" Turkish Journal of Mathematics 38: 706-727.

    • Crossref
    • Export Citation
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    Pişkin E. 2013. "Blow up of solutions for the Cauchy problem of the damped sixth-order Boussinesq equation" Theoretical Mathematics and Applications vol. 4 no. 3 61-71.

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    Dusunceli F. and Celik E. 2018. "Numerical Solution For High-Order Linear Complex Differential Equations with Variable Coefficients" Numerical Methods for Partial Differential Equations 34(5) 1645-1658. .

    • Crossref
    • Export Citation
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    Dusunceli F. and Celik E. 2017. "Numerical Solution for High-Order Linear Complex Differential Equations By Hermite Polynomials" Iğdır University Journal of the Institute of Science and Technology 7(4): 189-201.

    • Crossref
    • Export Citation
  • [19]

    Dusunceli F. and Celik E. 2017. "Fibonacci matrix Polynomial Method For Linear Complex Differential Equations" Asian Journal of Mathematics and Computer Research . 15(3): 229-238.

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    Dusunceli F. and Celik E. 2015. "An Effective Tool: Numerical Solutions by Legendre Polynomials for High-Order Linear Complex Differential Equations" British Journal of Applied Science and Technology 8(4): 348-355.

    • Crossref
    • Export Citation
  • [21]

    Ilhan O.A. Esen A. Bulut H. and Baskonus H.M. 2019. "Singular Solitons in the Pseudo-parabolic Model Arising in Nonlinear Surface Waves" Results in Physics 12 1712-1715.

    • Crossref
    • Export Citation
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    Baskonus H.M. 2019. "Complex Soliton Solutions to the Gilson-Pickering Model" Axioms 8(1) 18.

    • Crossref
    • Export Citation
  • [23]

    Cattani C. Sulaiman T.A. Baskonus H.M. and Bulut H. 2018. "On the soliton solutions to the Nizhnik-Novikov-Veselov and the Drinfel’d-Sokolov systems" Optical and Quantum Electronics 50(3) 138.

    • Crossref
    • Export Citation
  • [24]

    Baskonus H.M. and Bulut H. 2015. "On the Complex Structures of Kundu-Eckhaus Equation via Improved Bernoulli Sub-Equation Function Method" Waves in Random and Complex Media 254 720-728.

    • Crossref
    • Export Citation
  • [25]

    Bulut H. Yel G. and Baskonus H.M. 2016. An Application Of Improved Bernoulli Sub-Equation Function Method To The Nonlinear Time-Fractional Burgers Equation Turkish Journal of Mathematics and Computer Science 5 1-17.

  • [26]

    Dusunceli F. 2018. "Solutions for the Drinfeld-Sokolov Equation Using an IBSEFM Method" MSU Journal Of Science. 61505-510.

  • [27]

    Dusunceli F. 2019. "New Exponential and Complex Traveling Wave Solutions to the Konopelchenko-Dubrovsky Model" Advances in Mathematical Physics vol. 2019 Article ID 7801247 9 pages. https://doi.org/10.1155/2019/7801247.

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    Dusunceli F. 2019. "New Exact Solutions for the (3 + 1) Dimensional B-type Kadomtsev-Petviashvili Equation". Journal of science and Technology 12 (1) 463-468.

    • Crossref
    • Export Citation

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  • [1]

    Y. Zhang H. Dong X. Zhang and H. Yang Rational solutions and lump solutions to the generalized (3+1)-dimensional Shallow Water-like equation Computers and Mathematics with Applications vol. 73 no. 2 pp. 246-252 2017.

  • [2]

    Y.-N. Tang W.-X. Ma and W. Xu "Grammian and Pfaffian solutions as well as Pfaffianization for a (3+1)-dimensional generalized shallow water equation" Chinese Physics B vol. 21 no. 7 2012.

  • [3]

    B. Tian and Y.-T. Gao "Beyond travelling waves: A new algorithmfor solving nonlinear evolution equations" Computer Physics Communications vol. 95 no. 2-3 pp. 139-142 1996.

    • Crossref
    • Export Citation
  • [4]

    E. Zayed "travelling wave solutions for higher dimensional nonlinear evaluation equations using G’/G expansion method" Journal of Applied Mathematics and Informatics vol. 28 no. 1 2 pp. 383-395 2010.

  • [5]

    R. Sadat M. Kassem and Wen-Xiu Ma "Abundant Lump-Type Solutions and Interaction Solutions for a Nonlinear (3+1) Dimensional Model" Advances in Mathematical Physics vol. 2018 Article ID 9178480 8 pages 2018. https://doi.org/10.1155/2018/9178480.

  • [6]

    Akin L. 2017. "Some Weighted Martingale Inequalities On Rearrangement Invariant Quasi-Banach Function Spaces" MSU Journal of Science 5(2) 483-486.

  • [7]

    Akin L. and Zeren Y.2017. Approximation To Generalized Taylor Derivatives By Integral Operator Families MSU Journal of Science 5(2) 421-423.

  • [8]

    Baskonus H.M. and Bulut H. 2015. "An Effective Scheme for Solving Some Nonlinear Partial Differential Equation Arising In Nonlinear Physics" Open Physics 131 280-289.

  • [9]

    Baskonus H.M. Sulaiman T.A. and Bulut H. 2017. "New Solitary Wave Solutions to the (2+1)-Dimensional Calogero-Bogoyavlenskii-Schi and the Kadomtsev-Petviashvili Hierarchy Equations" Indian Journal of Physics91101237-1243.

    • Crossref
    • Export Citation
  • [10]

    Baskonus H.M. Bulut H. and Atas S.S. 2018. "Contour Surfaces in the (2+1)-dimensional Sine-Poisson Model" International Journal of Innovative Engineering Applications 2(2) 44-49.

  • [11]

    Modanlı M. 2018. "Two numerical methods for fractional partial differential equation with nonlocal boundary value problem" Advances in Difference Equations 2018:333. https://doi.org/10.1186/s13662-018-1789-2

  • [12]

    Modanlı M. and Akgül A. 2017. "Numerical solution of fractional telegraph differential equations by theta-method" Eur. Phys. J. Special Topics 226 3693-3703.

    • Crossref
    • Export Citation
  • [13]

    Polat N. and Pişkin E. 2015. "Existence and asymptotic behavior of solution of Cauchy problem for the damped sixth-order Boussinesq equation" Acta Mathematicae Applicatae Sinica English Series 31(3) 735-746.

    • Crossref
    • Export Citation
  • [14]

    Polat N. and Pişkin E. 2012. "Asymptotic behavior of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation" Applied Mathematics Letters 25 1871-1874.

  • [15]

    Pişkin E. and Polat N. 2014. "Existence global nonexistence and asymptotic behavior of solutions for Cauchy problem of a multidimensional generalized damped Boussinesq-type equation" Turkish Journal of Mathematics 38: 706-727.

    • Crossref
    • Export Citation
  • [16]

    Pişkin E. 2013. "Blow up of solutions for the Cauchy problem of the damped sixth-order Boussinesq equation" Theoretical Mathematics and Applications vol. 4 no. 3 61-71.

  • [17]

    Dusunceli F. and Celik E. 2018. "Numerical Solution For High-Order Linear Complex Differential Equations with Variable Coefficients" Numerical Methods for Partial Differential Equations 34(5) 1645-1658. .

    • Crossref
    • Export Citation
  • [18]

    Dusunceli F. and Celik E. 2017. "Numerical Solution for High-Order Linear Complex Differential Equations By Hermite Polynomials" Iğdır University Journal of the Institute of Science and Technology 7(4): 189-201.

    • Crossref
    • Export Citation
  • [19]

    Dusunceli F. and Celik E. 2017. "Fibonacci matrix Polynomial Method For Linear Complex Differential Equations" Asian Journal of Mathematics and Computer Research . 15(3): 229-238.

  • [20]

    Dusunceli F. and Celik E. 2015. "An Effective Tool: Numerical Solutions by Legendre Polynomials for High-Order Linear Complex Differential Equations" British Journal of Applied Science and Technology 8(4): 348-355.

    • Crossref
    • Export Citation
  • [21]

    Ilhan O.A. Esen A. Bulut H. and Baskonus H.M. 2019. "Singular Solitons in the Pseudo-parabolic Model Arising in Nonlinear Surface Waves" Results in Physics 12 1712-1715.

    • Crossref
    • Export Citation
  • [22]

    Baskonus H.M. 2019. "Complex Soliton Solutions to the Gilson-Pickering Model" Axioms 8(1) 18.

    • Crossref
    • Export Citation
  • [23]

    Cattani C. Sulaiman T.A. Baskonus H.M. and Bulut H. 2018. "On the soliton solutions to the Nizhnik-Novikov-Veselov and the Drinfel’d-Sokolov systems" Optical and Quantum Electronics 50(3) 138.

    • Crossref
    • Export Citation
  • [24]

    Baskonus H.M. and Bulut H. 2015. "On the Complex Structures of Kundu-Eckhaus Equation via Improved Bernoulli Sub-Equation Function Method" Waves in Random and Complex Media 254 720-728.

    • Crossref
    • Export Citation
  • [25]

    Bulut H. Yel G. and Baskonus H.M. 2016. An Application Of Improved Bernoulli Sub-Equation Function Method To The Nonlinear Time-Fractional Burgers Equation Turkish Journal of Mathematics and Computer Science 5 1-17.

  • [26]

    Dusunceli F. 2018. "Solutions for the Drinfeld-Sokolov Equation Using an IBSEFM Method" MSU Journal Of Science. 61505-510.

  • [27]

    Dusunceli F. 2019. "New Exponential and Complex Traveling Wave Solutions to the Konopelchenko-Dubrovsky Model" Advances in Mathematical Physics vol. 2019 Article ID 7801247 9 pages. https://doi.org/10.1155/2019/7801247.

  • [28]

    Dusunceli F. 2019. "New Exact Solutions for the (3 + 1) Dimensional B-type Kadomtsev-Petviashvili Equation". Journal of science and Technology 12 (1) 463-468.

    • Crossref
    • Export Citation
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