On the Complex and Hyperbolic Structures for the (2 + 1)-Dimensional Boussinesq Water Equation

Özpinar, Figen; Baskonus, Hacı Mehmet; Bulut, Hasan

doi:10.3390/e17127878

Cited by 49 publications

(24 citation statements)

References 19 publications

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“…In this section, we implement the MEFM to the two‐component second order KdV evolutionary system given by:

\begin{matrix} u_{t} + 3 v_{x x} & = & 0, \\ v_{t} - u_{x x} - 4 u^{2} & = & 0. \end{matrix}

Now, performing the wave transformation

u = U (η)

v = V (η)

η = x - k t

on Equation 6.1, we get the following single NODE:

3 U'' + 12 U^{2} + k^{2} U = 0.

Balancing the highest power nonlinear term

U^{2}

and highest derivative

U''

in Equation 6.2, gives the following relation between

M

and

N

;

N = M + 2 N, M \in ℤ^{+} .

Taking

M = 1

, produces

N = 3

. Using

M = 1

N = 3

along with Equation 2.4, produces;

U (η) = A_{0} + A_{1} e^{- Ω (η)}

…”

Section: Application Of Mefmmentioning

confidence: 99%

“…In this section, we implement the MEFM [40] to the two-component second order KdV evolutionary system [39] given by:…”

Section: Application Of Mefmmentioning

confidence: 99%

“…However, this study is aimed at investigating new analytical and numerical solutions to the two‐component second order KdV evolutionary system by using the modified exp

(- Ω (ξ))

‐expansion function method (MEFM) and the finite difference method, respectively. Various techniques have been applied to tackle the two‐component second order KdV evolutionary system such as the reduced differential transform method , the Homotopy perturbation method and Homotopy analysis method and so forth.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Numerical simulation and solutions of the two‐component second order KdV evolutionarysystem

Yokuş¹,

Baskonus²,

Sulaıman³

et al. 2017

Numerical Methods Partial

Self Cite

126

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In this study, with the aid of Wolfram Mathematica 11, the modified exp (− (η))-expansion function method is used in constructing some new analytical solutions with novel structure such as the trigonometric and hyperbolic function solutions to the well-known nonlinear evolutionary equation, namely; the two-component second order KdV evolutionary system. Second, the finite forward difference method is used in analyzing the numerical behavior of this equation. We consider equation (6.5) and (6.6) for the numerical analysis. We examine the stability of the two-component second order KdV evolutionary system with the finite forward difference method by using the Fourier-Von Neumann analysis. We check the accuracy of the finite forward difference method with the help of L 2 and L ∞ norm error. We present the comparison between the exact and numerical solutions of the two-component second order KdV evolutionary system obtained in this article which and support with graphics plot. We observed that the modified exp (− (η))-expansion function method is a powerful approach for finding abundant solutions to various nonlinear models and also finite forward difference method is efficient for examining numerical behavior of different nonlinear models. K E Y W O R D Sanalytical solutions, numerical solutions, the FDM, the MEFM, the twocomponent second order KdV evolutionary system Numer Methods Partial Differential Eq. 2018;34:211-227.wileyonlinelibrary.com/journal/num

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“…In this section, we implement the MEFM to the two‐component second order KdV evolutionary system given by:

\begin{matrix} u_{t} + 3 v_{x x} & = & 0, \\ v_{t} - u_{x x} - 4 u^{2} & = & 0. \end{matrix}

Now, performing the wave transformation

u = U (η)

v = V (η)

η = x - k t

on Equation 6.1, we get the following single NODE:

3 U'' + 12 U^{2} + k^{2} U = 0.

Balancing the highest power nonlinear term

U^{2}

and highest derivative

U''

in Equation 6.2, gives the following relation between

M

and

N

;

N = M + 2 N, M \in ℤ^{+} .

Taking

M = 1

, produces

N = 3

. Using

M = 1

N = 3

along with Equation 2.4, produces;

U (η) = A_{0} + A_{1} e^{- Ω (η)}

…”

Section: Application Of Mefmmentioning

confidence: 99%

“…In this section, we implement the MEFM [40] to the two-component second order KdV evolutionary system [39] given by:…”

Section: Application Of Mefmmentioning

confidence: 99%

“…However, this study is aimed at investigating new analytical and numerical solutions to the two‐component second order KdV evolutionary system by using the modified exp

(- Ω (ξ))

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Numerical simulation and solutions of the two‐component second order KdV evolutionarysystem

Yokuş¹,

Baskonus²,

Sulaıman³

et al. 2017

Numerical Methods Partial

Self Cite

126

View full text Add to dashboard Cite

show abstract

“…function method (MEFM) [14,15]. The ill-posed Boussinesq equation also known as bad Boussinesq equation was derived by J. Boussinesq [16] to describe the propagation of long waves on the surface of water with a small amplitude in none-dimensional nonlinear lattices and in nonlinear strings [17].…”

Section: Introductionmentioning

confidence: 99%

New soliton properties to the ill-posed Boussinesq equation arising in nonlinear physical science

Duran¹,

Askin²,

Sulaıman³

2017

Int. J. Optim. Control, Theor. Appl. (IJOCTA)

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In manuscript, with the help of the Wolfram Mathematica 9, we employ the modified exponential function method in obtaining some new soliton solutions to the ill-posed Boussinesq equation arising in nonlinear media. Results obtained with use of technique, and also, surfaces for soliton solutions are given. We also plot the 3D and 2D of each solution obtained in this study by using the same program in the Wolfram Mathematica 9.

show abstract

“…Nonlinear evolution equations often used to describe complex aspects in the field of nonlinear sciences such as Biological sciences, chemistry, mathematical physics, engineering and physics. For the past decades, various scholars have displayed their different effort for seeking the solutions of such types of equations, many analytical methods have been developed for this task such as sine-Gordon expansion method [1][2][3], the Bell-polynomial method [4], the new generalized Jacobi elliptic function expansion method [5], the Exp-function method [6], the modified Exp-function method [7], the (G /G)-expansion method [8][9][10], the sub equation method [11], the simplified Hirota's method [12], the simplest equation method [13], the modified simplest equation method [14][15][16], the improved (G /G)-expansion method [17], the multiple exp-function algorithm [18], the Lie group analysis and symmetry reductions [19]. In general, various methods have been developed to explore the search of different types of solutions to different kind of NLEEs [20][21][22][23][24][25][26].…”

Section: Introductionmentioning

confidence: 99%

On Some Complex Aspects of the (2+1)-dimensional Broer-Kaup-Kupershmidt System

Isik

Sulaıman

2017

ITM Web Conf.

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Abstract. The improved Bernoulli sub-equation function method is used in extracting some new exponential function solutions to the (2+1)-dimensional Broer-KaupKupershmidt system. It is of vital effort to look for more solutions of the (2+1)-dimensional Broer-Kaup-Kupershmidt system, which are very helpful for coastal and civil engineers to apply the nonlinear water models in a harbor and coastal design. All the obtained solutions satisfied the (2+1)-dimensional Broer-Kaup-Kupershmidt system. The two-and three-dimensional shapes of all the obtained solutions in this paper are also presented. All the computations and the graphics plots in this study are carried out with the aid of the Wolfram Mathematica 9.

show abstract

On the Complex and Hyperbolic Structures for the (2 + 1)-Dimensional Boussinesq Water Equation

Cited by 49 publications

References 19 publications

Numerical simulation and solutions of the two‐component second order KdV evolutionarysystem

Numerical simulation and solutions of the two‐component second order KdV evolutionarysystem

New soliton properties to the ill-posed Boussinesq equation arising in nonlinear physical science

On Some Complex Aspects of the (2+1)-dimensional Broer-Kaup-Kupershmidt System

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