Soliton Solutions for Some Nonlinear Partial Differential Equations in Mathematical Physics Using He’s Variational Method

Elboree, Mohammed K.

doi:10.1515/ijnsns-2018-0188

Cited by 14 publications

(7 citation statements)

References 31 publications

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“…In this section, we will use the He's variational method [47][48][49][50][51] to obtain the solitary wave solution of Equation 3.7. We assume that the solution of Equation 3.7 takes the following form:…”

Section: He's Variational Methodsmentioning

confidence: 99%

“…According to He's variational method, [47][48][49][50][51] there are ∂J ∂p = 0, ð4:3Þ ∂J ∂q = 0, ð4:4Þ which produce the following results: where k = a 6v c + 2d ð Þ. In light of Equations 4.7 and 4.8, we get the solution of Equation 3.7 as…”

Section: He's Variational Methodsmentioning

confidence: 99%

“…In this section, we will use the He's variational method 47–51 to obtain the solitary wave solution of Equation . We assume that the solution of Equation takes the following form:

ζ ((), γ) = p sech ((), qγ) .

Substituting Equation into Equation , we have

\begin{matrix} lefttrue \\ J = \int_{0}^{\infty} \{\frac{1}{2} p^{2} v {sech}^{2} (italicqγ) - \frac{1}{4} {italickp}^{4} {sech}^{4} (italicqγ) - \frac{1}{2} {italicbp}^{2} q^{2} {sech}^{2} (italicqγ) \tanh^{2} (italicqγ)\} dγ, \\ = \int_{0}^{\infty} \{\frac{p^{2} v}{2 q} {sech}^{2} (θ) - \frac{{kp}^{4}}{4 q} {sech}^{4} (θ) - \frac{{italicbp}^{2} q}{2} {sech}^{2} (θ) \tanh^{2} (θ)\} dθ \\ \begin{array}{c} = \end{array} \frac{p^{2} v}{2 q} \int_{0}^{\infty} {sech}^{2} (θ) dθ - \frac{{kp}^{4}}{4 q} \int_{0}^{\infty} {sech}^{4} (θ) dθ - \frac{{italicbp}^{2} q}{2} \int_{0}^{\infty} {sech}^{2} (θ) \tanh^{2} (θ) dθ \\ \begin{array}{c} = - \frac{p^{2}}{6 q} \end{array} ({italickp}^{2} + bq - 3 v) . \end{matrix}

According to He's variational method,…”

Section: He's Variational Methodsmentioning

confidence: 99%

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He's variational method for the time–space fractional nonlinear Drinfeld–Sokolov–Wilson system

Wang

2021

Math Methods in App Sciences

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The well-known Drinfeld-Sokolov-Wilson system is widely used to describe the flow of the shallow water. This paper considers the time-space fractional Drinfeld-Sokolov-Wilson system, where He's variational method together with the two-scale transform is applied to find its solitary solutions. The numerical results have been shown through graphs to demonstrate the applicability and efficiency of our approach. The main advantage of variational approach is that it can reduce the order of differential equation and make the equation more simple. The obtained results in this work are expected to shed a bright light on the study of fractional nonlinear partial differential equations.

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Section: He's Variational Methodsmentioning

confidence: 99%

Section: He's Variational Methodsmentioning

confidence: 99%

“…In this section, we will use the He's variational method 47–51 to obtain the solitary wave solution of Equation . We assume that the solution of Equation takes the following form:

ζ ((), γ) = p sech ((), qγ) .

Substituting Equation into Equation , we have

\begin{matrix} lefttrue \\ J = \int_{0}^{\infty} \{\frac{1}{2} p^{2} v {sech}^{2} (italicqγ) - \frac{1}{4} {italickp}^{4} {sech}^{4} (italicqγ) - \frac{1}{2} {italicbp}^{2} q^{2} {sech}^{2} (italicqγ) \tanh^{2} (italicqγ)\} dγ, \\ = \int_{0}^{\infty} \{\frac{p^{2} v}{2 q} {sech}^{2} (θ) - \frac{{kp}^{4}}{4 q} {sech}^{4} (θ) - \frac{{italicbp}^{2} q}{2} {sech}^{2} (θ) \tanh^{2} (θ)\} dθ \\ \begin{array}{c} = \end{array} \frac{p^{2} v}{2 q} \int_{0}^{\infty} {sech}^{2} (θ) dθ - \frac{{kp}^{4}}{4 q} \int_{0}^{\infty} {sech}^{4} (θ) dθ - \frac{{italicbp}^{2} q}{2} \int_{0}^{\infty} {sech}^{2} (θ) \tanh^{2} (θ) dθ \\ \begin{array}{c} = - \frac{p^{2}}{6 q} \end{array} ({italickp}^{2} + bq - 3 v) . \end{matrix}

According to He's variational method,…”

Section: He's Variational Methodsmentioning

confidence: 99%

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He's variational method for the time–space fractional nonlinear Drinfeld–Sokolov–Wilson system

Wang

2021

Math Methods in App Sciences

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“…The variational method proposed by professor Ji‐Huan He is a powerful method to seek the solitary wave solutions of many nonlinear partial differential equations 32–34 . In this section, we aim to get the solitary wave solution of Equation by He's variational method.…”

Section: Solitary Wave Solutionsmentioning

confidence: 99%

Solitary and periodic wave solutions of the generalized fourth‐order Boussinesq equation via He's variational methods

Wang

2021

Math Methods in App Sciences

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Yu‐Lan Ma, et al. (Mathematical Methods in the Applied Sciences, 2019,42(1)) make outstanding contributions for the soliton solutions of the generalized fourth‐order Boussinesq equation, which is used to describe the wave motion in fluid mechanics. But the periodic wave solution is not reported. So in this paper, He's variational methods are employed to find the solitary and periodic wave solutions of the generalized fourth‐order Boussinesq equation. The greatest advantage of the variational method is that it can reduce the order of the research equation, make the equation more simple, and obtain the optimal solution. Finally, the numerical results are shown in the form of a graph to prove the applicability and effectiveness of the method. The results reveal that variational method is simple and straightforward and can avoid the tedious calculation process, which is expected to shed a light to the new research frontiers of solitary wave theory.

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“…In this section, we aim to seek the solitary solution of equation (1.3) by the variational theory. According to He's variational method [51][52][53], we suppose the solution of equation (3.7) with the following form:…”

Section: Solitary Wave Solutionsmentioning

confidence: 99%

On the new exact traveling wave solutions of the time-space fractional strain wave equation in microstructured solids via the variational method

Wang¹

2021

Commun. Theor. Phys.

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In this paper, we mainly study the time-space fractional strain wave equation in microstructured solids. He’s variational method, combined with the two-scale transform are implemented to seek the solitary and periodic wave solutions of the time-space strain wave equation. The main advantage of the variational method is that it can reduce the order of the differential equation, thus simplifying the equation, making the solving process more intuitive and avoiding the tedious solving process. Finally, the numerical results are shown in the form of 3D and 2D graphs to prove the applicability and effectiveness of the method. The obtained results in this work are expected to shed a bright light on the study of fractional nonlinear partial differential equations in physics.

show abstract

Soliton Solutions for Some Nonlinear Partial Differential Equations in Mathematical Physics Using He’s Variational Method

Cited by 14 publications

References 31 publications

He's variational method for the time–space fractional nonlinear Drinfeld–Sokolov–Wilson system

He's variational method for the time–space fractional nonlinear Drinfeld–Sokolov–Wilson system

Solitary and periodic wave solutions of the generalized fourth‐order Boussinesq equation via He's variational methods

On the new exact traveling wave solutions of the time-space fractional strain wave equation in microstructured solids via the variational method

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