Traveling wave structures of some fourth-order nonlinear partial differential equations

Esen, Handenur; Özdemir, Neslihan; Seçer, Aydın; Bayram, Mustafa

doi:10.1016/j.joes.2021.12.006

Cited by 19 publications

(5 citation statements)

References 52 publications

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“…where μ, σ, τ are real numbers. When substituting equation (7) with equation (8) into equation (6) after determining the N value via the homogeneous balance principle, we get the equation system. When τ, υ, and Λ j which are acquired by solving this system and the equation ( 9) is substituted into equation equation (7), the analytical solutions of equation (4) are produced.…”

Section: Explanation Of the Nkmmentioning

confidence: 99%

“…Considering the rapid growth of symbolic computation systems, soliton solutions make it possible to analyze nonlinear physical phenomena; thus, examination of the soliton solutions for NLPDE has recently attracted the attention of researchers [1,2]. In the literature, there are diverse approaches applied by researchers to produce analytical solutions of NLPDE such as the first integral method [3,4], the extended Kudryashov technique [5], the Riccati Bernoulli sub-ODE scheme [6,7], the modified simple equation scheme [8], the extended rational sine-cosine and sinh-cosh method [9], the extended Poincare-Lighthill-Kuo method [10], the extended simplest equation method [11], traveling wave solution [12], the auxiliary ordinary differential equation method and the generalized Riccati method [13], the solitary wave solution technique with the…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Nonlinear complex generalized zakharov dynamical system inconformal sense utilizing new kudryashov method

Secer,

Bayram,

Ozdemir

et al. 2024

Phys. Scr.

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In this manuscript, we take into account the nonlinear complex generalized Zakharov dynamicalsystem, which models the spread of the Langmuir waves in ionized plasma in a conformalsense. Fractional wave transformation is enforced to convert a nonlinear fractional systemto a nonlinear ordinary differential equation system. The new Kudryashov method, whichwas recently introduced and is an efficient method, is implemented in the presented equationin order to acquire analytical solutions. The required constraint conditions are offered toensure the validity of the obtained solutions. To analyze the physical interpretations of someof the produced solutions, we illustrate some graphical representations. We derive brightand singular solitons. Furthermore, 2D views of the behavior of soliton are represented toinvestigate the effect of the values of the parameters in the proposed model and the fractionalparameter. Also, the modulation instability of the model is investigated to ensure theobtained results are stable.

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Section: Explanation Of the Nkmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nonlinear complex generalized zakharov dynamical system inconformal sense utilizing new kudryashov method

Secer,

Bayram,

Ozdemir

et al. 2024

Phys. Scr.

Self Cite

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“…Fourth-order PDEs are very useful in different fields of science, such as engineering [42][43][44], signal processing [45][46][47][48], nuclear physics [49,50], etc. As an example, these kind of PDEs arise in traveling waves of suspension bridges.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue of (p,q)-Biharmonic System along the Ricci Flow

Yan,

Li,

Saha

et al. 2024

Axioms

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“…Also, there are some different versions of the PINN method in the literature, such as multiple parallel subnets PINN (MPS-PINN) [9], improved PINN (IPINN) [10], coupled automatic-numeric PINN (CAN-PINN) [11] and energy conservation deep learning method [12] etc. On the other hand, researchers can also find some recent studies about obtaining soliton solutions to wave equations, such as [13,14]. The method is a machine learning technique, and since it is quite new, it also raises some questions.…”

Section: Introductionmentioning

confidence: 99%

Soliton Solutions of Some Ocean Waves Supported by Physics Informed Neural Network Method

Onder,

Sahiner,

Secer

et al. 2024

AIA

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In this study, we aim to obtain numerical results of the modified Benjamin-Bona-Mahony equation, Ostrovsky-Benjamin-Bona-Mahony equation and Mikhailov-Novikov-Wang equation via the physics-informed neural networks (PINN) method. The equations are modeled for shallow and long water waves, as well as fundamental and phenomenonal models in ocean engineering. According to the implementation, we obtained the PINN solutions of kink, bright, multisoliton (two-soliton) and mixed dark-bright soliton solutions. According to the inference from the obtained results, we achieved good results in some cases compared to other approximate solution methods in the literature. However, it was also observed that the best possible results could not be obtained in cases where the soliton type was intricate and layered. While the results were obtained, the number of hidden layers and the number of neural networks in the layers also varied. These results are shown in tables. Since it is known that the aforementioned models are not solved by the PINN method, we anticipate that the study will lead to other studies in the field of ocean engineering.

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Traveling wave structures of some fourth-order nonlinear partial differential equations

Cited by 19 publications

References 52 publications

Nonlinear complex generalized zakharov dynamical system inconformal sense utilizing new kudryashov method

Nonlinear complex generalized zakharov dynamical system inconformal sense utilizing new kudryashov method

Eigenvalue of (p,q)-Biharmonic System along the Ricci Flow

Soliton Solutions of Some Ocean Waves Supported by Physics Informed Neural Network Method

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