The Soliton Solutions for Some Nonlinear Fractional Differential Equations with Beta-Derivative

Özkan, Erdoğan Mehmet; Özkan, Ayten

doi:10.3390/axioms10030203

Cited by 17 publications

(4 citation statements)

References 53 publications

(65 reference statements)

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“…Using the traveling wave ansatz, Baiswas [17], studied the generalized Kawahara equation and derived a solitary wave solution for the family of the Kawahara equation. A lot of effective methods are applied to study the exact solutions, which makes the research more abundant [18][19][20][21][22][23]. Wang [24] used ansatz method to derive the exact solitary wave solution for the generalized Korteweg-de Vries-Kawahara (GKdV-K) equation.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method

Tian

Zhang

et al. 2022

Axioms

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In this article, we prove that the ⟨p,q⟩ condition holds, first by using the Fuchs index of the complex Kawahara equation, and then proving that all meromorphic solutions of complex Kawahara equations belong to the class W. Moreover, the complex method is employed to get all meromorphic solutions of complex Kawahara equation and all traveling wave exact solutions of Kawahara equation. Our results reveal that all rational solutions ur(x+νt) and simply periodic solutions us,1(x+νt) of Kawahara equation are solitary wave solutions, while simply periodic solutions us,2(x+νt) are not real-valued. Finally, computer simulations are given to demonstrate the main results of this paper. At the same time, we believe that this method is a very effective and powerful method of looking for exact solutions to the mathematical physics equations, and the search process is simpler than other methods.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method

Tian

Zhang

et al. 2022

Axioms

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“…A short list can be made as follows. The sub-equation method (see, [20][21][22][23]), the sinecosine method (see, [24]), Laplace transform method (see, [25]), the auxiliary equation method (see, [26]), direct algebraic method (see, [27]), Kudryashov method (see, [28]), the variational iteration method (see, [29]), the first integral method (see, [30]), F-expansion method (see, [31,32] ), ( ) ¢ G G 2 extension method (see, [33][34][35]), He's semi inverse method (see, [36]), ansatz approach (see, [37]) are some examples.…”

Section: Introductionmentioning

confidence: 99%

A view of solitary wave solutions to the fractional DNA Peyrard-Bishop equation via a new approach

Özkan

2024

Phys. Scr.

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In this study, the fractional impacts of the beta derivative and Mtruncated derivative are examined on the DNA Peyrard-Bishop dynamic model equation. To obtain solitary wave solutions for the model, the Sardar sub-equation approach was utilized. For a stronger comprehension of the model, the acquired solutions are graphically illustrated together with the fractional impacts of the beta andM-truncated derivatives. In addition to being simple and not needing any complicated computations, the approach has the benefit of getting accurate results.

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“…The whole of these FNEEs, FNPDEs, FNDDEs remains an ongoing task due to the fact that, the power of memory effect is of great interest in the modeling field's. From the mathematical point of view, the classical systems (integer-order derivatives modeling) are not thoughtfully to address this memory effect issue [34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51]. Obviously, different researchers demonstrated that, through the fractional-order derivatives (non-integer operators) it is possible to carry out more on memory effect.…”

Section: Introductionmentioning

confidence: 99%

Enhanced analysis of soliton-like pulses in space-time fractional beta-derivatives coupled nerve fibers with application insights

Fendzi-Donfack,

Fotoula,

Jantschi

et al. 2024

Preprint

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This work aims to delve exact physical wave solutions of nonlinear diﬀerential-diﬀerence equations with beta-derivative leading the conduction of nerve impulse dynamics in coupled nerve ﬁbers. The technique performed here names the polynomial expansion method. By employing a new variable through a fractional complex transformation, we turn the electrical model equation to a second-order elliptic nonlinear ordinary diﬀerential equation with two free parameters including the semi-discrete approximation. We extract short pulse, kink, anti-kink, singular kink/anti-kink and kink-rogue solitary waves as solutions of the derived elliptic nonlinear ordinary diﬀerential equations. By carrying out these various solutions, we peruse the dynamical behavior of the current nerve ﬁbers model by investigating their linear stability. We also study the dynamics of the model by applying the ﬁxed points theory and derive the related Jacobian matrix. Throughout the convenient physiological parameters values and signs, we display some 3D modulational instability zones. Furthermore, we also exhibit 3D and 2D graphs presenting the shapes of new pursued solitary waves. The numerous solutions families obtained and plotted help us to show the fractional-order parameter eﬀects in addition of the nerve ﬁber diameter on the amplitude and speed of the nerve impulse propagating inside the coupled nerve ﬁbers. The good or bad coordination of the movement inside the human parts can be explain through the amplitude varying, speed modiﬁcation of the nerve impulse ﬂowing into the coupled nerve ﬁbers. Aiming to this, we make the evolved mathematical framework accessible and highlight potential medical and biological applications. In addition, we enhance the understanding of nerve conduction mechanisms in myelinated ﬁbers.

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The Soliton Solutions for Some Nonlinear Fractional Differential Equations with Beta-Derivative

Cited by 17 publications

References 53 publications

All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method

All Traveling Wave Exact Solutions of the Kawahara Equation Using the Complex Method

A view of solitary wave solutions to the fractional DNA Peyrard-Bishop equation via a new approach

Enhanced analysis of soliton-like pulses in space-time fractional beta-derivatives coupled nerve fibers with application insights

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