Nonclassical symmetry analysis and heir-equations of forced Burger equation with time variable coefficients

Gülşen, Selahattin; Hashemi, Mir Sajjad; Alhefthi, Reem K.; Inc, Mustafa; Bicer, Harun

doi:10.1007/s40314-023-02358-y

Cited by 6 publications

(1 citation statement)

References 36 publications

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“…It's important to note that the specific importance of the fractional PGI equation depends on the context of the physical system being modeled. Several researchers have developed many efficient techniques over time to build analytical solutions, such as the G 1 ( ) ¢ -expansion method [10], the multiple exponential function method [11], the improved  -expansion scheme [12], novel Kudryashov method [13], the Hirota bilinear method [14], the SSE method [15],the first integral equation method [16], Heir equation method [17], Nucci reduction method [18,19], the Darboux transformation method [20], the extended simple approach [21] and Lie symmetry analysis method [22] to gain deeper insights into the behavior of complex systems and to refine their understanding of real-world phenomena. The lie symmetry analysis method was founded to solve the classical differential equations by Norwegian mathematician Sophus Lie at the end of the nineteenth century.…”

Section: Introductionmentioning

confidence: 99%

Soliton solutions and sensitive analysis to nonlinear wave model arising in optics

Lu,

Ahmad,

Akram

et al. 2024

Phys. Scr.

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In this study, we use analytical algorithms, specifically the auxiliary equation (AE) approach, the improved $F$-expansion method, and modified Sardar sub equation (MSSE) method to investigate complex wave structures for plentiful solutions associated with the fractional perturbed Gerdjikov-Ivanov (PGI) model with the $M$-fractional operator. The investigated model is a well-established mathematical model used to represent a variety of physical events in nonlinear dynamics and mathematical physics. By using the aforementioned techniques, we scrutinize some new optical wave solutions in the forms of dark, bright, periodic, combo, $W$-shaped, $M$-shape, $V$-shape, kink type, singular rational, exponential, trigonometric, and hyperbolic solutions. The acquired solutions address a wide range of optical solutions in the form of 3D-plots, contour plots, and 2D plots, declaring the free parameters of such optical soliton solutions and comprehending their dynamic behaviour. Also, the sensitive analysis of selected model is analyzed. \textcolor{green}{The main contribution of this study is to extract diverse solitary wave solutions of the adopted model. Some of the solutions are similar and some diverge from the previous solutions which justify the novelty of the study.} Finally, we discovered that the current technique provides a reliable instrument for investigating the analytic solutions of fractional differential equations. The proposed PGI model can be used to transmit ultra-fast pulses across optical fibers. This research goes beyond to the advancement of mathematical techniques for solving fractional differential equations and broadens their application to a wide range of real-world scientific and engineering problems.

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Section: Introductionmentioning

confidence: 99%

Soliton solutions and sensitive analysis to nonlinear wave model arising in optics

Lu,

Ahmad,

Akram

et al. 2024

Phys. Scr.

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Analytical treatment with the Nucci reduction technique on the p-forced nonlinear Klein–Gordon equation

Hashemi,

Gulsen,

Inc

et al. 2023

Opt Quant Electron

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Novel exact solutions to the perturbed Gerdjikov–Ivanov equation

Youssoufa,

Gulsen,

Hashemi

et al. 2024

Opt Quant Electron

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This paper introduces the Nucci reduction method, a novel and efficient approach for deriving exact solutions to the perturbed Gerdjikov–Ivanov equation, offering a significant advancement in the field. The suggested technique involves transforming the equation into real and imaginary components prior to application. We successfully obtained four distinct exact and explicit solutions, along with the corresponding first integrals. Explanations and presentations of solutions are given in a logical manner. We derive an analytical expression for the instability gain and examine its key features using linear stability analysis. Finally, we compare the correctness of the analytical and numerical solutions. We demonstrate the robustness and stability of solitary waves through numerical simulations.

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Nonclassical symmetry analysis and heir-equations of forced Burger equation with time variable coefficients

Cited by 6 publications

References 36 publications

Soliton solutions and sensitive analysis to nonlinear wave model arising in optics

Soliton solutions and sensitive analysis to nonlinear wave model arising in optics

Analytical treatment with the Nucci reduction technique on the p-forced nonlinear Klein–Gordon equation

Novel exact solutions to the perturbed Gerdjikov–Ivanov equation

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