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22 pages, 7850 KB

Open AccessArticle

Bifurcation Analysis and Solitons Dynamics of the Fractional Biswas–Arshed Equation via Analytical Method

by Asim Zafar, Waseem Razzaq, Abdullah Nazir, Mohammed Ahmed Alomair, Abdulaziz S. Al Naim and Abdulrahman Alomair

Mathematics 2025, 13(19), 3147; https://doi.org/10.3390/math13193147 - 1 Oct 2025

Viewed by 240

Abstract

This paper investigates soliton solutions of the time-fractional Biswas–Arshed (BA) equation using the Extended Simplest Equation Method (ESEM). The model is analyzed under two distinct fractional derivative operators: the

β

-derivative and the M-truncated derivative. These approaches yield diverse solution types, including [...] Read more.

This paper investigates soliton solutions of the time-fractional Biswas–Arshed (BA) equation using the Extended Simplest Equation Method (ESEM). The model is analyzed under two distinct fractional derivative operators: the

β

-derivative and the M-truncated derivative. These approaches yield diverse solution types, including kink, singular, and periodic-singular forms. Also, in this work, a nonlinear second-order differential equation is reconstructed as a planar dynamical system in order to study its bifurcation structure. The stability and nature of equilibrium points are established using a conserved Hamiltonian and phase space analysis. A bifurcation parameter that determines the change from center to saddle-type behaviors is identified in the study. The findings provide insight into the fundamental dynamics of nonlinear wave propagation by showing how changes in model parameters induce qualitative changes in the phase portrait. The derived solutions are depicted via contour plots, along with two-dimensional (2D) and three-dimensional (3D) representations, utilizing Mathematica for computational validation and graphical illustration. This study is motivated by the growing role of fractional calculus in modeling nonlinear wave phenomena where memory and hereditary effects cannot be captured by classical integer-order approaches. The time-fractional Biswas–Arshed (BA) equation is investigated to obtain diverse soliton solutions using the Extended Simplest Equation Method (ESEM) under the

β

-derivative and M-truncated derivative operators. Beyond solution construction, a nonlinear second-order equation is reformulated as a planar dynamical system to analyze its bifurcation and stability properties. This dual approach highlights how parameter variations affect equilibrium structures and soliton behaviors, offering both theoretical insights and potential applications in physics and engineering. Full article

(This article belongs to the Special Issue Analytical Methods and Qualitative Analysis for Differential Equations)

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16 pages, 6992 KB

Open AccessArticle

Truncated M-Fractional Exact Solutions, Stability Analysis, and Modulation Instability of the Classical Lonngren Wave Model

by Haitham Qawaqneh and Abdulaziz Khalid Alsharidi

Mathematics 2025, 13(19), 3107; https://doi.org/10.3390/math13193107 - 28 Sep 2025

Viewed by 285

Abstract

Many types of exact solutions to the truncated M-fractional classical Lonngren wave model are explored in this paper. The classical Lonngren wave model is a significant electronics equation. This model is used to explain the electronic signals within semiconductor materials, especially tunnel diodes. [...] Read more.

Many types of exact solutions to the truncated M-fractional classical Lonngren wave model are explored in this paper. The classical Lonngren wave model is a significant electronics equation. This model is used to explain the electronic signals within semiconductor materials, especially tunnel diodes. Through the application of a modified

(G^{'} / G^{2})

-expansion technique and an extended sinh-Gordon equation expansion (EShGEE) method, we obtained various wave solutions, including periodic, kink, singular, dark, bright, and dark–bright types, among others. To ensure that the solutions in question are stable, linear stability analysis is also carried out. Moreover, the stationary solutions of the concerning equation are studied through modulation instability. The obtained results are useful in various areas, including electronic physics, soliton physics, plasma physics, nonlinear optics, acoustics, etc. Both techniques are useful for solving nonlinear partial fractional differential equations. Both techniques provide exact solutions, which can be important for understanding complex phenomena. Both techniques are reliable and yield distinct types of exact soliton solutions. Full article

(This article belongs to the Special Issue Analytical Methods and Qualitative Analysis for Differential Equations)

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22 pages, 76128 KB

Open AccessArticle

Nonlinear Wave Structures, Multistability, and Chaotic Behavior of Quantum Dust-Acoustic Shocks in Dusty Plasma with Size Distribution Effects

by Huanbin Xue and Lei Zhang

Mathematics 2025, 13(19), 3101; https://doi.org/10.3390/math13193101 - 27 Sep 2025

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Abstract

This paper presents a detailed study of the

(3 + 1)

-dimensional Zakharov–Kuznetsov–Burgers equation to investigate shock-wave phenomena in dusty plasmas with quantum effects. The model provides significant physical insight into nonlinear dispersive and dissipative structures arising in charged-dust–ion environments, corresponding [...] Read more.

This paper presents a detailed study of the

(3 + 1)

-dimensional Zakharov–Kuznetsov–Burgers equation to investigate shock-wave phenomena in dusty plasmas with quantum effects. The model provides significant physical insight into nonlinear dispersive and dissipative structures arising in charged-dust–ion environments, corresponding to both laboratory and astrophysical plasmas. We then perform a qualitative, numerically assisted dynamical analysis using bifurcation diagrams, multistability checks, return maps, Poincaré sections, and phase portraits. For both the unperturbed and a perturbed system, we identify chaotic, quasi-periodic, and periodic regimes from these numerical diagnostics; accordingly, our dynamical conclusions are qualitative. We also examine frequency-response and time-delay sensitivity, providing a qualitative classification of nonlinear behavior across a broad parameter range. After establishing the global dynamical picture, traveling-wave solutions are obtained using the Paul–Painlevé approach. These solutions represent shock and solitary structures in the plasma system, thereby bridging the analytical and dynamical perspectives. The significance of this study lies in combining a detailed dynamical framework with exact traveling-wave solutions, allowing a deeper understanding of nonlinear shock dynamics in quantum dusty plasmas. These results not only advance theoretical plasma modeling but also hold potential applications in plasma-based devices, wave propagation in optical fibers, and astrophysical plasma environments. Full article

(This article belongs to the Special Issue Analytical Methods and Qualitative Analysis for Differential Equations)

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17 pages, 915 KB

Open AccessFeature PaperArticle

Solutions for Linear Fractional Differential Equations with Multiple Constraints Using Fractional B-Poly Bases

by Md. Habibur Rahman, Muhammad I. Bhatti and Nicholas Dimakis

Mathematics 2025, 13(19), 3084; https://doi.org/10.3390/math13193084 - 25 Sep 2025

Viewed by 292

Abstract

This study presents an innovative numerical method for solving linear fractional differential equations (LFDEs) using modified Bernstein polynomial bases. The proposed approach effectively addresses the challenges posed by the nonlocal nature of fractional derivatives, providing a robust framework for handling multiple initial and [...] Read more.

This study presents an innovative numerical method for solving linear fractional differential equations (LFDEs) using modified Bernstein polynomial bases. The proposed approach effectively addresses the challenges posed by the nonlocal nature of fractional derivatives, providing a robust framework for handling multiple initial and boundary value constraints. By integrating the LFDEs and approximating the solutions with modified fractional-order Bernstein polynomials, we derive operational matrices to solve the resulting system numerically. The method’s accuracy is validated through several examples, showing excellent agreement between numerical and exact solutions. Comparative analysis with existing data further confirms the reliability of the approach, with absolute errors ranging from 10⁻¹⁸ to 10⁻⁴. The results highlight the method’s efficiency and versatility in modeling complex systems governed by fractional dynamics. This work offers a computationally efficient and accurate tool for fractional calculus applications in science and engineering, helping to bridge existing gaps in numerical techniques. Full article

(This article belongs to the Special Issue Analytical Methods and Qualitative Analysis for Differential Equations)

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21 pages, 2434 KB

Open AccessFeature PaperArticle

Very Large Angular Oscillations (Up to 3π/4) of the Physical Pendulum—A Simple Trigonometric Analytical Solution

by Joao C. Fernandes

Mathematics 2025, 13(17), 2836; https://doi.org/10.3390/math13172836 - 3 Sep 2025

Viewed by 580

Abstract

The oscillatory properties of pendular motion, along with the associated energetic conditions, are used to induce analytical functions capable of simultaneously describing the angular position and velocity. To describe the angular position of a generic pendulum, for very large amplitudes of oscillation, we [...] Read more.

The oscillatory properties of pendular motion, along with the associated energetic conditions, are used to induce analytical functions capable of simultaneously describing the angular position and velocity. To describe the angular position of a generic pendulum, for very large amplitudes of oscillation, we used the numerical solutions obtained from the numerical resolution of the differential equation of motion. The solver software needed was built using the LabView 2019 platform, but any other ODE solver containing peak and valley detectors can be used. The fitting software and plots were performed with the ORIGIN 7.0 program, but also other equivalent programs can be used. For a non-damped pendulum, an analytical model is proposed, built from simple trigonometric functions, but containing the important physical information of the dependence between the period and amplitude of oscillation. The application of the proposed model, using the numerical solutions of the non-approximated differential equation of motion, shows very good agreement, less than 0.01%, for large amplitudes, up to 3π/4. Full article

(This article belongs to the Special Issue Analytical Methods and Qualitative Analysis for Differential Equations)

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26 pages, 4796 KB

Open AccessArticle

Novel Analytical Methods for and Qualitative Analysis of the Generalized Water Wave Equation

by Haitham Qawaqneh, Abdulaziz S. Al Naim and Abdulrahman Alomair

Mathematics 2025, 13(14), 2280; https://doi.org/10.3390/math13142280 - 15 Jul 2025

Viewed by 341

Abstract

For a significant fluid model and the truncated M-fractional (1 + 1)-dimensional nonlinear generalized water wave equation, distinct types of truncated M-fractional wave solitons are obtained. Ocean waves, tidal waves, weather simulations, river and irrigation flows, tsunami predictions, and more are all explained [...] Read more.

For a significant fluid model and the truncated M-fractional (1 + 1)-dimensional nonlinear generalized water wave equation, distinct types of truncated M-fractional wave solitons are obtained. Ocean waves, tidal waves, weather simulations, river and irrigation flows, tsunami predictions, and more are all explained by this model. We use the improved

(G^{'} / G)

expansion technique and a modified extended direct algebraic technique to obtain these solutions. Results for trigonometry, hyperbolic, and rational functions are obtained. The impact of the fractional-order derivative is also covered. We use Mathematica software to verify our findings. Furthermore, we use contour graphs in two and three dimensions to illustrate some wave solitons that are obtained. The results obtained have applications in ocean engineering, fluid dynamics, and other fields. The stability analysis of the considered equation is also performed. Moreover, the stationary solutions of the concerning equation are studied through modulation instability. Furthermore, the used methods are useful for other nonlinear fractional partial differential equations in different areas of applied science and engineering. Full article

(This article belongs to the Special Issue Analytical Methods and Qualitative Analysis for Differential Equations)

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17 pages, 1944 KB

Open AccessArticle

Abundant Elliptic, Trigonometric, and Hyperbolic Stochastic Solutions for the Stochastic Wu–Zhang System in Quantum Mechanics

by Wael W. Mohammed, Ekram E. Ali, Athar I. Ahmed and Marwa Ennaceur

Mathematics 2025, 13(5), 714; https://doi.org/10.3390/math13050714 - 22 Feb 2025

Cited by 1 | Viewed by 894

Abstract

In this article, we look at the stochastic Wu–Zhang system (SWZS) forced by multiplicative Brownian motion in the Itô sense. The mapping method, which is an effective analytical method, is employed to investigate the exact wave solutions of the aforementioned equation. The proposed [...] Read more.

In this article, we look at the stochastic Wu–Zhang system (SWZS) forced by multiplicative Brownian motion in the Itô sense. The mapping method, which is an effective analytical method, is employed to investigate the exact wave solutions of the aforementioned equation. The proposed scheme provides new types of exact solutions including periodic solitons, kink solitons, singular solitons and so on, to describe the wave propagation in quantum mechanics and analyze a wide range of essential physical phenomena. In the absence of noise, we obtain some previously found solutions of SWZS. Additionally, using the MATLAB program, the impacts of the noise term on the analytical solution of the SWZS were demonstrated. Full article

(This article belongs to the Special Issue Analytical Methods and Qualitative Analysis for Differential Equations)

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21 pages, 2438 KB

Open AccessFeature PaperArticle

Investigations of Modified Classical Dynamical Models: Melnikov’s Approach, Simulations and Applications, and Probabilistic Control of Perturbations

by Nikolay Kyurkchiev, Tsvetelin Zaevski, Anton Iliev, Vesselin Kyurkchiev and Asen Rahnev

Mathematics 2025, 13(2), 231; https://doi.org/10.3390/math13020231 - 11 Jan 2025

Cited by 3 | Viewed by 772

Abstract

We suggest a few kinds of extended classical oscillators in this study. We present a few specific modules for examining these oscillators’ behavior. This will be an essential component of a broader web-based scientific computing platform that is in the works. The modeling [...] Read more.

We suggest a few kinds of extended classical oscillators in this study. We present a few specific modules for examining these oscillators’ behavior. This will be an essential component of a broader web-based scientific computing platform that is in the works. The modeling and synthesis of radiating antenna designs is also taken into consideration as a potential use case for Melnikov functions. Additionally, we discuss strategies for achieving probabilistic control over system perturbations. Full article

(This article belongs to the Special Issue Analytical Methods and Qualitative Analysis for Differential Equations)

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