Search Results (20)

This work investigates the nonlinear flexural dynamics of a macroscale cantilever beam by combining analytical modeling, symbolic solution techniques, numerical simulation, and vision-based experiments. Starting from the Euler–Bernoulli equation with geometric and inertial nonlinearities, a reduced-order model is derived via a single-mode Galerkin projection, justified by the experimentally confirmed dominance of the fundamental bending mode. The resulting nonlinear ordinary differential equation is solved analytically using two symbolic methods rarely applied in structural vibration studies: the Extended Direct Algebraic Method (EDAM) and the Sardar Sub-Equation Method (SSEM). Comparison with high-accuracy numerical integration shows that EDAM reproduces the nonlinear waveform with high fidelity, including the characteristic non-sinusoidal distortion induced by mid-plane stretching. High-speed vision-based measurements provide displacement data for a physical cantilever beam undergoing free vibration. After calibrating the linear stiffness, analytical and experimental responses are compared in terms of the dominant oscillation frequency. The analytical model predicts the classical hardening-type amplitude–frequency dependence of an ideal Euler–Bernoulli cantilever, whereas the experiment exhibits a clear softening trend. This contrast reveals the influence of real-world effects, such as initial curvature, boundary compliance, or micro-slip at the clamp, which are absent from the idealized formulation. The combined analytical–experimental framework thus acts as a diagnostic tool for identifying competing nonlinear mechanisms in flexible structures and provides a compact physics-based reference for reduced-order modeling and structural health monitoring. Full article

This study investigates the convective–diffusive Cahn–Hilliard equation, a nonlinear model which is used in real-world applications to phase separation and material pattern formation. Using the modified Sardar sub-problem technique, which is an extension of the Sardar sub-equation approach, we derive multiple classes of exact soliton solutions, including bright, dark, kink, and periodic forms. The parametric behaviors of these solutions are examined and visualized through analytical plots generated in Mathematica and Maple. Furthermore, UV–Vis spectrophotometry is employed to examine the optical response of copper oxide (CuO) thin films. The films exhibited a sharp absorption edge around 380–410 nm and an optical band gap of approximately 2.3 eV, confirming their semiconducting nature. The experimentally observed periodic transmission characteristics are linked with the theoretical soliton profiles predicted by the model. Overall, the proposed analytical and experimental framework establishes a clear connection between nonlinear wave theory and thin-film optical characterization, providing new insights into soliton transformation phenomena in complex material systems. Full article

This work investigates the coupled Ivancevic option-pricing model, a nonlinear wave alternative to the Black–Scholes model. By utilizing the recently developed Kumar-Malik method, modified Sardar sub-equation method and the generalized Arnous method, the substantial results of this research are the successful derivation of novel exact soliton solutions, including bright, singular, dark, combined dark–bright, singular-periodic, complex solitons, exponential and Jacobi elliptic functions. A detailed analysis of option price wave functions and modulation instability analysis is conducted, with the conditions for valid solutions outlined. Additionally, a mathematical framework is established to capture market price fluctuations. Numerical simulations, illustrated through 2D, 3D and contour graphs, highlight the effects of parameter variations. Our findings demonstrate the effectiveness of the coupled Ivancevic model as a fractional nonlinear wave system, providing valuable insights into stock volatility and returns. This study contributes to creating new option-pricing models, which affect financial market analysis and risk management. Full article

The present paper examines a novel exact solution to nonlinear fractional partial differential equations (FDEs) through the Sardar sub-equation method (SSEM) coupled with Jumarie’s Modified Riemann–Liouville derivative (JMRLD). We take the (3+1)-dimensional space–time fractional modified Korteweg-de Vries (KdV) -Zakharov-Kuznetsov (ZK) equation as a case study, which describes some intricate phenomena of wave behavior in plasma physics and fluid dynamics. With the implementation of SSEM, we yield new solitary wave solutions and explicitly examine the role of the fractional-order parameter in the dynamics of the solutions. In addition, the sensitivity analysis of the results is conducted in the Galilean transformation in order to ensure that the obtained results are valid and have physical significance. Besides expanding the toolbox of analytical methods to address high-dimensional nonlinear FDEs, the proposed method helps to better understand how fractional-order dynamics affect the nonlinear wave phenomenon. The results are compared to known methods and a discussion about their possible applications and limitations is given. The results show the effectiveness and flexibility of SSEM along with JMRLD in forming new categories of exact solutions to nonlinear fractional models. Full article

Gravity water waves in the shallow-ocean scenario described by generalized Boussinesq–Broer–Kaup–Whitham (gBBKW) equations are discussed. The residual symmetry and Bäcklund transformation associated with the gBBKW equations are systematically constructed. The time and space evolution of wave velocity and height are explored. Additionally, it is demonstrated that the gBBKW equations are solvable through the consistent Riccati expansion method. Leveraging this property, a novel Bäcklund transformation, solitary wave solution, and soliton–cnoidal wave solution are derived. Furthermore, miscellaneous novel solutions of gBBKW equations are obtained using the modified Sardar sub-equation method. The impact of variations in the diffusion power parameter on wave velocity and height is quantitatively analyzed. The exact solutions of gBBKW equations provide precise description of propagation characteristics for a deeper understanding and the prediction of some ocean wave phenomena. Full article

This study uses a conformable derivative of order $\beta$ to investigate a fractional Whitham–Broer–Kaup ($\mathcal{FWBK}$) model. This model has significant uses in several scientific domains, such as plasma physics and nonlinear optics. The enhanced modified Sardar sub-equation $\mathcal{EMSSE}$ approach is applied to achieve precise analytical solutions, demonstrating its effectiveness in resolving complex wave photons. Bright, solitary, trigonometric, dark, and plane waves are among the various wave dynamics that may be effectively and precisely determined using the $\mathcal{FWBK}$ model. Furthermore, the study explores the chaotic behaviour of both perturbed and unperturbed systems, revealing illumination on their dynamic characteristics. By demonstrating its validity in examining wave propagation in nonlinear fractional systems, the effectiveness and reliability of the suggested method in fractional modelling are confirmed through thorough investigation. Full article

This work is concerned with Hirota bilinear, ${exp}_a$ function, and Sardar sub-equation methods to find the breather-wave, 1-Soliton, 2-Soliton, three-wave, and new periodic-wave results and some exact solitons of the special (1 + 1)-dimensional Korteweg–de Vries (KdV) equation. The model of concern is a partial differential equation that is used as a mathematical model of waves on shallow water surfaces. The results are attained as well as verified by Mathematica and Maple softwares. Some of the obtained solutions are represented in three-dimensional (3-D) and contour plots through the Mathematica tool. A stability analysis is performed to verify that the results are precise as well as accurate. Modulation instability is also performed for the steady-state solutions to the governing equation. The solutions are useful for the development of corresponding equations. This work shows that the methods used are simple and fruitful for investigating the results for other nonlinear partial differential models. Full article

New types of truncated M-fractional wave solitons to the simplified Modified Camassa–Holm model, a mathematical physics model, are obtained. This model is used to explain the unidirectional propagation of shallow water waves. The required solutions are obtained by utilizing the simplest equation, the Sardar subequation, and the generalized Kudryashov schemes. The obtained results consist of the dark, singular, periodic, dark-bright, and many other analytical solitons. Dynamical behaviors of some obtained solutions are represented by two-dimensional (2D), three-dimensional (3D), and Contour graphs. An effect of fractional derivative is shown graphically. The results are newer than the existing results of the governing equation. Obtained solutions have much importance in the various areas of applied science as well as engineering. We concluded that the utilized methods are helpful and applicable for other partial fractional equations in applied science and engineering. Full article

The current work investigates a recently introduced unidirectional wave model, applicable in science and engineering to understand complex systems and phenomena. This investigation has two primary aims. First, it employs a novel modified Sardar sub-equation method, not yet explored in the literature, to derive new solutions for the governing model. Second, it analyzes the complex dynamical structure of the governing model using bifurcation, chaos, and sensitivity analyses. To provide a more accurate depiction of the underlying dynamics, they use quantum mechanics to explain the intricate behavior of the system. To illustrate the physical behavior of the obtained solutions, 2D and 3D plots, along with a phase plane analysis, are presented using appropriate parameter values. These results validate the effectiveness of the employed method, providing thorough and consistent solutions with significant computational efficiency. The investigated soliton solutions will be valuable in understanding complex physical structures in various scientific fields, including ferromagnetic dynamics, nonlinear optics, soliton wave theory, and fiber optics. This approach proves highly effective in handling the complexities inherent in engineering and mathematical problems, especially those involving fractional-order systems. Full article

Solitary waves, inherent in nonlinear wave equations, manifest across various physical systems like water waves, optical fibers, and plasma waves. In this study, we present this type of wave solution within the integrable Mikhailov–Novikov–Wang (MNW) equation, an integrable system known for representing localized disturbances that persist without dispersing, retaining their form and coherence over extended distances, thereby playing a pivotal role in understanding nonlinear dynamics and wave phenomena. Beyond this innovative work, we examine the stability and modulation instability of its gained solutions. These new solitary wave solutions have potential applications in telecommunications, spectroscopy, imaging, signal processing, and pulse modeling, as well as in economic systems and markets. To derive these solitary wave solutions, we employ two effective methods: the improved Sardar subequation method and the (℧′/℧, 1/℧) method. Through these methods, we develop a diverse array of waveforms, including hyperbolic, trigonometric, and rational functions. We thoroughly validated our results using Mathematica software to ensure their accuracy. Vigorous graphical representations showcase a variety of soliton patterns, including dark, singular, kink, anti-kink, and hyperbolic-shaped patterns. These findings highlight the effectiveness of these methods in showing novel solutions. The utilization of these methods significantly contributes to the derivation of novel soliton solutions for the MNW equation, holding promise for diverse applications throughout different scientific domains. Full article

The Sharma–Tasso–Olver (STO) equation is a nonlinear, double-dispersive, partial differential equation that is physically important because it provides insights into the behavior of nonlinear waves and solitons in various physical areas, including fluid dynamics, optical fibers, and plasma physics. In this paper, the STO equation is generalized to a fractional equation by using Atangana (or Atangana–Baleanu) fractional space and time beta-derivatives since they have been found to be useful as a model for a variety of traveling-wave phenomena. Exact solutions are obtained for the integer-order and fractional-order equations by using the Sardar subequation method and an appropriate traveling-wave transformation. The exact solutions are obtained in terms of generalized trigonometric and hyperbolic functions. The exact solutions are derived for the integer-order STO and for a range of values of fractional orders. Numerical solutions are also obtained for a range of parameter values for both the fractional and integer orders to show some of the types of solutions that can occur. As examples, the solutions are obtained showing the physical behavior, such as the solitary wave solutions of the singular kink-type and periodic wave solutions. The results show that the Sardar subequation method provides a straightforward and efficient method for deriving new exact solutions for fractional nonlinear partial differential equations of the STO type. Full article

Fractional calculus with symmetric kernels is a fast-growing field of mathematics with many applications in all branches of science and engineering, notably electromagnetic, biology, optics, viscoelasticity, fluid mechanics, electrochemistry, and signals processing. With the use of the Sardar sub-equation and the Bernoulli sub-ODE methods, new trigonometric and hyperbolic solutions to the time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation have been constructed in this paper. Notably, the definition of our fractional derivative is based on the Jumarie’s modified Riemann–Liouville derivative, which offers a strong basis for our mathematical explorations. This equation is widely utilized to report a variety of fascinating physical events in the domains of classical mechanics, plasma physics, fluid dynamics, heat transfer, and acoustics. It is presumed that the acquired outcomes have not been documented in earlier research. Numerous standard wave profiles, such as kink, smooth bell-shaped and anti-bell-shaped soliton, W-shaped, M-shaped, multi-wave, periodic, bright singular and dark singular soliton, and combined dark and bright soliton, are illustrated in order to thoroughly analyze the wave nature of the solutions. Painlevé analysis of the proposed study is also part of this work. To illustrate how the fractional derivative affects the precise solutions of the equation via 2D and 3D plots. Full article

The time-fractional coupled Drinfel’d–Sokolov–Wilson (DSW) equation is pivotal in soliton theory, especially for water wave mechanics. Its precise description of soliton phenomena in dispersive water waves makes it widely applicable in fluid dynamics and related fields like tsunami prediction, mathematical physics, and plasma physics. In this study, we present novel soliton solutions for the DSW equation, which significantly enhance the accuracy of describing soliton phenomena. To achieve these results, we employed two distinct methods to derive the solutions: the Sardar subequation method, which works with one variable, and the $\left(\frac{{\mathsf{\Omega }}^{\prime }}{\mathsf{\Omega }}, \frac{1}{\mathsf{\Omega }}\right)$ method which utilizes two variables. These approaches supply significant improvements in efficiency, accuracy, and the ability to explore a broader spectrum of soliton solutions compared to traditional computational methods. By using these techniques, we construct a wide range of wave structures, including rational, trigonometric, and hyperbolic functions. Rigorous validation with Mathematica software 13.1 ensures precision, while dynamic visual representations illustrate soliton solutions with diverse patterns such as dark solitons, multiple dark solitons, singular solitons, multiple singular solitons, kink solitons, bright solitons, and bell-shaped patterns. These findings highlight the effectiveness of these methods in discovering new soliton solutions and supplying deeper insights into the DSW model’s behavior. The novel soliton solutions obtained in this study significantly enhance our understanding of the DSW equation’s underlying dynamics and offer potential applications across various scientific fields. Full article

An essential mathematical structure that demonstrates the nonlinear short-wave movement across the ferromagnetic materials having zero conductivity in an exterior region is known as the fractional stochastic Kraenkel–Manna–Merle system. In this article, we extract abundant wave structure closed-form soliton solutions to the fractional stochastic Kraenkel–Manna–Merle system with some important analyses, such as bifurcation analysis, chaotic behaviors, sensitivity, and modulation instability. This fractional system renders a substantial impact on signal transmission, information systems, control theory, condensed matter physics, dynamics of chemical reactions, optical fiber communication, electromagnetism, image analysis, species coexistence, speech recognition, financial market behavior, etc. The Sardar sub-equation approach was implemented to generate several genuine innovative closed-form soliton solutions. Additionally, phase portraiture of bifurcation analysis, chaotic behaviors, sensitivity, and modulation instability were employed to monitor the qualitative characteristics of the dynamical system. A certain number of the accumulated outcomes were graphed, including singular shape, kink-shaped, soliton-shaped, and dark kink-shaped soliton in terms of 3D and contour plots to better understand the physical mechanisms of fractional system. The results show that the proposed methodology with analysis in comparison with the other methods is very structured, simple, and extremely successful in analyzing the behavior of nonlinear evolution equations in the field of fractional PDEs. Assessments from this study can be utilized to provide theoretical advice for improving the fidelity and efficiency of soliton dissemination. Full article

Recently, nonlinear fractional models have become increasingly important for describing phenomena occurring in science and engineering fields, especially those including symmetric kernels. In the current article, we examine two reliable methods for solving fractional coupled nonlinear Schrödinger models. These methods are known as the Sardar-subequation technique (SSET) and the improved generalized tanh-function technique (IGTHFT). Numerous novel soliton solutions are computed using different formats, such as periodic, bell-shaped, dark, and combination single bright along with kink, periodic, and single soliton solutions. Additionally, single solitary wave, multi-wave, and periodic kink combined solutions are evaluated. The behavioral traits of the retrieved solutions are illustrated by certain distinctive two-dimensional, three-dimensional, and contour graphs. The results are encouraging, since they show that the suggested methods are trustworthy, consistent, and efficient in finding accurate solutions to the various challenging nonlinear problems that have recently surfaced in applied sciences, engineering, and nonlinear optics. Full article