This work examines the fractional generalized Korteweg-de-Vries-Zakharov-Kuznetsov equation (gKdV-ZKe) by utilizing three well-known analytical methods, the modified $$\left( \frac{G^{'}}{G^2}\right)$$G ′ G 2 -expansion method, $$\left( \frac{1}{G^{'}}\right)$$1 G ′ -expansion method and the Kudryashov method. The gKdV-ZK equation is a nonlinear model describing the influence of magnetic field on weak ion-acoustic waves in plasma made up of cool and hot electrons. The kink, singular, anti-kink, periodic, and bright soliton solutions are observed. The effect of the fractional parameter on wave shapes have been analyzed by displaying various graphs for fractional-order values of $$\beta$$β . In addition, we utilize the Hamiltonian property to observe the stability of the attained solution and Galilean transformation for sensitivity analysis. The suggested methods can also be utilized to evaluate the nonlinear models that are being developed in a variety of scientific and technological fields, such as plasma physics. Findings show the effectiveness simplicity, and generalizability of the chosen computational approach, even when applied to complex models.

This study employs spatial optimization principles to investigate the nonlinear vibration of a flexibly supported Euler–Bernoulli beam, a (1 + 1)-dimensional system subjected to axial loads. The modified Khater method, a crucial tool in mechanical engineering, is utilized to analyze analytical solutions, which include a symmetric spatial representation of the waveform as an integral part of each solution. Notably, periodic soliton solutions for the nonlinear model closely align with numerical and approximate analytical solutions, demonstrating the accuracy of our modeling approach. Density diagrams, contour diagrams, and Poincaré maps depicting the obtained analytical solutions are presented to elucidate their accuracy and provide visual confirmation of the optimized engineering model’s physical significance. The planar dynamical system is derived through the Galilean transformation by employing mathematical models and appropriate parameter values, thereby further refining problem understanding. Sensitivity analysis is conducted, and phase portraits with equilibrium points are illustrated by analyzing a special case of the investigated dynamical system, emphasizing its symmetrical properties. Lastly, we perform a global analysis to identify periodic, quasi-periodic, and chaotic behaviors, with an extra weak forcing term confirmed by Poincaré maps and a two-dimensional symmetric basin of the largest Lyapunov exponent.

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.

This paper exhibits a brief study of Marangoni convective, Activation energy, non-uniform heat generation, thermal-bioconvection and Oxytactic microorganisms, and time-dependent and stagnation point flow of MHD trihybrid nanofluid around a rotating sphere based on the extended model, referred to as the YOM (Yamada-Ota model) and XM (Xue model). The porous medium impact of Darcy-Forchheimer is included in the momentum equation. Some acceptable transformations are implemented to translate the governing system of PDEs (partial differential equations) and the relevant boundary conditions for computation. This model aims to evaluate the efficiency of two well-known trihybrid nanofluid models, Yamada-Ota and Xue. The shooting approach (BVP4C) is applied to derive the numerical estimates for the relevant differential expression system. The detailed analysis and simulation of trihybrid nanofluid flow patterns are facilitated by the variation in changing factors, which include temperature, velocity, concentration, and gyrotactic and Oxytactic microorganisms the profiles. The results indicate that the YOM performs significantly superior than the XM flow model in terms of heat transmission ability.

In the study of molecular atoms, numerous topological indices possess useful properties. This paper examines different types of topological indices, including the first Zagreb connection index, second Zagreb connection index, forgotten index, redefined Zagreb index, general randic index, and symmetric division degree index, to analyze the chemical and metallic structure. Furthermore, we explore the Zagreb Connection indices of Concealed non-kekulean benzenoid hydrocarbons (CNBH) and the Face Centered Cubic Lattice (FCC). The results are presented through mathematical quantities and visualized using 3D plots.