Ahead of Publication / Just Accepted

We show that numerical simulations of particular solutions of the Gross–Pitaevskii equation with parabolic potential can have features of both regular and chaotic dynamics, depending on initial conditions. Such behavior is a characteristic feature of systems with Kolmogorov–Arnold–Moser chaos. We also report analytical arguments that could support such interpretation.

Rotational dusty fluid flow refers to the motion of a mixture of fluid and solid particles in a rotating frame of reference. Various industrial processes, such as oil drilling, chemical processing, and materials manufacturing, have applications of rotational dusty fluid flow. Studying the dynamics of rotational dusty fluid flow is crucial for optimizing these industrial processes and improving their efficiency. The research focuses on understanding the behavior of a dusty fluid in a horizontal channel subjected to the combined influences of rotation and a magnetic field. The flow is driven by a constant pressure gradient and the movement of the upper plate, with the fluid flowing between two parallel plates. To analyze this system, a set of coupled partial differential equations governing the motion of the fluid and dust particles is developed. These equations account for both primary and secondary velocity components of the fluid and dust. To solve them, the study employs a meshfree radial basis function pseudospectral method. This advanced numerical technique is known for its flexibility in solving complex systems of partial differential equations without requiring structured grids, enabling high accuracy even in scenarios with irregular geometries or boundary conditions. The computed velocity profiles are then used to evaluate the pumping power needed to sustain flow in the absence of the pressure gradient. Results are presented through graphical analysis, showcasing the effects of key fluid parameters such as the Coriolis frequency parameter, dust particle concentration parameter, Reynolds number, Ekman number, ion slip parameter, and Hall parameter. Notably, the findings reveal that an increase in the Coriolis frequency reduces the primary velocity while increasing the secondary velocity. This behavior arises because the Coriolis force, which acts perpendicular to the flow direction, distorts the velocity profile, creating a complex interplay between rotational and flow dynamics.

This paper proposes a new transverse-torsional coupled planetary gear transmission model with a self-adjustment backlash control model, which offers an effective approach to reducing vibration and controlling system stability. The nonlinear governing equation is derived, and the system dynamics behaviors and double-parameter stability region are explored. Several valuable conclusions can be drawn. First, bifurcation diagrams under different parameters are obtained. Colorful bifurcation characteristics are observed, and different parameters cause remarkable differences in the system dynamic response. Second, different initial values can lead to dramatically different motion states, and the change in system input parameters can significantly affect the attractor’s structure. Third, solution domain structures under different parameter combinations are obtained and comprehensively analyzed. A flame-shaped structure is observed and evolves with the change of parameters. Fourth, double-parameter stable regions are presented and carefully analyzed, providing an effective visual tool to study system stability and the variation trend caused by parameter changes. The research on bifurcation characteristics, initial attractor, solution domain structure, and double-parameter stability significantly reveals the colorful nonlinear behaviors of the planetary gear system. The study in this paper could be utilized as the optimization method for the design of gear transmission system and a guide for field problems.

In this paper, we present a concise overview of fractal calculus and explore the solution of non-homogeneous fractal differential equations. We analyze fractal homogeneous linear systems with initial conditions, introducing the fundamental matrix and special fundamental matrix, and demonstrate their applications in solving systems and analyzing the Jordan form of matrices. We propose the method of undetermined coefficients for solving non-homogeneous fractal linear differential equations and introduce the method of variation of parameters as a supplementary technique. To illustrate these methods, we apply them to the differential equations of resistor–inductor–capacitor (RLC) circuits, successfully solving the corresponding fractal differential equations. Additionally, we provide examples, solve systems with initial conditions, and present the results through plotted graphs.

The (2 + 1)-dimensional stochastic coupled nonlinear Schrödinger system with multiplicative white noise has significant effect in several fields as statistical mechanics, ecological systems, nonlinear optics, plasma physics, optical-fiber communications, and so forth. Through this article we will extract variety forms of the analytical solutions that represent the dynamical behaviors of the optical soliton solutions and the other rational solutions as well as the soliton velocity emerged by each method to the suggested model. For this purpose, we will utilize four various techniques namely the generalized Kudryashov schema, the Paul–Painleve approach schema, extended simple equation schema and Ricatti–Bernoulli sub order schema that will be applied individually to explore these various types of these optical soliton solutions. The suggested methods will be implemented independently and parallel. Multiple types of optical soliton solutions are detected in forms like the bright-dark, periodic solitons, W-like soliton, some other traveling wave solutions in forms trigonometric, hyperbolic, Jacobi elliptic rational function and solitary wave solutions are also explored. The 2-D and 3D Figure simulations of the dynamical soliton behaviors of all obtained solutions have been documented. All achieved solutions as well as 2D and 3D plots have been established by Mathematica program.

This study presents the development of a composite controller by combining the optimal robust controller(ORC) and a back-stepping controller, in presence of unmatched model uncertainty, to achieve vibration reduction and joint position tracking in a two link flexible manipulator (TLFM). The dynamics of the TLFM have been modelled using the assumed mode method, incorporating mode shapes to accurately represent the flexible links’ vibrations. Two mode shapes have been considered for each link. One source of uncertainty in the system has been identified as the payload mass. The manipulator responds differently when the payload mass connected to the end-effector is changed, demonstrating the effectiveness of the controller. The closed-loop stability is established utilizing the Lyapunov method, and trajectory convergence has been demonstrated through Barbalat’s Lemma. Unknown gains present in the controller’s formulation have been fine-tuned using the FOX optimization algorithm (FOXOA) and arithmetic optimization algorithm (AOA). The simulation results showcase the impact of both optimization techniques on the controller’s performance, validating its robustness and efficacy in the presence of uncertainties. In the simulation section, it has been found that the performance of the TLFM with FOXOA is better compared to AOA.

This study presents a ϕ 6 -model technique for the Ivancevic option pricing model (IOPM) in the sense of M-truncated fractional derivative to successfully derive traveling wave solutions. In addition to other new findings, periodic, dark, and bright topological solutions are found. To the best of our knowledge, this research has not been previously addressed in the literature. The work improves our comprehension of the model by revisiting the concept of solitary waves. Prior research has already employed a variety of techniques to derive analytical solutions, which has contributed to identify novel soliton solutions within this framework. To improve visualization, we also assign particular parameter values to generate 2D, 3D, and contour plots for some of the discovered solutions. By revealing significant information about the dynamics and patterns of the solutions, these illustrations help to clarify the behavior of the model. Furthermore, the effect of the fractional parameter on wave pattern propagation is also investigated. The outcomes show that our suggested approach is a useful mathematical tool for resolving non-linear evolution equations (NLEES) by producing fresh, more thorough solutions.