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In this paper, we establish iterative Hille-type criteria for advanced functional half-linear dynamic equations of the second order. These results extend and improve recent criteria established by multiple authors for the same equation and encompass classical criteria. We provide an example to demonstrate the significance of the results obtained. [ABSTRACT FROM AUTHOR]

The aim of this study is to refine the known Riccati transformation technique to provide new oscillation criteria for solutions to second-order dynamic equations over time. It is important to note that the convergence or divergence of some improper integrals on time scales depends not only on the integration function but also on the integration time scale. Therefore, there has been a motivation to find new oscillation criteria that can be applicable regardless of whether ∫ ζ 0 ∞ Δ ξ a (ξ) is convergent or divergent, in contrast to what has been followed in most previous works in the literature. We have provided an example to illustrate the significance of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we study the jerk vector that is the rate of change of the acceleration vector over time. In three‐dimensional space, the decomposition of the jerk vector is a new concept in the field. This decomposition expresses the jerk vector as the sum of three unique components in specific directions: the tangential direction, the radial direction in the osculating plane, and the radial direction in the rectifying plane. The snap vector is the rate of change of the jerk vector over time. In this paper, the authors examine non‐relativistic particles moving along non‐lightlike Frenet curves at low speeds compared to the speed of light in Minkowski 3‐space. They resolve the jerk and snap vectors using Frenet–Serret frames. Additionally, the cases for motion along non‐lightlike Frenet planar curves in the Minkowski 3‐space are given as corollaries. To help understand these results, the paper provides some illustrative examples [ABSTRACT FROM AUTHOR]

This paper focuses on establishing new criteria to guarantee the oscillation of solutions for second-order differential equations with a superlinear and a damping term. New sufficient conditions are presented, aimed at analysing the oscillatory properties of the solutions to the equation under study. To prove these results, we employed various analysis methods, establishing new relationships to address certain problems that have hindered previous research. Consequently, by applying the principles of comparison and the Riccati transformation, we obtained findings that develop and complement those reported in earlier literature. The significance of our results is illustrated with several examples. [ABSTRACT FROM AUTHOR]

The generalized nonlinear Schrödinger equation with M-truncated derivatives (GNLSE-MTD) is studied here. By using generalized Riccati equation and mapping methods, new elliptic, hyperbolic, trigonometric, and rational solutions are discovered. Because the GNLSE is widely employed in communication, heat pulse propagation in materials, optical fiber communication systems, and nonlinear optical phenomena, the resulting solutions may be used to analyze a wide variety of important physical phenomena. The dynamic behaviors of the various derived solutions are interpreted using 3-D and 2-D graphs to explain the affects of M-truncated derivatives. We can deduce that the surface shifts to the left when the order of M-truncated derivatives decreases. [ABSTRACT FROM AUTHOR]