The current research aims to investigate thermodynamic responses to thermal media based on a modified mathematical model in the field of thermoelasticity. In this context, it was considered to present a new model with a fractional time derivative that includes Caputo-Fabrizio and Atangana-Baleanu fractional differential operators within the framework of the two-phase delay model. The proposed mathematical model is employed to examine the problem of an unbounded material with a spherical hole experiencing a reduced moving heat flow on its inner surface. The problem is solved analytically within the modified space utilizing the Laplace transform as the solution mechanism. An arithmetic inversion of the Laplace transform was performed and presented visually and tabularly for the studied distributions. In the tables, specific comparisons are introduced to evaluate the influences of different fractional operators and thermal properties on the response of all the fields examined.

In this article, we derive an optimized relationship between the solution and its corresponding function for second- and fourth-order neutral differential equations (NDE) in the canonical case. Using this relationship, we obtain new monotonic properties of the second-order equation. The significance of this paper stems from the fact that the asymptotic behavior and oscillation of solutions to NDEs are substantially affected by monotonic features. Based on the new relationships and properties, we obtain oscillation criteria for the studied equations. Finally, we present examples and review some previous theorems in the literature to compare our results with them.

This work investigated the asymptotic performance of nonoscillatory solutions to the functional differential equation (FDE) ${\left(\alpha \left(u\right){\left|y^{\mathrm{\prime }}\left(u\right)\right|}^{\kappa -1}{y}^{\mathrm{\prime }}\left(u\right)\right)}^{\mathrm{\prime }}+p\left(u\right)F\left(y^{\mathrm{\prime }}\left(\delta \left(u\right)\right)\right)+q\left(u\right)G\left(y\left(\rho \left(u\right)\right)\right)=0$, which involves a delayed damping term. Using Riccati and comparison methods, we extended the previous results to the nonlinear case of the considered equation. Furthermore, the new criteria improved upon the previous ones by removing some constraints on the delay functions. Then, for the linear case, we derived new criteria that take into account all parameters of the equation. The examples and comparisons provided illustrate the importance and novelty of our results.

In this study, we investigate the qualitative properties of solutions to a general model of difference equations (DEs), which includes the flour beetle model as a particular case. We investigate local and global stability and boundedness, as well as the periodic behavior of the solutions to this model. Moreover, we present some general theorems that help study the periodicity of solutions to the DEs. The presented numerical examples support the finding and illustrate the behavior of the solutions for the studied model. A significant agricultural pest that is extremely resistant to insecticides is the flour beetle. Therefore, studying the qualitative characteristics of the solutions in this model greatly helps in understanding the behavior of this pest and how to resist it or benefit from it. By applying the general results to the flour beetle model, we clarify the conditions of global stability, boundedness, and periodicity.

The study of the oscillatory behavior of a general class of neutral Emden-Fowler differential equations is the focus of this work. The main motivations for studying the oscillatory behavior of neutral equations are their many applications as well as the richness of these equations with exciting analytical issues. We obtained novel oscillation conditions in Kamenev-type criteria for the considered equation in the canonical case. We improve the monotonic and asymptotic characteristics of the non-oscillatory solutions to the considered equation and then utilize these characteristics to refine the oscillation conditions. We present, through examples and discussions, what demonstrates the novelty and efficiency of the results compared to previous relevant findings in the literature. In addition, we numerically represent the solutions of some special cases to support the theoretical results.