The oscillation of a class of fourth-order nonlinear damped delay differential equations with distributed deviating arguments is the subject of this research. We propose a new explanation of the fourth-order equation oscillation in terms of the oscillation of a similar well-studied second-order linear differential equation without damping. The extended Riccati transformation, integral averaging approach, and comparison principles are used to provide some additional oscillatory criteria. An example demonstrates the efficacy of the acquired criteria.

This work aims to derive new inequalities that improve the asymptotic and oscillatory properties of solutions to fourth-order neutral differential equations. The relationships between the solution and its corresponding function play an important role in the oscillation theory of neutral differential equations. Therefore, we improve these relationships based on the modified monotonic properties of positive solutions. Additionally, we set new conditions that confirm the absence of positive solutions and thus confirm the oscillation of all solutions of the considered equation. We finally explain the importance of the new inequalities by applying our results to some special cases of the studied equation, as well as comparing them with previous results in the literature.

The main purpose of this research was to use the comparison approach with a first-order equation to derive criteria for non-oscillatory solutions of fourth-order nonlinear neutral differential equations with p Laplacian operators. We obtained new results for the behavior of solutions to these equations, and we showed their symmetric and non-oscillatory characteristics. These results complement some previously published articles. To find out the effectiveness of these results and validate the proposed work, two examples were discussed at the end of the paper.

The goal of this paper was to study the oscillations of a class of fourth-order nonlinear delay differential equations with a middle term. Novel oscillation theorems built on a proper Riccati-type transformation, the comparison approach, and integral-averaging conditions were developed, and several symmetric properties of the solutions are presented. For the validation of these theorems, several examples are given to highlight the core results.

Hybrid nanofluids combine two or more nano properties with a base fluid such as water ethylene. Usually, this helps enhance the heat transfer rate; in this article, using new similarity transformations created by Lie group analysis, the governing nonlinear partial differential equations are transformed into a system of connected nonlinear ordinary differential equations. The resulting design is numerically solved using a BVP4C solver with the shooting method (MATLAB). The magneto hydrodynamic flow of an incompressible fluid and the rate of heat and mass transfer were investigated for two cases, with various nanoparticle shapes including cylindrical, spherical, and platelet. Case 1 was CNT (1%), graphene (1%), and aluminum oxide (1%), and Case 2 was copper (1%), silver (1%), and cobalt ferrite (1%). When the Hartmann number rises, velocity and temperature exhibit inverse behavior: the velocity profile increases, and the temperature profile decreases. When the suction rises, the velocity and temperature profiles both increase. Optimization techniques were used from response surface methodology (RSM) to set factorial variables so that the response met the desired maximum or minimum value. Factorial methods like ANOVA were used to model the response, but they were expanded to simulate the effects in terms of extrapolation.

In recent times, heat and mass transportation have had some of the most recognized and attractive research areas in computational fluid dynamics. It is useful in the modeling of the flow of nuclear reactors, bioinformatics, the medical discipline, etc. Driven by the execution of the flow in the manufacturing application, the goal of the present analysis is to explore the novel effect of micropolar fluid configured by an exponentially elongated sheet positioned horizontally in a porous channel. The impact of activation energy, internal heating, and heat and mass transfer features are integrated into the revised flow model. A mathematical framework for different flow fields is developed in order to highlight the significant aspects of the thermal and concentration slip effects evaluated on the extended plat surface, with the aid of appropriate transformation factors to diminish the nonlinear fundamental flow equations (PDEs) to a system of (ODEs). Precise numerical treatment for a wide range of pertinent parameters is adopted to solve the nonlinear system through a built-in algorithm in the MATHEMATICA platform. The features of prominent emerging parameters against various flow fields are viewed and addressed through plotted visuals. The influence of the factors on skin friction, heat, and mass coefficients offered through 3D animation is evaluated. The temperature profile improves with ascending values of Brownian parameter and thermophoretic diffusion force but diminishes with subject expansions in Prandtl number and thermal slip parameter. It has been noticed that the concentration outlines increase for reaction rate and activation energy parameters but dwindle for expending values of porosity parameter, Lewis number, and concentration slip parameter. Skin fraction values increase due to the growing nature of the micropolar and second-grade fluid parameters. Nusselt numbers upsurge for increasing thermophoretic diffusion parameters while exhibiting a declining trend for Brownian motion parameters.