Khalil Salim Al-Ghafri

This paper investigates the oscillatory behavior of a class of higher-order nonlinear neutral differential equations with sublinear terms and time delays. By employing generalized Riccati transformations, new results on oscillation are established, extending and refining existing criteria while relaxing previously imposed restrictions. The analysis is based on a detailed examination of solution properties, and theoretical findings are illustrated through numerical examples. The proposed approach enhances the available analytical tools for studying neutral differential equations with delays and offers potential implications for applications in physics, engineering, and the modeling of complex systems.

Singularly perturbed integro-partial differential equations with reaction–diffusion behavior present significant challenges due to boundary layers arising from small perturbation parameters, which complicate the development of accurate and efficient numerical schemes for physical and engineering models. In this study, a uniformly convergent higher-order method is proposed to address these challenges. The approach applies the implicit Euler method for temporal discretization on a uniform mesh and central differences on a Shishkin mesh for spatial approximation, and utilizes the trapezoidal rule for evaluating integral terms; further, extrapolation techniques are incorporated in both time and space to increase accuracy. Numerical analysis demonstrates that the base scheme achieves first-order convergence, while extrapolation enhances convergence rates to second-order in time and fourth-order in space. Theoretical results confirm uniform convergence with respect to the perturbation parameter, and comprehensive numerical experiments validate these analytical claims. Findings indicate that the proposed scheme is reliable, efficient, and particularly effective in attaining fourth-order spatial accuracy when solving singularly perturbed integro-partial differential equations of reaction–diffusion type, thus providing a robust numerical tool for complex applications in science and engineering.

The present study discusses the investigation of optical solitons in birefringent fibers with cubic-quartic dispersion. The nonlinear effect in this medium is considered because of non-local law in presence of perturbation terms. The model is dealt with analytical strategy using the traveling wave hypothesis to reach an ordinary differential equation. To detect various exact solutions, the later equation is handled by projective Riccati equations method. Miscellaneous optical soliton solutions such as bright, dark, singular and M-shaped solitons are derived based on parameter restrictions. The visual representations of these wave structures are illustrated to reveal their complex physical aspects. It is noted that the soliton amplitude and width undergo remarkable variations due to the influence of dispersion, non-local law and nonlinear steeping. The results obtained can pave the way for improving the applications in the field of nonlinear optics.

The present study discusses the investigation of optical solitons in birefringent fibers with cubic-quartic dispersion. The nonlinear effect in this medium is considered because of non-local law in presence of perturbation terms. The model is dealt with analytical strategy using the traveling wave hypothesis to reach an ordinary differential equation. To detect various exact solutions, the later equation is handled by projective Riccati equations method. Miscellaneous optical soliton solutions such as bright, dark, singular and M-shaped solitons are derived based on parameter restrictions. The visual representations of these wave structures are illustrated to reveal their complex physical aspects. It is noted that the soliton amplitude and width undergo remarkable variations due to the influence of dispersion, non-local law and nonlinear steeping. The results obtained can pave the way for improving the applications in the field of nonlinear optics.

The objective of the present study is to detect chirped optical solitons of the perturbed Chen–Lee–Liu equation with full nonlinearity. By virtue of the traveling wave hypothesis, the discussed model is reduced to a simple form known as an elliptic equation. The latter equation, which is a second-order ordinary differential equation, is handled by the undetermined coefficient method of two forms expressed in terms of the hyperbolic secant and tangent functions. Additionally, the auxiliary equation method is applied to derive several miscellaneous solutions. Various types of chirped solitons are revealed such as W-shaped, bright, dark, gray, kink and anti-kink waves. Taking into consideration the existence conditions, the dynamical behaviors of optical solitons and their corresponding chirp are illustrated. The modulation instability of the perturbed CLL equation is examined by means of the linear stability analysis. It is found that all solutions are stable against small perturbations. These entirely new results, compared to previous works, can be employed to understand pulse propagation in optical fiber mediums and dynamic characteristics of waves in plasma.

The present work is devoted to investigate the chirped bright and dark optical solitons of fractional Kaup-Newell equation (KNE) in birefringent fibers. The study is carried out analytically by the traveling wave hypothesis with the conformable derivative which reduces the governing model to an ordinary differential equation (ODE). The obtained equation is handled with the aid of an exotic integration scheme that utilizes the Jacobi elliptic equation in the form of a first-order nonlinear ODE with three-degree terms. Taking the modulus of Jacobi elliptic function to unity, distinct types of bright and dark optical solitons are derived with their corresponding chirping. The fractional order derivative is noted to have a significant influence on the pulse propagation. Additionally, the nonlinearity amount causes also marked variations in the amplitude and width of solitons. The modulation instability of the KNE is reported by implementing the linear stability analysis which confirms that all solutions are stable. The revealed results can be capitalized in improving the relevant physical and engineering applications in the field of birefringent fiber.

As a pivotal branch within the realm of differential equations, the theory of oscillation holds a crucial position in the exploration of natural sciences and the construction of modern control theory Frameworks. In this manuscript, we obtain some oscillatory properties for the noncanonical even-order differential equation with a p-Laplacian-like operator by using comparison with first-order differential inequality. Some examples are discussed to illustrate the effectiveness of our main results.

In this study, we use the integral averaging methodology, comparison with second-order differential equations, and the Riccati technique to determine the Philos-type and Hille–Nehari-type oscillation conditions of fourth-order advanced differential equations with canonical operators. In essence, these techniques supplement and generalize a wide range of established oscillation conditions. Two instance cases demonstrate the importance of our outcomes and their significant improvement.

In this work, we find new oscillation criteria for fourth-order advanced differential equations with a p-Laplace-type operator. We established our results through a comparison method with integral averaging and Riccati techniques to obtain new oscillatory properties for the considered equation. Our criteria substantially simplify and complement a number of existing ones. We give some examples to illustrate the significance of the obtained results.

‎The main aim of this paper is to obtain new criteria for oscillating all solutions of second-order differential equations with distributed deviating arguments and superlinear neutral terms‎. ‎Using the comparative and integral averaging techniques‎, ‎we find new conditions for oscillation that generalize and add to some of the already found results‎. ‎There are examples to show how important the main results are‎.