Numerous scientific disciplines such as fluid dynamics, non linear optics and laser physics are the sources of non-linear evolution equations. The modified Korteweg-de Vries-Zakharov-Kuznetsov equation can be applied to analyze the evolution of ion-acoustic perturbations in magnetized plasma made up of two negative ion ingredients that have distinct temperatures. The analysis of shallow water waves gives rise to the time-regularized long-wave equation. In this work, traveling wave solutions to the non-linear evolution equations are obtained by using the modified auxiliary equation method and the simple mapping method. This study demonstrates the presence of traveling wave solutions for the modified Korteweg-de Vries-Zakharov-Kuznetsov equation and the time-regularized long-wave equation. An ordinary differential equation is transformed into a non linear form by applying the traveling wave solutions, which arise from Lie symmetry with infinite dimensions. The results show how rich the analyzed models are in explicit solutions. This leads to the discovery of exact traveling wave solutions for the proposed study including periodic and quasi-periodic solitons, bright and dark solitons, kink, anti-peakon, bell shape, anti-bell shape, W-shape, and M-shape soliton solutions by using the proposed methods. Adding the solutions into the original equation allows for an analysis of their accuracy. It shows how the free parameters affect the amplitudes and wave characteristics. The study provides thorough two-dimensional (2D) and three-dimensional (3D) graphical illustrations of the outcomes that improve comprehension of their physical attributes and show how effectively the recommended approaches work to solve challenging non-linear equations. It is crucial to remember that the suggested techniques are capable, reliable, and engaging analytical instruments for resolving non linear partial differential equations.

This research work focused on the modified Khater method to obtain the optical soliton solutions of the ion sound and Langmuir wave equation. The modified Khater approach, which produces a wide variety of solutions for the model under consideration, is regarded as one of the most modern and accurate analytical approach for non-linear evolution equations. The wave transformation was used to translate the governing model into an ordinary differential equation. The optical soliton solutions that were developed exhibit many waveforms, including singular periodic shape, sharp dark soliton, kink, and anti-kink soliton solutions. The outcomes may have a significant impact on applications in mathematical physics and engineering. Using the wave transformation, the dynamical system of the governing equation was obtained, and the theory of the planar dynamical system was used to carry out its bifurcation. The existence of chaotic behaviors in the suggested model was examined by taking into account a perturbed term in the resulting dynamical system. Furthermore, the dynamical system's sensitivity analysis was analyzed.

This work presents two new numerical methods for finding the zeros of nonlinear equations, based on the Adomian decomposition approach and the quadrature rule. Using a predictor–corrector scheme, the proposed iterative methods provide a high order of convergence and do not require second derivatives, thereby reducing computational time. The methods are tested on several functions, and their efficiency is demonstrated through comparisons with various established techniques. To support the numerical findings, graphical analyses are provided. Furthermore, a study of the basins of attraction is conducted to validate the stability of the proposed methods.

Ma-Minda-type analytic functions have been a central topic in geometric function theory due to their importance in studying properties such as univalence, convexity, and starlikeness. However, the class of Ma-Minda functions associated with kidney-shaped domains has not been systematically investigated before, leaving a clear research gap. In this paper, we introduced and studied the geometric properties of class ${\mathcal{S}}_k^{\ast }$. Our approach combined rigorous geometric analysis with Python programming (version 3.13.2) and graphical illustrations, providing both theoretical depth and computational support. Several inclusion relations and radius problems for ${\mathcal{S}}_k^{\ast }$ were established in comparison with other known classes. Furthermore, we examined various problems related to the initial coefficient bounds, with particular emphasis on their sharpness.

This paper discusses an elaborate investigation of the dimensionless time-dependent paraxial equation based on its varied soliton solutions obtained through an improved modified extended tanh function method. The approach is capable of producing kink, bell-shaped, singular wave, periodic, singular periodic wave, bright and dark solitary wave, breather soliton, singular bell, M-shaped, W-shaped, and V-pattern solitons expressed as hyperbolic and trigonometric functions. Visualization via three dimensional (3D) surface plots, two dimensional (2D) cross-sections and contour plots allows a thorough examination of the wave morphology. Nonlinear dynamics are investigated using bifurcation, chaotic, sensitivity, and stability analyses that uncover complex solution behaviors and stability regimes. Modulation instability analysis verifies the stability of soliton structures under perturbations. The work demonstrates the effectiveness of the improved modified extended tanh function method for solving nonlinear partial differential equations and enhances theoretical knowledge of paraxial wave phenomena.

In this investigation, a generalized form of the Gregory function, called the $\beta$-Gregory function, is derived. This function includes several well-known analytic functions as special cases. We then consider a new symmetric class of analytic and bi-univalent functions related to the $\beta$-Gregory function. The geometric and analytic properties of this class are investigated, and inclusion relationships with other known subclasses are established under suitable sufficient conditions. Furthermore, we derive accurate estimates of the coefficients of this class and discuss the associated Fekete-Szegö inequality, showing how the results obtained contribute to generalizing and improving several previous works in bi-univalent functions.