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Nematicons upgrade the recognition of light localization in the reorientation of non-local media. the current research employs a powerful integral scheme using a different procedure, namely, the modified simple equation method (MSEM), to analyze nematicons in liquid crystals from the controlling model. The expanded MSEM is investigated to enlarge the applicability of the standard one. The suggested expansion depends on merging the MSEM and the ansatz method. The new generalized nonlinear n-times quadruple power law is included. With the aid of the symbolic computational package Mathematica, new explicit complex hyperbolic, periodic, and more exact spatial soliton solutions are derived. Moreover, the related existence constraints are obtained. To show the dynamical properties of some of the obtained nematicons, three-dimensional profiles with corresponding contours are depicted with the choice of appropriate values of arbitrary parameters. The fractional impacts in various applicable senses are analyzed to investigate the generality of the considered model. [ABSTRACT FROM AUTHOR]

Ordinary differential equations (ODEs) are fundamental tools for modeling and understanding a wide range of chemistry, physics, and biological phenomena. However, solving complex ODEs often presents significant challenges, necessitating advanced numerical approaches beyond traditional analytical techniques. Thus, a novel machine learning (ML)-based method for solving and analyzing ODEs is proposed in the current investigation. In this study, we utilize a feed-forward neural network (FNN) with five fully connected layers trained on data samples generated from the exact solutions of specific ODEs. To show the efficacy of our suggested method, we will conduct a thorough evaluation by comparing the anticipated solutions of the FNN with the exact solutions for some ODEs. Furthermore, we analyze the absolute error and present the loss functions for some ODE examples, providing valuable insights into the model's performance and potential areas for further development. [ABSTRACT FROM AUTHOR]

Complex fuzzy sets (CFSs) have recently emerged as a potent tool for expanding the scope of fuzzy sets to encompass wider ranges within the unit disk in the complex plane. This study explores complex fuzzy numbers and introduces their application for the first time in the literature to address a complex fuzzy partial differential equation that involves a complex fuzzy heat equation under Hukuhara differentiability. The researchers utilize an implicit finite difference scheme, namely the Crank–Nicolson method, to tackle complex fuzzy heat equations. The problem's fuzziness arises from the coefficients in both amplitude and phase terms, as well as in the initial and boundary conditions, with the Convex normalized triangular fuzzy numbers extended to the unit disk in the complex plane. The researchers take advantage of the properties and benefits of CFS theory in the proposed numerical methods and subsequently provide a new proof of the stability under CFS theory. A numerical example is presented to demonstrate the proposed approach's reliability and feasibility, with the results showing good agreement with the exact solution and relevant theoretical aspects. [ABSTRACT FROM AUTHOR]

The current study suggests a new generalisation of highly dispersive nonlinear Schrödinger-type equation (NLSE) with perturbation terms. With polynomial refractive index, known by cubic–quintic–septic (CQS) law and Hamiltonian-type cubic perturbation terms, the new model includes eighth-order dispersion term. The generalised Riccati simplest equation method (RSEM) and the modified simplest equation method (MSEM) are successfully utilised to analytically process the fractional version of the considered NLSE. A diverse collection of bright, dark and singular optical solitons under some constraints, in hyperbolic, periodic and rational-exponential forms are derived. Graphical interpretations of some obtained solutions are displayed. The two considered schemes, with different algorithms, show an influential mathematical tool for processing nonlinear fractional evolution equations. [ABSTRACT FROM AUTHOR]

Periodic and soliton solutions are derived for the (1+1)-dimensional van der Waals gas system in the viscosity-capillarity regularization form. The system is handled via the e -ϕ(ξ) -expansion method. The obtained solutions have been articulated by the hyperbolic, trigonometric, exponential and rational functions with arbitrary constants. Mathematical analysis and numerical graphs are provided for some solitons, periodic and kink solitary wave solutions to visualize the dynamics of equations. Obtained results reveal that the method is very influential and effective tool for solving nonlinear partial differential equations in applied mathematics. [ABSTRACT FROM AUTHOR]

In this study, we investigate the interactions of a hybrid nanofluid on a curved surface that is being stretched. The magnetic field is perpendicular to the flow and interacts with a mixture of molybdenum disulfide and argentum nanoparticles suspended in pure water, forming a hybrid nanomaterial. Our investigation considers heat transport analysis under different conditions, such as magnetohydrodynamic, Darcy-Forchheimer porous medium flow, Joule heating, and a convective boundary condition. We employ numerical and statistical methods to study the problem's intricacies comprehensively. Our findings indicate that Darcy-Forchheimer flow includes viscous and inertial forces, which results in higher flow rates and reduced skin friction. Additionally, the convective boundary condition leads to uniform temperature distribution within the hybrid material due to rapid internal heat transfer relative to surface resistance, significantly increasing the heat transfer rate., Competing Interests: The authors declare that there is no conflict of interest amongst them., (This journal is © The Royal Society of Chemistry.)