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This paper conducts an extensive comparative analysis of numerical methods employed in modeling blood flow through arteries with Magnetohydrodynamics (MHD) and hybrid nanofluids. The study investigates the effectiveness and precision of distinct numerical approaches: Akbari Ganji's Method (AGM), Fourth-Order Runge–Kutta (RK4), Finite Volume Method (FVM), and the Finite Element Method (FEM). These methods are essential for comprehending the intricate fluid dynamics that arise in the presence of magnetic fields and hybrid nanofluids a phenomenon relevant in numerous medical applications. Blood flow is subjected to a homogeneous magnetic field in a radial direction while the magneto-hemodynamics effect is taken into account. A variety of medical, physiological, and surgical procedures, as well as the regulation of blood pressure, heat distribution, wound healing, diagnostic imaging, and drug discovery, depend on blood flow through arteries to carry out vital functions such as oxygen and nutrition delivery, organ maintenance, and wound healing. Our findings highlight that while each method has strengths, their applicability varies based on the problem's characteristics and computational resource constraints. This analysis aids researchers and practitioners in selecting the most suitable method for their modeling requirements, advancing numerical techniques for complex fluid dynamics involving MHD and hybrid nanofluids. [ABSTRACT FROM AUTHOR]

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 research aims to implement the numerical results for the Holling third kind of functional response delay differential model utilizing a stochastic framework based on Levenberg-Marquardt backpropagation neural networks (LVMBPNNs). The nonlinear model depends upon three dynamics, prey, predator, and the impact of the recent past. Three different cases based on the delay differential system with the Holling 3rd type of the functional response have been used to solve through the proposed LVMBPNNs solver. The statistic computing framework is provided by selecting 12%, 11%, and 77% for training, testing, and verification. Thirteen numbers of neurons have been used based on the input, hidden, and output layers structure for solving the delay differential model with the Holling 3rd type of functional response. The correctness of the proposed stochastic scheme is observed by using the comparison performances of the proposed and reference data-based Adam numerical results. The authentication and precision of the proposed solver are approved by analyzing the state transitions, regression performances, correlation actions, mean square error, and error histograms. [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]

In this paper, we examined four different forms of generalized (2+1)-dimensional Boussinesq–Kadomtsev–Petviashvili (B-KP)-like equations. In this connection, an accurate computational method based on the Riccati equation called sub-equation method and its Bäcklund transformation is employed. Using this method, numerous exact solutions that do not exist in the literature have been obtained in the form of trigonometric, hyperbolic, and rational. These solutions are of considerable importance in applied sciences, coastal, and ocean engineering, where the B–KP-like equations modeled for some significant physical phenomenon. The graph of the bright and dark solitons is presented in order to demonstrate the influence of different physical parameters on the solutions. All of the findings prove the stability, effectiveness, and accuracy of the proposed method. [ABSTRACT FROM AUTHOR]

The main objective of our analysis is to modify the reduced differential transform method (RDTM) to be applicable for a wide range of nonlinear partial differential equations. The Adomian polynomials are employed to overcome the huge size and complexity of computing reduced differential transforms, and new generalizations of transformed formulas are established. Our new approach provides a series solution with effortless components to compute. A new sufficiency convergence proof of series solution is introduced. The (1 + m)-dimensional telegraph and Burgers equations are tested to illustrate robustness and reliability of our modification. [ABSTRACT FROM AUTHOR]