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This investigation explores the analytical solutions to the time-fractional multi-dimensional Navier–Stokes (NS) problem using advanced approaches, namely the Aboodh residual power series method and the Aboodh transform iteration method, within the context of the Caputo operator. The NS equation governs the motion of fluid flow and is essential in fluid dynamics, engineering, and atmospheric sciences. Given the equation’s extensive and diverse applicability across several disciplines, we are motivated to conduct a thorough analysis to understand the complex dynamics associated with the nonlinear events it describes. For this purpose, we effectively handle the challenges posed by fractional derivatives by utilizing the Aboodh approach. This will enable us to obtain accurate analytical approximations for the time fractional multi-dimensional NS equation. By conducting thorough analysis and computational simulations, we provide evidence of the efficiency and dependability of the suggested methodologies in accurately representing the dynamic behavior of fractional fluid flow systems. This work enhances our comprehension of the utilization of fractional calculus in fluid dynamics and provides valuable analytical instruments for examining intricate flow phenomena. Its interdisciplinary nature ensures that the findings are applicable to various scientific and engineering fields, making the research highly versatile and impactful.

A numerical framework is established for a two-dimensional steady flow of the magnetized Jeffery fluid model over elongated/shrinking sheets, with potential applications such as coating sheets, food products, fiber optics, drilling fluids, and the manufacturing processes of thermoplastic polymers. The model also demonstrates the influence of chemical reaction, magnetic field, and stability analysis which provide a novel contribution to this study. To ensure the ease and effectiveness of this analysis, we transform the set of difference equations governing the system into ordinary equations using the similarity transformation. The reliability of the solution is examined by using stability analysis. The Navier–Stokes equations have been transformed into self-similar equations by adopting appropriate similarity transformations and subsequently solved numerically using the bvp4c (three-stage Labatto-three-A formula) approach. The comparison between the derived asymptotic solutions and previously documented numerical results is quite remarkable. The self-similar equations display a duality of solutions within a limited range of the shrinking parameter, as observed from the data. For each stretching scenario, there is a unique solution. Hence, an examination of temporal stability has been conducted through linear analysis to establish the most fundamentally viable solution. The determination of stability in the analysis is based on the sign of the smallest eigenvalue, which indicates whether a solution is unstable or stable. The analysis of stability reveals that the first solution, which describes the primary flow, remains stable. Through the utilization of graphs, we thoroughly examine and discuss the influence of emerging factors. The numerical results obtained from this analysis demonstrate multiple solutions within a certain range of M1≥Mci{M}_{1}\ge {M}_{{ci}}, i=1,2,3i=1,2,3, and no solution in the range M1

This article presents a new approach for solving the fuzzy fractional Degasperis–Procesi (FFDP) and Camassa–Holm equations using the iterative transform method (ITM). The fractional Degasperis–Procesi (DP) and Camassa–Holm equations are extended from the classical DP and Camassa–Holm equations by incorporating fuzzy sets and fractional derivatives. The ITM is a powerful technique widely used for solving nonlinear differential equations. This approach transforms the fuzzy fractional differential equations into a series of ordinary differential equations, which are then solved iteratively using a recursive algorithm. Numerical simulations demonstrate the proposed approach’s accuracy and effectiveness. The results show that the ITM provides an efficient and accurate method for solving the FFDP and Camassa–Holm equations. The proposed method can be extended to solve other fuzzy fractional differential equations.

This study demonstrates the use of fractional calculus in the field of epidemiology, specifically in relation to dengue illness. Using noninteger order integrals and derivatives, a novel model is created to examine the impact of temperature on the transmission of the vector–host disease, dengue. A comprehensive strategy is proposed and illustrated, drawing inspiration from the first dengue epidemic recorded in 2009 in Cape Verde. The model utilizes a fractional-order derivative, which has recently acquired popularity for its adaptability in addressing a wide variety of applicable problems and exponential kernel. A fixed point method of Krasnoselskii and Banach is used to determine the main findings. The semi-analytical results are then investigated using iterative techniques such as Laplace-Adomian decomposition method. Computational models are utilized to support analytical experiments and enhance the credibility of the results. These models are useful for simulating and validating the effect of temperature on the complex dynamics of the vector–host interaction during dengue outbreaks. It is essential to note that the research draws on dengue outbreak studies conducted in various geographic regions, thereby providing a broader perspective and validating the findings generally. This study not only demonstrates a novel application of fractional calculus in epidemiology but also casts light on the complex relationship between temperature and the dynamics of dengue transmission. The obtained results serve as a foundation for enhancing our understanding of the complex interaction between environmental factors and infectious diseases, leading the way for enhanced prevention and control strategies to combat global dengue outbreaks.

In this research, we use the homotopy perturbation method (HPM) combined with the Elzaki transform to investigate the fractional Biswas–Milovic equation (BME) within the framework of the Caputo operator. The fractional BME is a significant mathematical model with applications in various scientific and engineering fields, including physics, biology, and chemistry. However, its fractional nature introduces analytical complexities. By integrating the HPM with the Elzaki transform, we aim to provide an effective approach for obtaining accurate solutions to this equation. The combination of these mathematical techniques allows us to explore the behavior of the fractional BME in a comprehensive manner. The research outcomes are supported by numerical results and comparisons, demonstrating the reliability and efficiency of the proposed methodology. This study contributes to advancing the tools for solving fractional equations and enhances our understanding of the intricate dynamics described by the fractional BME.

In this article, we propose a novel new iteration method and homotopy perturbation method (HPM) along with the Elzaki transform to compute the analytical and semi-analytical approximations of fractional Airy’s-type partial differential equations (FAPDEs) subjected to specific initial conditions. A convergent series solution form with easily commutable coefficients is used to examine and compare the performance of the suggested methods. Using Maple graphical method analysis, the behavior of the estimated series results at various fractional orders ς\varsigma and its modeling in two-dimensional (2D) and three-dimensional (3D) spaces are compared with actual results. Also, detailed descriptions of the physical and geometric implications of the calculated graphs in 2D and 3D spaces are provided. As a result, the obtained solutions of FAPDEs that are subject to particular initial values quite closely match the exact solutions. In this way, to solve FAPDEs quickly, the proposed approaches are considered to be more accurate and efficient.

The human immunodeficiency virus, which attacks the immune system and especially targets CD4 cells that are crucial for immunological defense against infections, is the cause of the severe illness known as acquired immunodeficiency syndrome (AIDS). This condition has the potential to take a patient’s life. Understanding the dynamics of AIDS and evaluating potential methods of prevention and treatment have both significantly benefited from the use of mathematical modeling. This research article proposes a unique technique that solves a model system of differential equations representing diverse populations, such as susceptible populations, acute populations, asymptomatic populations, and symptomatic populations or populations with AIDS. The method uses an artificial neural network (ANN) to do this. A specific Caputo–Fabrizio derivative is included in the suggested method to validate the system’s stability via the use of Krassnoselskii’s and Banach’s fixed-point approach in combination with the exponential kernel. In order to solve the differential equations and get the required data, the Laplace Adomian Decomposition (LAD) technique is used. Training the ANN involves obtaining simulated data from LAD and doing it within the context of a supervised learning framework. The performance of the ANN is assessed by comparing its predicted solutions to the LAD solutions. This allows for the calculation of the average error for each of the system’s functions. This study presents a potentially useful computational tool for understanding the dynamics of AIDS and delivering important insights into the design of new prevention and treatment methods.

Momentum and heat transmission influence the coated physical characteristics of wire product. As a result, understanding the polymeric movement and heat mass distribution is crucial. An increase in thermal efficiency is necessary for the wire covering technology. So, the aim of this work is to investigate the influence of nanomaterials on the heat and mass transport processes in wire coating analyses. A thin film nanofluid is used to investigate heat and mass transfer in three dimensions over a rotating inclined disc. Both the suction and injection effects of nanofluids and the thermal radiation of their fluxes are taken into account. By employing similarity variables, the set of governing equations can be transformed into a differential equation system. The necessary differential equation system is solved using the Haar wavelet collocation method. Plots and observations of the velocity distribution, concentration, and thermal fields within the boundary layer across an inclining, steadily rotating plane are made. Flow characteristics change as a result of varying embedded factors such as S,Sc,Nb,Pr,S,{\rm{Sc}},{N}_{{\rm{b}}},\Pr , and thermophoretic parameters. Evidence suggests that as the number of rotation parameters grows, the thermal boundary layer weakens.

In this study, we use a dual technique that combines the Laplace residual power series method (LRPSM) and the new iteration method, both of which are combined with the Caputo operator. Our primary goal is to solve two unique but difficult partial differential equations: the foam drainage equation and the nonlinear time-fractional Fisher’s equation. These equations, which are crucial in modeling complex processes, confront analytical complications, owing to their fractional derivatives and nonlinear behavior. We develop exact and efficient solutions by merging these unique methodologies, which are supported by thorough figures and tables that demonstrate the precision and trustworthiness of our methodology. We not only shed light on the solutions to these equations, but also demonstrate the prowess of the LRPSM and the new iteration method as powerful tools for grappling with complex mathematical and physical models, significantly contributing to advancements in various scientific domains.

This investigation explores two numerical approaches: the optimal auxiliary function method (OAFM) and the new iterative method (NIM). These techniques address the physical fractional-order Klein-Gordon equations (FOKGEs), a class of partial differential equations (PDEs) that model various physical phenomena in engineering and diverse plasma models. The OAFM is a recently introduced method capable of efficiently solving several nonlinear differential equations (DEs), whereas the NIM is a well-established method specifically designed for solving fractional DEs. Both approaches are utilized to analyze different variations in FOKGE. By conducting numerous numerical experiments on the FOKGE, we compare the accuracy, efficiency, and convergence of these two proposed methods. This study is expected to yield significant findings that will help researchers study various nonlinear phenomena in fluids and plasma physics.

The study of magnetohydrodynamics (MHD) and thermal radiation over‐stretching and shrinking sheets has significant applications in a wide range of fields, from chemical manufacturing to transport engine cooling systems, electronic chip cooling, plasma, the nuclear‐powered sector, saltwater, etc. The present study aims to numerically and theoretically analyze the movement of thermally magnetized Williamson nanofluid owing to an extended/contracting sheet. The regulating flow equations are converted to self‐similar equations through similarity transformation, and then numerically solved by bvp4c using MATLAB software. A comparison of the present numerical study with the published study is quite impressive. The investigation demonstrates that the self‐similar equations disclose the two branches for the limited shrinking factor range. For the stretching case, only one solution exists. As a result, the most fundamentally feasible solution has been determined by the linear assessment of temporal stability. For the aim of stability analysis, the lowest eigenvalue sign indicates the stability or instability of a solution. Through stability analysis, it is witnessed that the first solution (first branch) is stable. Due to the presence of the Lorentz force effect, it is perceived that the velocity curves decline in the whole channel. The point to be noted here is that the velocity of the lower branch compared to the upper branch is higher. The reduced Skin Friction is increased in the first branch and declines in the second branch for the two different values of magnetic factor. [ABSTRACT FROM AUTHOR]

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This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated with nanobiosciences and biophysics based on the transmission line model of microtubules for nanoionic currents. The equation introduced here in this form is suitable for critical nanoscience concerns like cell signaling and might continue to explain some of the basic cognitive functions in neurons. We employ advanced procedures to replicate the previously detected solitary waves. We offer our solutions in graphical forms, such as 3D and contour plots, using Mathematica. We can generalize the elementary method to other nonlinear equations in physics, requiring only a few steps. [ABSTRACT FROM AUTHOR]

In the present research work, we construct and examine the self-similarity of optical solitons by employing the Riccati Modified Extended Simple Equation Method (RMESEM) within the framework of non-integrable Coupled Nonlinear Helmholtz Equations (CNHEs). This system models the transmission of optical solitons and coupled wave packets in nonlinear optical fibers and describes transverse effects in nonlinear fiber optics. Initially, a complex transformation is used to convert the model into a single Nonlinear Ordinary Differential Equation (NODE), from which hyperbolic, exponential, rational, trigonometric, and rational hyperbolic solutions are produced. In order to better understand the physical dynamics, we offer several 3D, contour, and 2D illustrations for the independent selections of physical parameter values. These illustrations highlight the graphic behaviour of some optical solitons and demonstrate that, under certain constraint conditions, acquired optical solitons lose their stability when they approach an axis and display periodic-axial perturbations, which lead to the generation of optical fractals. As a framework, the generated optical solitons have several useful applications in the field of telecommunications. Furthermore, our suggested RMESEM demonstrates its use by broadening the spectrum of optical soliton solutions, offering important insights into the dynamics of the CNHEs, and suggesting possible applications in the management of nonlinear models. [ABSTRACT FROM AUTHOR]

This study explores the application of advanced mathematical techniques to solve fractional differential equations, focusing particularly on the fractional diffusion equation. The fractional diffusion equation, used to simulate a range of physical and engineering phenomena, poses considerable difficulties when applied to fractional orders. Thus, by utilizing the mighty powers of fractional calculus, we employ the variational iteration method (VIM) with the Elzaki transform to produce highly accurate approximations for these specific differential equations. The VIM provides an iterative framework for refining solutions progressively, while the Elzaki transform simplifies the complex integral transforms involved. By integrating these methodologies, we achieve accurate and efficient solutions to the fractional diffusion equation. Our findings demonstrate the robustness and effectiveness of combining the VIM and the Elzaki transform in handling fractional differential equations, offering explicit functional expressions that are beneficial for theoretical analysis and practical applications. This research contributes to the expanding field of fractional calculus, providing valuable insights and useful tools for solving complex, nonlinear fractional differential equations across various scientific and engineering disciplines. [ABSTRACT FROM AUTHOR]

This study examines the heat transfer properties of a recently created hybrid nanofluid in contrast to a traditional nanofluid. The aim is to improve the transfer of heat in the flow of the boundary layer by employing this novel hybrid nanofluid. Our study investigates the impact of the Lorentz force on a three-dimensional stretched surface. We utilize a new model that incorporates thermo-physical factors. A quantitative parametric study is performed to investigate the influence of different physical parameters, enabling meaningful comparisons. The results demonstrate that the hybrid nanofluid exhibits a higher heat transfer rate compared to the conventional fluid, even in the presence of a magnetic field. Moreover, the efficiency of heat transfer can be enhanced by modifying the concentration of nanoparticles in the hybrid nanofluid.

The optimal auxiliary function method (OAFM) is introduced and used in the analysis of a nonlinear system containing coupled Schrödinger–KdV equations, all within the framework of the Caputo operator. The OAFM, known for its efficiency in solving nonlinear issues, is used to obtain approximate solutions for the coupled equations’ complicated dynamics. Numerical and graphical assessments prove the suggested method’s correctness and efficiency. This study contributes to the understanding and analysis of coupled Schrödinger–KdV equations and their many applications by providing insights into the behavior of nonlinear systems within mathematical physics.

This research paper aims to investigate the peristaltic transport of a nanofluid (NF) in a tapered asymmetric channel. Initially, the governing equations for the balance of mass, momentum, temperature, and volume fraction for the NF using Reiner–Philippoff (RP) based NF are formulated. Subsequently, these equations are employed to analyze long wavelength and small Reynolds number scenarios. The numerical results for various flow features are thoroughly examined and discussed. Dual solutions have been examined for some factors. So, stability assessment is implemented to find stable solution. Novelty of the existing is to investigate the peristaltic motion of Buongiorno's NF model and its stability which has not investigated in the previous literatures. It has been demonstrated that modifying the RPF parameter leads to a transition in the fluid's velocity, changing it from a dilatant liquid to a Newtonian fluid and from Newtonian to pseudoplastic. The findings indicate that the temperature curves rise as Brownian motion and thermophoretic factors increase, while they decrease as the Prandtl number increases. Furthermore, a concise mathematical and graphical analysis is carried out to examine the impact of each key parameter on the flow characteristics. [ABSTRACT FROM AUTHOR]

The main objective of this work is to investigate the characteristics and behavior of the ion-acoustic cnoidal waves (IACWs) in an electron–positron–ion magnetoplasma having inertial positive ions with anisotropic thermal pressure and inertialess Maxwellian positrons and electrons. We utilize the reductive perturbation technique to reduce the fluid governing equations of the present model into the Korteweg–de Vries (KdV) equation in order to achieve this objective. We calculate the periodic solution of the KdV equation, also referred to as the cnoidal wave. We investigate the impact of various related parameters, including ion pressure anisotropy, positron concentrations, and temperature ratio, on the properties of IACWs. This study, particularly in the near-Earth magnetosheath and magnetosphere, may offer an insightful analysis of space and astrophysical plasma systems displaying ion pressure anisotropy. [ABSTRACT FROM AUTHOR]