An accurate THINC scheme for interface capturing based on homotopy analysis method

Modified homotopy perturbation method for solving singular integral equations of the first kind

Abstract In this research, we conduct a comprehensive analysis of the steady, two-dimensional laminar flow of a non-Newtonian Williamson nanofluid confined between two inclined and fixed parallel plates, subjected to a uniform transverse magnetic field. The nanofluid dynamics are modeled using the Wakif-Buongiorno’s model, which effectively incorporates essential nanoparticle transport mechanisms such as Brownian motion and thermophoresis. By employing appropriate dimensionless variables, the governing equations are systematically transformed into their dimensionless forme, facilitating analytical and numerical treatment. The main contributions of this study lie in the theoretically well-detailed mathematical formulation of the problem. To solve the resulting system of nonlinear ordinary differential equations, we utilize the semi-analytical, the Homotopy Perturbation Method (HPM), and we critically examine its applicability and limitations within the context of our problem. Furthermore, we apply the midpoint Richardson extrapolation technique implemented via the Midrich function in MAPLE software to enhance the accuracy of our solutions. The influence of various dimensionless parameters on the temperature and concentration profiles is systematically investigated, and the findings yield important insights with practical implications for engineering applications, especially in fields where magnetohydrodynamic nanofluid flows are of industrial relevance.

Neutron stars represent one of the most compact astrophysical objects in the universe, characterized by extremely high densities and strong gravitational fields. The study of their internal structure requires the framework of general relativity, where the Einstein field equations and the Tolman–Oppenheimer–Volkoff (TOV) equation govern the equilibrium configuration of matter. However, these equations are highly nonlinear and difficult to solve analytically. In this work, the Homotopy Perturbation Method (HPM) is employed to obtain approximate analytical solutions for relativistic neutron star models. The method provides a systematic and rapidly convergent approach to handle nonlinear differential equations without the need for small perturbation parameters. The derived solutions are analyzed for physical acceptability by examining energy conditions, pressure and density profiles, causality constraints, and stability criteria. The results demonstrate that the Homotopy Perturbation Method offers an efficient and reliable semi-analytical technique for modeling neutron stars and understanding their structural properties under relativistic conditions.

Permanent gravity waves propagating in deep water, spanning amplitudes from infinitesimal to their theoretical limiting values, remain a classical yet challenging problem due to its inherent nonlinear complexities. Traditional analytical and numerical methods encounter substantial difficulties near the limiting wave condition due to singularities at sharp wave crests. In this study, we propose a novel hybrid framework combining the homotopy analysis method (HAM) with machine learning (ML) to efficiently compute convergent series solutions of Stokes waves in deep water for arbitrary wave amplitudes from small to theoretical limiting values, which show excellent agreement with established benchmarks. We introduce a neural network trained using only 20 representative cases whose series solution are given by means of HAM, which can rapidly predict series solutions across arbitrary steepness levels, substantially improving computational efficiency. Additionally, we develop a neural network to gain the inverse mapping from the conformal coordinates $(\theta , r)$ to the physical coordinates $(x,y)$ , facilitating explicit and intuitive representations of series solutions in physical plane. This HAM–ML hybrid framework represents a powerful and efficient approach to compute convergent series in a whole range of physical parameters for water waves with arbitrary wave height including even limiting waves. In this way we establish a new paradigm to quickly obtain convergent series solutions of complex nonlinear systems for a whole range of physical parameters, thereby significantly broadening the scope of series solutions that can be easily gained by means of HAM even for highly nonlinear problems in science and engineering.

One popular regularization technique for handling both linear and nonlinear ill-posed problems is homotopy perturbation. In order to solve nonlinear ill-posed problems, we investigate an iteratively-regularized simplified version of the Homotopy perturbation approach in this study. We examine the method's thorough convergence analysis under typical circumstances, focusing on the nonlinearity and the convergence rate under a H\"{o}lder-type source condition. Lastly, numerical simulations are run to confirm the method's effectiveness.

This study presents a mathematical model for the transport of non-Newtonian nanofluids in an inclined ciliated converging microchannel. The analysis focuses on the combined effects of cilia length variation, electroosmotic effect, and temperature-dependent viscosity and thermal conductivity. The governing equations were derived using the Debye-Hückel approximation along with lubrication theory. These equations were then solved semi-analytically using the Homotopy Perturbation Method (HPM) in MATHEMATICA. The resulting solutions were visualized by plotting graphs in MATLAB. The results indicate that increased cilia length leads to a reduction in axial velocity but lowers the external pressure required to maintain flow, allowing for precise adjustments to transport dynamics. Variable viscosity and thermal conductivity improve flow and heat transfer under mild obstruction. The applied electric field accelerates the fluid by offsetting the drag caused by cilia, thereby enhancing overall transport efficiency. These findings illustrate the capability of cilia to serve as moderators of flow and transport in applications like targeted drug delivery, lab-on-a-chip diagnostics, microscale heat exchangers, and bio-inspired pumping systems.

This research explores the impact of natural convection on magnetized MoS 2 –GO/H 2 O hybrid nanofluid flow on a variable porous stretching sheet, considering velocity and thermal slip effects. The fluid rotates around z-axis with magnetic effects in the normal direction to the fluid’s motion. The thermal features are controlled through the use of thermal radiation and dissipative effects in the energy equation. The governing equations have been solved through the use of the homotopy analysis method (HAM) in dimensionless form. It has been highlighted in this work that an increase in the concentration of GO nanoparticles from 0.00 to 0.04 results in a 13.46% rise in Nusselt number. In contrast, the same rise in volumetric fraction for GO–MoS 2 nanoparticles yields a higher enhancement of 17.62% in Nusselt number. It has been discovered in this work that GO–MoS 2 hybrid nanoparticles are more effective than pure GO nanoparticles in boosting the thermal flow rate, making them a more suitable choice for applications requiring enhanced heat transfer performance. To validate the solution method applied in this study, its results have matched with previously established findings and showed strong agreement across all comparisons. The outcomes of this work can be applied in manufacturing processes involving porous media, like polymer extrusion, metal forming and biomedical devices where precise thermal control is required.

This study investigates the unsteady, incompressible, electrically conducting nanofluid flow between two parallel disks subjected to simultaneous radial stretching and squeezing, with emphasis on comparing Newtonian (viscous) and non-Newtonian (second-grade) nanofluids. The physical model incorporates the effects of Joule heating, viscous dissipation and linear Rosseland thermal radiation under convective heat and mass boundary conditions. The governing equations are transformed using similarity transformations and solved through the Homotopy Analysis Method (HAM). The influence of key parameters on physical quantities is comprehensively examined. Graphical and statistical comparisons (t-statistics and p-values) reveal that the magnetic field reduces fluid velocity but increases temperature and skin friction due to the Lorentz force. Squeezing enhances velocity while lowering temperature, and thermophoresis elevates both temperature and concentration, unlike Brownian motion, which reduces concentration. Regression analysis validates the model’s accuracy and identifies the dominant roles of the squeezing, thermophoresis and Biot numbers in enhancing transport phenomena. This work fills a notable research gap by addressing the comparison of two different fluids that are undergoing squeezing. The findings of this study have practical applications in lubrication, biomedical devices and microfluidic technologies.

The ongoing work explores the thermal attributes of a polar hybrid nanofluid containing CdTe and ZnO nanoparticles dispersed in water through a surface undergoing expansion packed within a porous matrix. The inclusion of magnetization in conjunction with radiation enriches the flow phenomena. Additionally, it is unusual to neglect the role of dissipative heat energy; therefore, both the viscous, Joule and Darcy dissipation enrich the thermal properties. The innovative aspect of this study lies in incorporating Cattaneo–Christov thermal relaxation for energy transport, with velocity slip influencing the overall analysis. Dissipative heat along with thermal radiation significantly enhances energy transport, which improves heat dissipation in photothermal therapy. The developed mathematical framework is nondimensionalized to enable the implementation of suitable similarity transformations. Moreover, the transport processes are examined computationally through the “homotopy analysis method (HAM)”, The validation with the traditional fluid in a particular case is observed and deployed.

Dehumidification induced thermal behaviour and efficiency analysis of T shaped porous metallic fin: A semi analytical approach using Homotopy Perturbation Method

Abstract In this paper, we find the solution for two-dimensional fractional diffusion equation using an efficient scheme namely, q-homotopy analysis transform method (q-HATM). To validate the competence and applicability of the proposed scheme, we consider two examples. The proposed algorithm is the joint application of the homotopy analysis method and Laplace transform (LT). For the obtained series solution, the proposed method offers ¯ h -curves that illustrate convergence region. The numerical behaviour is presented to ensure the reliability and accuracy of the proposed method. Further, the nature of the achieved results is presented for a diverse order of the fractional derivative. The achieved results show that, the considered method is easy to implement and very effective to study the behaviour of nonlinear models.

Application of the homotopy perturbation method to the Jeffery–Hamel flow of nanofluids between two non-parallel planar walls

Modified homotopy perturbation technique for solving third-order nonlinear Lane–Emden equations

This paper investigates the nonlinear horizontal vibration of a cold rolling system induced by the gyroscope precession effect-a critical yet underexplored issue affecting strip quality and rolling stability. A nonlinear dynamic model is developed by incorporating the axial excitation force and the elastic deformation of the work roll based on d’Alembert principles. The primary parametric resonance response, corresponding to the first Arnold tongue, is analyzed using the multi-scale method and validated experimentally. To further understand the systematic dynamic behavior, the homotopy analysis method is employed to trace the evolution of energy orbits, revealing bifurcation and jump phenomena as the frequency ratio varies. A devil’s staircase pattern emerges, indicating multiple frequency-locked regions. These nonlinear features are further validated through cell mapping techniques, which depict the transformation of modal energy manifolds. Moreover, by introducing active control inputs, a constraint space for control parameters is designed to induce amplitude death within the maximum Arnold tongue region. The findings contribute to a deeper understanding of the resonance mechanism and offer a theoretical basis for stabilizing precision cold rolling systems via nonlinear control strategies.Supplementary InformationThe online version contains supplementary material available at 10.1038/s41598-025-27552-2.

Computational analysis of entropy in a chemically reactive micropolar fluid flow within a magnetically influenced microchannel: optimal homotopy analysis method

Abstract The study addresses the flow dynamics of an incompressible non‐Newtonian Ree‐Eyring nanofluid. It explores an innovative interaction of electromagnetic squeezing flow with microorganisms, including Hall current effects and nonlinear heat generation. The flow traverses a permeable region within a channel formed by two parallel plates, influenced by microorganisms, Hall currents, nonlinear heat sources, and a constant response rate. The thermal transfer due to Ohmic dissipation is also examined in the flow. The examination of the density of microorganisms and dispersion of nanoparticles under electromagnetic squeezing advances the knowledge of linked magneto‐bio‐convective nanofluid dynamics. The controlling equations are restructured as nonlinear ordinary differential equations by suitable similarity transformations. The theoretical analysis addresses the ordinary differential equations governing equations of motion via the Homotopy perturbation method. It is discovered that the middle of the channel is a vital point in fluid flow, and the impact of any parameter is reflected there, but the impact is similar near the edges. The heat broadcast gets better when all pertinent parameters increase, except for Hall parameter. Nanomaterials are condensed with the growth of most of their related parameters. Microbes almost diminish in density and decrease in existence with augmentation of their related parameters.

Heat transfer analysis of steady laminar 2D flow of CNTs-blood-based nanofluid over a moving permeable plate with viscous dissipation and thermal radiation

ABSTRACT The Bagley–Torvik equations with 1/2 or 3/2 order derivatives describe the motion of real bodies in Newtonian fluids. In this work, we use fractional generalized homotopy analysis method (FGHAM) to determine the solution of inhomogeneous Bagley–Torvik equation of fractional order , which is the particular case of nonlinear fractional differential equation. In application of the method, we take two specific values of the function and exhibit the process. Further, the graphical interpretations are also proposed for the obtained solution.

This study investigates the peristaltic bioconvection flow of a third-grade nanofluid in the presence of autocatalysis chemical reactions and entropy generation, employing an analytical approach through Homotopy Analysis Method (HAM). The analysis highlights the influence of key governing parameters, including the Prandtl number, Brownian diffusion, thermophoresis, and the bioconvection Rayleigh number on flow, heat and mass transport as well as gyrotactic microorganisms. The results demonstrate that autocatalysis substantially alters the concentration field, while entropy generation provides valuable insight into the competition between thermal irreversibility and viscous dissipation. Improved thermophoresis and Brownian motion parameters are shown to promote heat transfer, and the bioconvection parameter regulates microorganism distribution. The findings contribute to an understanding of transport processes in complex non-Newtonian fluid systems.