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In this paper, a hybrid method called variational iteration transform method has been implemented to solve fractional-order Navier–Stokes equation. Caputo operator describes fractional-order derivatives. The solutions of three examples are presented to show the validity of the current method without using Adomian and He’s polynomials. The results of the proposed method are shown and analyzed with the help of figures. It is shown that the proposed method is found to be efficient, reliable, and easy to implement for various related problems of science and engineering.

This paper studies the heat transfer analysis caused due to free convection in a vertically oscillating cylinder. Exact solutions are determined by applying the Laplace and finite Hankel transforms. Expressions for temperature distribution and velocity field corresponding to cosine and sine oscillations are obtained. The solutions that have been obtained for velocity are presented in the forms of transient and post-transient solutions. Moreover, these solutions satisfy both the governing differential equation and all imposed initial and boundary conditions. Numerical computations and graphical illustrations are used in order to study the effects of Prandtl and Grashof numbers on velocity and temperature for various times. The transient solutions for both cosine and sine oscillations are also computed in tables. It is found that, the transient solutions are of considerable interest up to the times t = 15 for cosine oscillations and t = 1.75 for sine oscillations. After these moments, the transient solutions can be neglected and, the fluid moves according with the post-transient solutions.

In the present work, the magnetohydrodynamic flow and heat transfer of a micropolar tri-hybrid nanofluid between two porous surfaces inside a rotating system has been examined. A tri-hybrid nanofluid is a new idea in the research area, which gives a better heat transfer rate as compared to hybrid nanofluid and nanofluid. We also incorporated the thermal radiation effects and Hall current in this article. The similarity techniques are used to reduce the governing nonlinear PDEs to a set of ODEs. For the numerical solution of the considered problem, we have used the MATLAB-based Bvp4c method. The results are presented for tri-hybrid Fe 3 O 4 -Al 2 O 3 -TiO 2 /H 2 O nanofluid. The main focus of this study is to examine the magnetohydrodynamic heat transfer and tri-hybrid nanofluid flow in a rotating system between two orthogonal permeable plates by taking into account the Hall current and thermal radiation effects. The obtained results have been explained with the help of graphical illustrations and tables. It is observed that the heat transfer rate of tri-hybrid nanofluid is greater than as compared to hybrid nanofluid and nanofluid. The increasing behavior is also noticed in micro rotational velocity for augmented values of R 0 , H a , and β . The larger values of ϕ 1 , ϕ 2 , and ϕ 3 result in the decrement of SFC and increment in Nusselt number in both (suction and injection) cases.

Under the influence of an alternating magnetic field, flow and heat transfer of a ferrofluid flow over a flexible revolving disc are examined. The flow is hampered by the external magnetic field, which is dependent on the alternating magnetic field's frequency. The current work examines the heat transfer and three-dimensional flow of fluid with high viscosity on a spinning disc that is stretched in a radial direction. The governing equations' symmetries are computed using Lie group theory. In the problem, there is a resemblance that can accomplish with radially stretching velocities divided into two categories, specifically, linear and power-law, by imposing limits from the boundary conditions. The literature has already covered linear stretching, but this is the first discussion of power-law stretching. The governing partial differential is turned into an ordinary differential equations system using additional similarity transformations, which are then numerically handled. The results are presented for hybrid alumina–copper/ethylene glycol ( Al 2 O 3 - Cu / EG ) nanofluid. The calculated findings are novel, and it has been seen that they accord quite well with those of the earlier extended literature. It has been found that hybrid nanofluid flow outperforms nanofluid flow in terms of Nusselt number or heat transfer rate. The heat transmission in the fluid is reduced as the Prandtl number is increased. The heat transfer increases as dimensionless magnetic field intensity ξ increases. Also, axial velocity and radial velocity decrease as magnetic field intensity increases. As the ferromagnetic interaction parameter is raised, the efficiency of heat transmission decreased. For non-linear stretching with stretching parameter 0 < m < 1, the velocity decreases with the increase in m.

This paper explores anisotropic spherical structures within metric f ( R ) gravity, where R is the Ricci scalar, extending general relativity to include functions of R modified gravitational effects. We analyze compact stars using Finch–Skea solutions under the models; f ( R ) = R + α R 2 , f = R + α R 2 ( 1 + γ R ) , and f = R + α R e - R γ - 1 . These models help us examine how gravity modifications affect the internal structure and behavior of neutrons and strange stars. We investigate material variables such as density, pressures, anisotropy, and forces (gravitational, hydrostatic, anisotropic) through graphical analysis. The physical viability of the stellar models is assessed by evaluating energy conditions (NEC, WEC, SEC, and DEC) and the equation of state (EoS) parameter. We also examine the role of anisotropy in stability and structure, comparing the results with general relativity to highlight the implications of f ( R ) gravity on compact stars. This study aims to enhance the understanding of how modified gravity theories could impact the properties of compact astrophysical objects.

The present research article is related to the analytical investigation of some nonlinear fractional-order Fisher’s equations. The homotopy perturbation technique and Shehu transformation are implemented to discuss the fractional view analysis of Fisher’s equations. For a better understanding of the proposed procedure, some examples related to Fisher’s equations are presented. The identical behavior of the derived and actual solutions is observed. The solutions at different fractional are calculated, which describe some useful dynamics of the given problems. The proposed technique can be modified to study the fractional view analysis of other problems in various areas of applied sciences.

The present model deals with the consequence of Dufour, activation energy, and generation of heat on electromagnetohydrodynamic flow of hyperbolic tangent nanofluid via a stretching sheet. This offers a broad significance in several engineering fields. With adequate similarity variables, the regulating governing equations of PDEs are renovated into nonlinear ODEs. The numerical output of the produced ordinary differential equations is conducted with MATLAB bvp4c. The influence of increasing features on temperature, velocity, concentration patterns, drag force coefficient, Sherwood number and Nusselt number is depicted graphically and numerically. Hence, the resultant conclusions are confirmed utilising contrast with earlier output. Interestingly, the activation energy retards the nanofluid's tangential hyperbolic concentration distribution and the rise in temperature of the hyperbolic tangential nanofluid flow is traceable to an increase in the Dufour effect, However, the electromagnetohydrodynamic variable increases the velocity distribution, which influences the Power law index. Conclusively, the rate of heat transfer is inhibited when the thermophoresis parameter, heat source and the Weissenberg number are enhanced.

The nonlinear coupled Riemann wave equation serves as a mathematical framework for analyzing the interaction between short and long waves in various physical phenomena. Its importance lies in capturing both soliton-like behaviors and complex instabilities that arise in nonlinear media. Despite this significance, exact solutions for this equation, particularly in fractional-order forms, remain limited. In this article, numerous soliton solutions of the NLCRW equation are derived by applying novel modified ( G ′/ G 2 )-expansion method, thereby advancing the state of the art in analytical wave modeling. By employing several fractional derivatives, including the M-Truncated, β, and Conformable operators, the method produces diverse solution families such as hyperbolic, trigonometric and rational forms. Comparative 2D and 3D visualizations further highlight M-type and singular periodic solitary wave structures across these derivatives. In addition, the dynamic behavior of a perturbed nonlinear Hamiltonian system is investigated using bifurcation analysis, Poincaré sections, and Lyapunov exponents. The bifurcation diagrams and phase portraits reveal regime transitions and underscore the critical role of system parameters. Sensitivity and multistability analyses confirm the influence of initial conditions on long-term dynamics. These results provide insights relevant to ion-acoustic waves in plasma, shallow-water wave propagation, and the transmission of optical pulses, where nonlocal interactions and memory effects play an essential role.

The Zakharov–Kuznetsov–Benjamin–Bona–Mahony equation (ZKBBME) is a crucial mathematical model used in fractional quantum mechanics, optical fiber signal processing, ion-acoustic waves in plasma, water waves driven by gravity, turbulent flow, fluid flow waves, and for describing many other real-world phenomena. This article employs the modified exp-function method and exp -expansion method, along with a truncated M-fractional wave transformation, to investigate new rational, trigonometric, hyperbolic, and exponential function solutions. Assigning specific parameter values generates diverse wave shapes most significantly, a new combined wave type called the compacton-kink and a class of peakon waves, which has not yet been documented in previous research of this model. 2-dimensional, 3-dimensional, contour, density, and polar plots illustrate the physical properties of soliton solutions, demonstrating the method’s suitability for analyzing a range of nonlinear fractional models with truncated M-fractional derivative (TMFD). Furthermore, utilizing the Galilean transformation to transform the equation into a planar dynamical system, bifurcation theory is applied to investigate its bifurcation and equilibrium points. The findings show that the TMFD framework captures intricate nonlinear wave dynamics and considerably enriches the ZKBBME solution space. These results advance our knowledge of wave structures in engineering and applied physics models controlled by fractional-order nonlinear partial differential equations (FNLPDEs).

Thermal performance can be enhanced due to the mixing of nanoparticles in base fluid. This research discusses the involvement of ternary hybrid nanoparticles in the mixture of pseudo-plastic fluid model past over a two dimensional porous stretching sheet. Modelling of energy equation is carried out in the presence of external heat source or sink and viscous dissipation. The flow presenting equations and derived in Cartesian coordinate system under usual boundary layer theory in the form of complex coupled partial differential equations (PDEs). The derived PDEs have been converted into corresponding ordinary differential equations (ODEs) with the engagement of suitable transformation. The engineers, scientists and mathematicians have great interest in the solution of differential equations because to understand the real physics of the problem. Here, finite element scheme has been used to approximate the solution of the converted problem. The contribution of several emerging parameters on solution have been displayed through graphs and discussed. It is recommended that the finite element method can be engaged to approximate the solution of nonlinear problems arising in modelling the problem in mathematical physics.