The present investigation addresses the flow of hybrid (nickel–zinc ferrite and ethylene glycol) nanoliquid with entropy optimization and nonlinear thermal radiation coatings past a curved stretching surface. Analysis was carried out in the presence of magnetohydrodynamic, heat generation/absorption, and convective heat and mass flux conditions. Solution of the modeled problem was attained numerically using MATLAB built-in function bvp4c. Impacts of prominent parameters on betrothed distributions were depicted through graphs and were well supported by requisite discussions. Numerically calculated values of Sherwood number were established in a tabulated form and were scrutinized critically. An excellent concurrence was achieved when results of the presented model were compared with previously published result; hence, dependable results are being presented. It was observed that concentration field diminished with increasing values of curvature parameter, though the opposite trend was noticed for velocity and temperature distributions. Further, it was detected that Nusselt number decreased with augmented values of radiation and curvature parameters.

In this paper, we present an efficient solution method for solving fractional system partial differential equations (FSPDEs) using the Laplace residual power series (LRPS) method. The LRPS method is a powerful technique for solving FSPDEs, as it allows for the efficient computation of the solution in the form of a power series expansion. The method is based on the Laplace transform and the residual power series, and is applied to a system of coupled FSPDEs. The method is validated using several test problems, and the results show that the LRPS method is a reliable and efficient method for solving FSPDEs.

The present study characterizes the flow of three-dimensional viscoelastic magnetohydrodynamic (MHD) nanofluids flow with entropy generation analysis past an exponentially permeable stretched surface with simultaneous impacts of chemical reaction and heat generation/absorption. The analysis was conducted with additional effects nonlinear thermal radiation and convective heat and mass boundary conditions. Apposite transformations were considered to transform the presented mathematical model to a system of differential equations. Analytical solutions of the proposed model were developed via a well-known homotopy analysis scheme. The numerically calculated values of the dimensionless drag coefficient, local Nusselt number, and mass transfer Nusselt number are presented, with physical insights. The graphs depicting the consequences of numerous parameters on involved distributions with requisite deliberations were also a part of this model. It is seen that the Bejan number is an increasing function of the thermal radiation parameter.

This study presents a comprehensive numerical investigation into the effects of thermal radiation and chemical reactions on magnetohydrodynamic (MHD) nanofluid bioconvection influenced by gyrotactic microorganisms, with direct applications in bioreactors, heat exchangers, smart cooling, and bioengineered fluid systems. The complex interplay among thermal gradients, magnetic fields, nanoparticle motion, and microorganism activity is modelled through governing equations for velocity, temperature, concentration, and microorganism density, transformed via similarity techniques and solved using the classical fourth-order Runge–Kutta (RK4) method coupled with the shooting technique. Key numerical findings reveal that increasing Brownian motion enhances thermal uniformity, while more substantial gyrotactic effects intensify bioconvection by concentrating microorganisms near the boundary layer. According to parametric analysis, increasing the Hartmann number (Ha) from 0 to 3 reduces skin friction at the wall and suppresses nanofluid velocity by approximately 22% and the Nusselt number by approximately 15%, due to Lorentz force damping. Heating is intensified by thermal radiation (R); an increase in R from 0.5 to 1.5 results in a temperature increase of about 21%. While thermophoresis (Nt) encourages particle migration to cooler locations, lowering the Sherwood number by up to 14%, the Brownian motion parameter (Nb) increases nanoparticle diffusion, increasing temperature profiles by \(\approx \) 12% but decreasing near-wall concentration by \(\approx \) 8%. While higher Peclet numbers (Pe) reinforce bioconvection currents and increase microorganism clumping by approximately 17%, higher Lewis numbers (Le) steepen microorganism density gradients near the wall. By increasing the density of microorganisms close to the wall by approximately 18%, the gyrotactic parameter (\(\Omega \)) improves bioconvective mixing. An artificial neural network (ANN) with the Levenberg–Marquardt algorithm for training backpropagation is employed to enhance forecasting capability, achieving \(R^2=0.9996\) with mean squared errors \(< 10^{-5}\) for skin friction, Nusselt number, and Sherwood number. These results provide design insights for applications in microfluidics and thermal systems by establishing quantitative correlations between governing factors and transport characteristics, demonstrating the potential of the ANN for intelligent and optimized control of MHD nanofluid bioconvection systems.

In this investigation, the fractional Hirota–Satsuma coupled Korteweg–de Vries (KdV) problem is solved using two modern semi-analytic techniques known as the Aboodh residual power series method (ARPSM) and Aboodh transform iteration method (ATIM). The two suggested approaches are briefly explained, along with how to use them to solve the fractional Hirota–Satsuma coupled KdV problem. Some analytical approximate solutions for the current problem are derived using the proposed techniques until the second-order approximation. To ensure high accuracy of the derived approximation, they are analyzed numerically and graphically and compared with the exact solutions of the integer cases. The offered techniques demonstrate more accuracy in their outcomes compared to other alternatives. The numerical results show that ARPSM and ATIM are highly accurate, practical, and beneficial for solving nonlinear equation systems. The current results are expected to help many physics researchers in modeling their different physical problems, especially those interested in plasma physics.