In the given research, our main goal is to explore newsoliton solutions in (2+1)-dimensional Heisenberg ferromagnetic spin chain (HFSC) equation. This was achieved via the extended (G'/G2)-expansion method. For this, we first analyse the given (2 + 1)-dimensional HFSC system of a partial differential equation (PDE) that is obtained by separating the equation into real and imaginary parts. The solution of our equation is determined as rapidly convergent sequences effectively computed by using Mathematica software. Furthermore, a special class of nonlinear wave solutions like periodic, straight, decreasing, dark-singular soliton and dark bright singular soliton were obtained in Heisenberg ferromagnetic dynamics. The amplitude of the soliton is determined by giving different values of parameters. With a suitable choice of parameters, 3D and 2D graphical illustrations are reported. The methodology applied is effective for obtaining exact solutions for various fractional partial differential equations in complex medium. The numerical module of the results obtained is studied, with fascinating figures showing the physical significance of the solutions.

This study investigates the hotness transport and entropy creation of a time-dependent Prandtl–Eyring half and half nanofluid (P-EHNF). The flow and heat transport features of P-EHNF are analysed by subjecting the nanofluid to the slippage heated surface. The effects of the shapes of the nanosolid molecule, porosity material, Cattaneo–Christov heat flux, and radiative transition are likewise associated with this assessment. In a system of partial differential equalities (PDEs), the transcendent stream conditions are planned. Keller box mathematical strategy is a technique used to recognise the self-similar answer for recipes changed into ordinary differential equalities (ODEs) utilising legitimate changes. Two types of nanoparticles, copper (Cu) and graphene oxide (GO)with engine oil (EO) as the base liquid are considered in this review. Significant outcomes for the various factors are drawn up graphically in the streaming, energy, surface friction, Nusselt sum and entropic estimation. The imperative consequence of this investigation is that the heat transmission pace of P-EHNF (GO–Cu/EO) is more than the customary nanofluid (Cu–EO). Likewise, heat transport is the highest for circular-shaped and the lowest for lamina-shaped nano strong particles. With the upgrade of nanoparticles size φ, the entropy is supported in the model. A similar impact likewise occurs with progress in radiative stream N$_r$ and Prandtl–Eyring variable α$^∗$.