This manuscript deals with the Heisenberg ferromagnet-type integrable Akbota equation (AE), which refers to a set of differential equations that are integrable and linked together, and they possess solitary waves. AE is a basic gear for investigating nonlinear dynamics in the fields of optics, magnetism, and differential geometry of curves and surfaces. It is extensively used to represent optical solitons in nonlinear optical fibers, which are crucial for fiber-optic communication owing to their capacity to maintain form across considerable distances. The dynamical behavior of AE is explored by constructing accurate closed-form traveling wave solutions. For this purpose, the Kumar-Malik method, the new Kudryashov method, and the Riccati equation method are utilized. The resulting solutions consist of trigonometric, hyperbolic, and rational functions. By employing these methodologies, precise analytical remedies for soliton waves are derived, which include kink, bright, and dark solitons. To get a better understanding of the physical aspects of these solutions, we depict them via several visual representations. 3D-surface graphs, 2D-line graphs, and contour and density plots, in addition to theoretical derivations.

Nanofluid flow over a rotating disk has several applications in engineering and industrial sectors, such as in cooling systems, heat exchangers, aerospace systems, and renewable energy systems. In the current analysis, the couple stress nanofluid flow over a rotating disk is reported. The nanofluid consists of ethylene glycol and titanium aluminum vanadium (Ti6Al4V) nanoparticles (NPs). The unique properties of Ti6Al4V-NPs, such as biocompatibility, high strength, high boiling point (1604–1660 °C), and high corrosion resistance, make them more suitable for automobile industries. For the heat and mass transfer, the Cattaneo–Christov concept is introduced. In addition, the fluid flow is subjected to magnetic field, Hall current, thermal radiation, and Joule heating. The modeled equations are restructured into the dimensionless system of ordinary differential equations (ODEs) by using the similarity approach. The system of ODEs is further numerically solved through a MATLAB package based on the finite difference method (BVP4c). The results are presented in figures. It has been observed that the energy and curves of the nanofluid decline with the influence of thermal and solutal time relaxation parameters, respectively.

Despite the historical position of the F-expansion method as a method for acquiring exact solutions to nonlinear partial differential equations (PDEs), this study highlights its superiority over alternative auxiliary equation methods. The efficacy of this method is demonstrated through its application to solve the convective–diffusive Cahn–Hilliard (cdCH) equation, describing the dynamic of the separation phase for ternary iron alloys (Fe–Cr–Mo) and (Fe–X–Cu). Significantly, this research introduces an extensive collection of exact solutions by the auxiliary equation, comprising fifty-two distinct types. Six of these are associated with Weierstrass-elliptic function solutions, while the remaining solutions are expressed in Jacobi-elliptic functions. I think it is important to emphasize that, exercising caution regarding the statement of the term ’new,’ the solutions presented in this context are not entirely unprecedented. The paper examines numerous examples to substantiate this perspective. Furthermore, the study broadens its scope to include soliton-like and trigonometric-function solutions as special cases. This underscores that the antecedently obtained outcomes through the recently specific cases encompassed within the more comprehensive scope of the present findings.

In this paper, we introduce a biparametrized multiplicative integral identity and employ it to establish a collection of inequalities for multiplicatively convex mappings. These inequalities encompass several novel findings and refinements of established results. To enhance readers’ comprehension, we offer illustrative examples that highlight appropriate choices of multiplicatively convex mappings along with graphical representations. Finally, we demonstrate the applicability of our results to special means of real numbers within the realm of multiplicative calculus.

The heat transfer augmentation, solar power systems, medical equipment, semiconductor cooling, aerospace, and automotive industries all use ternary hybrid nanofluids (THNFs). The current study is mainly about a magnetized THNF flow that cannot be squished around a spinning sphere that has different viscosity, thermal conductivity, and shape (brick, platelets, cylinder, and blade). The heat transport simulation incorporates the principles of viscous dissipation and joule heating. Water is mixed with silver, magnesium oxide, and iron trioxide to make the THNF. Similarity substitution converts model equations to ordinary differential equations (ODEs). Runge–Kutta fourth order numerically estimates the non-dimensional set of ODEs. For certain emergent parameters, velocity, temperature, entropy generation, Nusselt number, and skin friction are computed and analyzed. The research shows that entropy generation increases with brinkman number, nanoparticle volume fraction and magnetic parameters and reduces with temperature difference parameter. Increasing the unsteadiness parameter upsurges velocity in the x-direction, but decreases it in the z-direction and temperature curve. Skin friction upsurges in the x-direction and declines in the z-direction with rotation. Platelet-shaped nanoparticles usually outperform blade, brick, and cylinder shapes. When mass suction $( S )$ is elevated from 1.0 to 2.0, the heat transfer rate increases by 47.25% for the brick form, 47.26% for the platelets shape, 35.08% for the cylinders shape, and 37.65% for the blades shape. Comparing the results to prior literature shows excellent agreement.

This research focuses on investigating soliton solutions for the space-time fractional modified third-order Korteweg-de Vries equation using the auxiliary equation method. The Korteweg-de Vries equation is renowned for its application in modeling shallow-water waves, oceanographic dynamics, and ion-acoustic waves in plasma. By employing a traveling wave transformation, the fractional-order partial differential equation is converted into a nonlinear ordinary differential equation. This study used conformable derivatives to solve the fractional differential equation. Soliton solutions, including bright, dark, singular, periodic singular, combined bright-dark, and combined dark-singular forms, are derived through the application of auxiliary equation method to the reduced equation.