In this work, numerous features of the nonlinear Van der Waals gas system in the sense of viscosity capillarity are analytically and theoretically investigated. A novel trustworthy integrating approach, namely the Riccati modified extended simple equation method (RMESEM), is utilized to determine the physical nature of the system. This technique successfully generates a novel range of kink, anti-kink, cuspon kink, and bell-shaped bright and dark hump solitary wave solutions in the form of exponential, periodic, hyperbolic, rational, and rational-hyperbolic functions. Using suitable parameter values that satisfy constraint criteria, we generate a set of 2D graphs to improve comprehension and graphically depict these solitary wave solutions. The efficiency and adaptability of our method in solving a range of nonlinear problems in mathematical science and engineering are validated by our computational research.

This study includes an innovative approach to optimizing heat transfer in ternary nanofluids embedded with three nanoparticles. This research is valuable for engineering, biomedical and industrial applications, and its findings can be utilized to improve energy-efficient heat exchangers, cooling systems, and biomedical thermal therapies. This study investigates the nanoparticle-based heat transport characteristics of ternary nanofluids under the influence of magnetization, thermal convection, and heat generation with nonisothermal and nonisosolutal geometries. The basic equations and assumptions are modeled using the cross fluid model. This model accounts for the non-Newtonian behavior of the ternary nanofluid, providing a more realistic representation of fluid at higher and lower shear rates. The impact of thermal radiation and chemical reactions on the temperature and concentration fields is also examined to enhance thermal efficiency and reaction kinetics. To solve the governing partial differential equations (PDEs), a hybrid computational approach is employed by integrating a multilayer neural network scheme with the bvp4c solver. The artificial neural network (ANN) is trained on numerical solutions obtained from bvp4c, providing an intelligent framework for predicting the thermal and concentration profiles with enhanced accuracy and computational efficiency. The numerical performance of the ANN is evaluated in terms of error reduction over 35 training epochs, achieving a final error of 1.7539×10−5. An increasing trend of the velocity of the ternary nanofluid is seen in both the wedge and cone cases for higher solutal Grashof numbers and cross-index parameters. A higher wall concentration parameter establishes a stronger concentration gradient between the wall and the bulk nanofluid, promoting diffusive transport away from the boundary. This results in a reduction in concentration levels within the core  fluid region.