Ebrahim Momoniat

The Lane-Emden equation, a nonlinear second-order ordinary differential equation, plays a fundamental role in theoretical physics and astrophysics, particularly in modeling the structure of stellar interiors. Also referred to as the polytropic differential equation, it describes the behavior of self-gravitating polytropic spheres. In this study, we present a novel approach to the solution of the eigenvalue problem which arises when considering the Lane-Emden equation for n = 0, 1, 2, 3, 4 using Physics-Informed Neural Networks (PINNs). The novelty of this work is that, we not only solve the Lane-Emden equation via PINNS but we also determine the eigenvalue, r, which is the stellar radius. Hyperparameter tuning was conducted using Bayesian optimization in the Optuna framework to identify optimal values for the number of hidden layers, number of neurons, activation function, optimizer, and learning rate for each value of n. The results show that, for n = 0, 1, PINNs achieve near-exact agreement with theoretical eigenvalues (errors < 0.000806%). While for more nonlinear cases, n = 2, 3 and n=4, PINNs yield errors below 0.0009% and 0.05% respectively, validating their robustness.

This paper investigates the collective impacts of magnetic field, Brownian diffusion, thermophoresis, and buoyancy force on the Casson – Williamson nanoliquid over a rough slender cylinder. The novelty of this paper is to inspect the effect of inclined magnetic field (in particular, at an angle of 45 ⁰ to the liquid flow) over an oxytactic bioconvective nanoliquid flow. To simulate the problem, a set of nonlinearly coupled partial differential equations is employed, and nonsimilar transformations are used to reduce them into nondimensional form. For mathematical simplification, an implicit finite difference scheme and Quasilinearization technique are used. The outcomes for the influences of several significant parameters on the mass and heat transmission properties are illustrated through graphs. Temperature and velocity profiles are pronounced to be more for augmenting values of thermophoresis and Brownian motion parameters. The heat transmission and coefficient of skin friction are enhanced with the surface roughness parameter, while skin friction increases and the heat transmission decreases with the magnetic parameter. The mass transmission rate of oxygen liquid, and microbial density number enhance with the higher bioconvection Lewis number and Peclet number. In a comparison with earlier published works, the results are discovered to be highly concordant.