Understanding the internal structure and stability of compact stellar objects (e.g., Neutron stars), particularly neutron stars, remains a central and unresolved problem in modern astrophysics. Despite significant progress, key challenges persist in accurately determining the equation of state (EoS) at supra-nuclear densities, where matter exhibits extreme behavior not reproducible in terrestrial laboratories. Recent observational breakthroughs—such as precise mass-radius measurements and GW signals from neutron star mergers—have imposed stringent constraints, yet theoretical models within GR and modified gravity frameworks often yield competing predictions. Additionally, the roles of density profile, pressure anisotropy, strong magnetic fields, phase transitions, and exotic matter components further complicate realistic modelling. Our this project addresses these challenges by developing numerically tractable and physically viable models of compact stars, aiming to reconcile theoretical predictions with observational data while exploring the implications of alternative gravitational theories on stellar stability, structure, and evolution.

A central challenge in contemporary mathematical astrophysics is to develop a consistent theoretical framework that can simultaneously explain multi-messenger observations of BH, including EHT, GW, electromagnetic emissions, and high-energy particle signals. Since the landmark detection of GW by the LIGO Scientific Collaboration and Virgo Collaboration, alongside horizon-scale imaging by the EHT, it has become evident that isolated analyses are insufficient to capture the full physical picture of BH environments. Current problems include reconciling strong-field gravity predictions with observational data, understanding the role of accretion dynamics and magnetic fields in multi-wavelength emissions, and identifying robust signatures of modified gravity in observed signals. This project aims to address these issues by constructing analytically tractable and numerically viable models that integrate relativistic dynamics with observational constraints, thereby providing a unified interpretation of BH phenomena in the era of multi-messenger astrophysics.

Understanding gravitational collapse remains one of the most fundamental and unresolved problems in modern astrophysics and gravitation. Despite the well-established framework of GR, several critical questions persist regarding the end state of collapse, the nature of singularities, and the conditions under which BH or alternative objects form (such as ECO, etc). In particular, issues related to cosmic censorship, stability of collapsing configurations, and the role of anisotropy, dissipation, and exotic matter fields are still actively debated. Our this project addresses these challenges through a combined analytical and numerical approach, focusing on new EoS, exact/numerical solutions, dynamical evolution of relativistic fluids, and stability analysis under realistic physical conditions. By integrating mathematical modeling with computational simulations, we aim to provide deeper insights into collapse dynamics, horizon formation, and possible deviations arising from modified gravity frameworks, thereby contributing to a more comprehensive understanding of high-energy astrophysical phenomena.