Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 9 2 2025 10 29 On the Nonlinear Equation of State in Black Hole Thermodynamics 270 279 1935 10.22128/ansne.2025.3060.1154 EN Sudhaker Upadhyay Department of Physics, K.L.S. College, Nawada, Magadh University, Bodh Gaya, Bihar 805110, India 0000-0002-3880-7315 Journal Article 2025 09 16 The study of black hole thermodynamics has revealed profound connections between gravitation, quantum theory, and statistical mechanics. In many instances, the key physical information is encoded in nonlinear algebraic or transcendental equations that relate horizon radius, temperature, and pressure. In this work, we examine a specific nonlinear equation arising from the extended phase space of charged anti-de Sitter (AdS) black holes. By analyzing its structure and obtaining approximate and exact solutions, we highlight the physical implications for the thermodynamic stability of black holes. Our results clarify the role of nonlinearities in determining critical points and phase transitions analogous to the van der Waals fluid. https://ansne.du.ac.ir/article_1935_583f1b6a01e175779b565368da40f087.pdf