Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 10 1 2025 12 17 Non-Linear Equation Approach to Black Hole Phase Transitions 115 124 2018 10.22128/ansne.2025.3133.1173 EN Yerlan Myrzakulov Department of General \& Theoretical Physics, L. N. Gumilyov Eurasian National University, Astana, 010008, Kazakhstan 0000-0003-0160-0422 Journal Article 2025 11 08 We investigate the role of non-linear equations in characterizing black hole phase transitions within extended thermodynamic frameworks. By formulating the critical conditions as the solution of a non-linear equation derived from the equation of state, we provide an analytical and numerical study of black hole thermodynamics near criticality. Our analysis demonstrates that the non-linear structure encodes universal behavior analogous to the Van der Waals fluid, and it offers a systematic way of locating transition points. The method extends naturally to higher-dimensional and charged AdS black holes, highlighting the generality of this approach. Non-Linear Equation Approach black hole Phase Transitions https://ansne.du.ac.ir/article_2018_2b04e23a27a2bccee5476956fb4cf7af.pdf