non-linear equation approach to black hole phase transitions

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.