analytical and numerical approaches to nonlinear equations in holographic black hole backgrounds

We investigate nonlinear differential equations arising in holographic models of black hole spacetimes within the ads/cft correspondence. emphasis is placed on analytical approximations and numerical solutions of scalar perturbations and nonlinear wave equations in asymptotically anti-de sitter black holes. we show how effective potentials lead to boundary value problems governed by nonlinear ordinary differential equations. analytical methods, such as matched asymptotic expansions and perturbative ans{a}tze, are compared with spectral numerical approaches. applications to holographic thermalization, quasinormal spectra, and cosmological analogues are discussed. our results highlight the interplay between nonlinear analysis and physical interpretation in string-inspired models of gravity.