Adaptive Spectral Homotopy Analysis with Convergence Control for High-Precision Solution of Bratu-Type Equations

Purpose : This study addresses the computational challenges in solving Bratutype equations, particularly near critical parameters where traditional methods fail.We develop a robust numerical framework to achieve machine-precision accuracy across the entire solution spectrum. Methods : An adaptive spectral homotopy analysis method (ASHAM) is proposed, integrating three innovations: (1) Chebyshev-Gauss-Lobatto spectral discretization for exponential convergence, (2) Homotopy deformation with Bell polynomial expansion of nonlinear terms, and (3) Adaptive ℏ-optimization via residual minimization using Brent’s method. Results : For Bratu’s boundary value problem, ASHAM achieves 10 ⁻¹² maximum absolute error with λ = 2 in 0.15 seconds, outperforming state-of-the-art methods by 3 orders of magnitude. Near criticality (λ _c = 3.513830719), it maintains 5.7×10 ⁻¹⁰ accuracy where existing techniques diverge. The method also demonstrates 98% computational speedup versus wavelet approaches. Conclusion : ASHAM provides unprecedented accuracy and efficiency for Bratutype equations through its convergence-optimized spectral framework. The methodology establishes a new paradigm for singular nonlinear boundary value problems with broad applications in combustion theory and nanotechnology.