August 21, 2025Open Access

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

Key Points

ASHAM achieves unprecedented accuracy of 10−12 for bratu-type equations, demonstrating remarkable effectiveness.
With λ = 2, the method performs calculations in 0.15 seconds, surpassing existing techniques by 3 orders of magnitude.
Using a robust spectral discretization approach, the method integrates homotopy techniques for enhanced convergence.
ASHAM not only enhances accuracy but also offers a 98% computational speedup compared to wavelet methods.

Abstract

Abstract 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 (λc = 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.

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