Classical vs. spectral Adomian decomposition: A comparative solution of the strongly nonlinear corneal shape equation

This study presents a comparative analysis of the classical Adomian decomposition method (ADM) and the spectral Adomian decomposition method (SADM) for solving the strongly nonlinear boundary value problem governing the human corneal shape. The model, a second-order nonlinear ordinary differential equation, lacks a known closed-form solution, necessitating robust numerical or semi-analytical approaches. We provide a detailed algorithmic implementation for both methods, where SADM enhances the classical ADM framework by integrating Chebyshev pseudospectral techniques. Numerical simulations across six distinct parameter configurations, spanning weak to strong nonlinearity, demonstrate the superior performance of SADM. The solutions were validated against established theoretical bounds, confirming their physical realism and methodological robustness. Comparative assessments revealed that SADM achieves accelerated convergence, higher accuracy, and significantly reduced errors compared to classical ADM, MATLAB’s bvp4c solver, and other methods in the literature. Crucially, SADM maintains exceptional precision across all nonlinear regimes, establishing it as a highly efficient and accurate computational framework for complex boundary value problems in ophthalmic biomechanics. • This is the first application of ADM and SADM to the corneal shape model. • The SADM application shows superior accuracy and convergence over ADM. • The SADM solution maintains high precision across all tested nonlinear regimes. • High computational efficiency supports potential real-time clinical use. • A rigorous convergence analysis for both proposed methods is established.