Robust Computational Methods for Solving Strongly Nonlinear Initial Value Problems in Combustion and Astrophysics

In this study, we propose two robust computational schemes for solving the well-known strongly nonlinear Bratu and Lane–Emden-type equations, with applications to combustion and astrophysics. The proposed quasi-linearized Picard iteration method (QPIM) combines the quasi-linearization technique with the Picard iteration, while its piecewise extension is uniform and significantly enhances the convergence region and improves stability near singularity. In QPIM, quasi-linearization transforms a nonlinear equation into a sequence of linearized problems, which are then efficiently solved using the Picard approach. Unlike existing approaches such as Adomian decomposition, Taylor wavelets, or block Nyström methods, the proposed schemes achieve highly accurate solutions with only a few iterations, even over large intervals. Comprehensive numerical experiments on Bratu and Lane–Emden equations demonstrate that QPIM and piecewise QPIM yield up to several orders of magnitude smaller errors compared to existing methods, confirming their superiority in terms of accuracy, efficiency, and ease of implementation.