Hybrid Analytical-Numerical Investigation of the Bratu Equation with Emphasis on Convergence

This study presents a comprehensive investigation of the Bratu equation, a nonlinear boundary value problem with significant applications in combustion theory, chemical reactor modeling, and thermal self-ignition phenomena. Characterized by its exponential nonlinearity and bifurcation behavior, the Bratu equation is analyzed using both exact and approximate methodologies. Lie symmetry analysis is utilized to obtain exact analytical solutions and to reveal the structural and invariant characteristics of the equation. In parallel, a semi-analytical framework is developed wherein a set of auxiliary differential equations is constructed, and their linear combination is optimized via the weighted residual method to yield approximate solutions. The convergence behavior of the proposed method is thoroughly investigated and confirmed for diverse values of the nonlinear coefficient [Formula: see text]. Additionally, MATLAB’s built-in boundary value problem solvers are employed to compute numerical solutions, serving as benchmarks for validating the analytical and semi-analytical findings. The study identifies the critical bifurcation parameter [Formula: see text] and offers detailed insights into the qualitative behavior and multiplicity of solutions. The findings contribute to a deeper understanding of the Bratu equation’s solution landscape and its sensitivity to nonlinear parameters.