Improved Analytical Technique Based on Laplace Transform and Homotopy Perturbation Method for Solving Fractional Bratu-Type Equations

This study presents a modified Laplace transform homotopy perturbation method (MLT-HPM) for obtaining approximate solutions for fractional-order Bratu-type ordinary differential equations involving Caputo fractional derivatives. The proposed modification introduces a specific rule for selecting the initial solution, replacing the conventional random choice. This modification significantly improves the convergence rate, resulting in more accurate and computationally efficient approximate solutions. Several numerical examples of fractional Bratu-type differential equations are provided to demonstrate the accuracy, reliability, and effectiveness of the proposed approach. Comparative analyses reveal that MLT-HPM achieves high-precision results with substantially fewer iterations than the standard LT-HPM, underscoring its superior computational performance and practical applicability.