This study presents an enhanced analytical approximation method for solving the fractional Phi-four formula. We offer an alternate foundation for applying the LRPS technique to handle fractional differentiation problems that arise in many physical occurrences. This technique is based on residue function and the expanded Taylor series formula. The suggested course of action produces favorable results. The robustness, effectiveness, and ease of use of the proposed approach are demonstrated for all classes of fractional issues that arise in the analytical and technical areas. The efficiency of the suggested method in resolving numerous types of fractional differential equations is demonstrated with a case study. The case study provides empirical proof that the recommended method works well for solving a variety of issues involving fractions. Finally, compared to existing methods like q-HATM and HAM, our approach yields effective and favorable outcomes