In this article, we investigate the utilization of Riemann–Liouville’s fractional integral and the Caputo derivative in the application of the Optimal Auxiliary Function Method (OAFM). The extended OAFM is employed to analyze fractional non-linear coupled ITO systems and non-linear KDV systems, which feature equations of a fractional order in time. We compare the results obtained for the ITO system with those derived from the Homotopy Perturbation Method (HPM) and the New Iterative Method (NIM), and for the KDV system with the Laplace Adomian Decomposition Method (LADM). OAFM demonstrates remarkable convergence with a single iteration, rendering it highly effective. In contrast to other existing analytical approaches, OAFM emerges as a dependable and efficient methodology, delivering high-precision solutions for intricate problems while saving both computational resources and time. Our results indicate superior accuracy with OAFM in comparison to HPM, NIM, and LADM. Additionally, we enhance the accuracy of OAFM through the introduction of supplementary auxiliary functions.

Background: This study investigated the potential of chicken head and foot extract (CHFE) as a low-cost alternative to commercial microbiological media for cultivating clinically important bacteria. Methods: The growth characteristics, colony morphology, and antibiotic susceptibility profiles of selected pathogens, including Staphylococcus aureus, Escherichia coli, Pseudomonas aeruginosa, Klebsiella pneumoniae, Proteus mirabilis, Acinetobacter baumannii, Salmonella typhi, and Shigella dysenteriae, were assessed on CHFE agar and broth. The growth of S. aureus, E. coli, and P. aeruginosa on CHFE media was comparable to that on Nutrient agar and broth, with characteristic pigment production and colony morphology. However, the growth of A. baumannii and S. dysenteriae was relatively low on CHFE medium. Supplementation of CHFE broth with 1% glucose or peptone resulted in enhanced growth, surpassing that in Nutrient broth by 20-30%. Results: Principal-component analysis demonstrated strong relationships between optical density (OD₆₀₀) and colony-forming units (CFU/mL) in both media. Statistical analysis revealed that peptone supplementation had the most pronounced effect on growth improvement. The study highlights the potential of CHFE as a cost-effective medium for clinical diagnostics in resource-limited settings, although further optimization is required to support the growth of fastidious organisms. Conclusion: The utilization of poultry waste in microbiological media can add value to this waste while reducing the reliance on expensive commercial media in low-income countries.