Analysis of fractional logistic integro-differential equations with time scales in medical sciences and industry

This study investigates the theoretical and practical challenges in solving nonlinear pantograph integro-differential equations of fractional order on arbitrary time scales. It aims to establish the existence and uniqueness of solutions by incorporating two Riemann-Liouville fractional derivatives and applying advanced analytical tools, such as Schauder’s fixed point theory and the Banach contraction principle. Furthermore, the research demonstrates the practicality of the results through a MATLAB-based numerical example, graphical visualizations, and real-world applications of the pantograph equation, with a particular focus on industrial and medical instances to highlight its modeling relevance in these domains.