A Residual Power Series Framework in Fractional Convection–Diffusion Model with Convergence Analysis and Applications

This work investigates a novel residual power series technique (RPST) to analyze time-fractional convection–diffusion equations (TFC-DEqs) that model transport processes with memory and anomalous diffusion effects. The approach is appropriate for both linear and nonlinear cases because it avoids linearization, discretizations, and other simplifying assumptions. The reliability of the method is demonstrated by the results, which show strong agreement with exact solutions and rapid convergence as the fractional order approaches one. The error analysis also verifies that such results can be obtained with high accuracy by considering only a few terms, which demonstrates the efficiency of the method. A significant outcome of this research is the development of a reliable and efficient analytical framework that can accommodate complex transport phenomena without resorting to assumptions. The results are significant as they offer insights and show promise for potential applications in engineering and applied science, particularly for nonlocal and memory-dependent systems.