Caputo-Type Fractional Differential Equations: Recent Advances and Review

Abstract

This review examines recent advances in Caputo-type fractional differential equations, focusing on theoretical foundations, existence and uniqueness results, and progress in numerical methods. Particular emphasis is placed on discretization schemes, convergence analysis, and high-order hybrid techniques that enhance accuracy and efficiency. Applications in science and engineering such as viscoelasticity, anomalous transport, population dynamics, and epidemiology demonstrate the broad utility of Caputo-type models in capturing memory effects and long-range interactions. The review also explores nonlinear formulations, optimal control, inverse problems, and stochastic extensions, underscoring both the versatility and the challenges of Caputo-based approaches. Finally, the review outlines open questions and future research directions, offering a roadway for advancing fractional calculus theory and applications in complex systems.