Fractional-Order Ordinary Differential Equations: Theory and Applications

Fractional calculus extends classical calculus by allowing differentiation and integration of arbitrary real or complex order. Fractional-order ordinary differential equations (FODEs) have become an important mathematical tool for modelling systems with memory and hereditary characteristics. Unlike classical differential equations, fractional models incorporate non-local dynamics, enabling more accurate representations of physical and biological processes. This paper presents a comprehensive study of fractional-order ordinary differential equations, including their theoretical foundations, solution techniques, and practical applications. The fundamental definitions of fractional derivatives such as the Riemann–Liouville and Caputo derivatives are discussed. Existence and uniqueness theorems are established using fixed-point theory. Analytical and numerical methods for solving fractional differential equations are also explored. MATLAB-based numerical simulations illustrate the dynamic behaviour of fractional systems for different fractional orders. The study further highlights applications of fractional differential equations in viscoelasticity, anomalous diffusion, electrical circuits, and biological systems. The results demonstrate that fractional models provide more flexible and realistic frameworks for describing complex dynamic phenomena.