Structural Optimization of Integer-Order Approximations for Fractional Differential Equations

This study presents a computational framework designed to analyze the intrinsic structure of fractional differential equations (FDEs) by mapping them onto optimal ordinary differential equation (ODE) architectures. While FDEs are instrumental in modeling non-local dynamics and memory effects, analyzing their fundamental structural properties—distinct from obtaining numerical solutions—remains a complex challenge. The proposed methodology transforms FDEs involving iterated Caputo derivatives into a higher-order representation comprised of two distinct elements: a parameterizable integer-order polynomial component and a residual fractional power series.

Crucially, this approach posits that the integer-order component is not a static derivation but a flexible, parameterizable architecture. By employing global optimization techniques, specifically Particle Swarm Optimization (PSO), this framework searches for a polynomial structure that minimizes a dual objective: the deviation from a high-fidelity reference solution and the magnitude of the truncated residual series. This process effectively identifies the most parsimonious ODE that captures the dominant dynamics of the fractional system.

Numerical experiments conducted on both a linear FDE and a nonlinear fractional Riccati equation demonstrate the framework's efficacy. The results reveal that the optimal integer-order representation often requires a higher polynomial degree than the forcing function of the original FDE to compensate for the transformation of the fractional operator. For instance, a quadratic ODE architecture was identified as the optimal representation for a linear FDE. This work provides a novel tool for model reduction and structural analysis, offering deeper insights into the interplay between local integer-order dynamics and non-local fractional memory effects.