TY  - EJOU
AU  - Telksniene, Inga 
AU  - Telksnys, Tadas 
AU  - Marcinkevičius, Romas 
AU  - Navickas, Zenonas 
AU  - Čiegis, Raimondas 
AU  - Ragulskis, Minvydas 

TI  - Framework for the Structural Analysis of Fractional Differential Equations via Optimized Model Reduction
T2  - Computer Modeling in Engineering \& Sciences

PY  - 2025
VL  - 145
IS  - 2
SN  - 1526-1506

AB  - Fractional differential equations (FDEs) provide a powerful tool for modeling systems with memory and non-local effects, but understanding their underlying structure remains a significant challenge. While numerous numerical and semi-analytical methods exist to find solutions, new approaches are needed to analyze the intrinsic properties of the FDEs themselves. This paper introduces a novel computational framework for the structural analysis of FDEs involving iterated Caputo derivatives. The methodology is based on a transformation that recasts the original FDE into an equivalent higher-order form, represented as the sum of a closed-form, integer-order component G(y) and a residual fractional power series Ψ(x). This transformed FDE is subsequently reduced to a first-order ordinary differential equation (ODE). The primary novelty of the proposed methodology lies in treating the structure of the integer-order component G(y) not as fixed, but as a parameterizable polynomial whose coefficients can be determined via global optimization. Using particle swarm optimization, the framework identifies an optimal ODE architecture by minimizing a dual objective that balances solution accuracy against a high-fidelity reference and the magnitude of the truncated residual series. The effectiveness of the approach is demonstrated on both a linear FDE and a nonlinear fractional Riccati equation. Results demonstrate that the framework successfully identifies an optimal, low-degree polynomial ODE architecture that is not necessarily identical to the forcing function of the original FDE. This work provides a new tool for analyzing the underlying structure of FDEs and gaining deeper insights into the interplay between local and non-local dynamics in fractional systems.
KW  - Fractional differential equations; Caputo derivative; fractional power series; ordinary differential equation; model reduction; structural optimization; particle swarm optimization

DO  - 10.32604/cmes.2025.072938