Open Access

ARTICLE

Framework for the Structural Analysis of Fractional Differential Equations via Optimized Model Reduction

Inga Telksniene¹, Tadas Telksnys², Romas Marcinkevičius³, Zenonas Navickas², Raimondas Čiegis¹, Minvydas Ragulskis^2,*

1 Mathematical Modelling Department, Faculty of Fundamental Sciences, Vilnius Gediminas Technical University, Saulėtekio al. 11, Vilnius, LT-10223, Lithuania
2 Department of Mathematical Modelling, Kaunas University of Technology, Studentu 50-147, Kaunas, LT-51368, Lithuania
3 Department of Software Engineering, Kaunas University of Technology, Studentu 50-415, Kaunas, LT-51368, Lithuania

* Corresponding Author: Minvydas Ragulskis. Email:

(This article belongs to the Special Issue: Analytical and Numerical Solution of the Fractional Differential Equation)

Computer Modeling in Engineering & Sciences 2025, 145(2), 2131-2156. https://doi.org/10.32604/cmes.2025.072938

Received 07 September 2025; Accepted 20 October 2025; Issue published 26 November 2025

Abstract

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.

---

Keywords

---