| Title |
Framework for the structural analysis of fractional differential equations via optimized model reduction |
| Authors |
Telksnienė, Inga ; Telksnys, Tadas ; Marcinkevičius, Romas ; Navickas, Zenonas ; Čiegis, Raimondas ; Ragulskis, Minvydas |
| DOI |
10.32604/cmes.2025.072938 |
| Full Text |
|
| Is Part of |
Computer modeling in engineering & sciences.. Henderson, NV : Tech Science Press. 2025, vol. 145, iss. 2, p. 2131-2156.. ISSN 1526-1492. eISSN 1526-1506 |
| Keywords [eng] |
fractional differential equations ; Caputo derivative ; fractional power series ; ordinary differential equa- tion ; model reduction ; structural optimization ; particle swarm optimization |
| Abstract [eng] |
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. |
| Published |
Henderson, NV : Tech Science Press |
| Type |
Journal article |
| Language |
English |
| Publication date |
2025 |
| CC license |
|
