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Fractal Fract., Volume 9, Issue 9 (September 2025) – 5 articles

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31 pages, 8499 KiB

Open AccessArticle

Systemic Risk Contagion in China’s Financial–Real Estate Network: Modeling and Forecasting via Fractional-Order PDEs

by Weiye Sun, Yulian An and Yijin Gao

Fractal Fract. 2025, 9(9), 557; https://doi.org/10.3390/fractalfract9090557 (registering DOI) - 24 Aug 2025

Abstract

Modeling risk evolution in financial networks presents both practical and theoretical challenges, particularly during periods of heightened systemic stress. This issue has gained urgency recently in China as it faces unprecedented financial strain, largely driven by structural shifts in the real estate sector and broader economic vulnerabilities. In this study, we combine Fractional-order Partial Differential Equations (FoPDEs) with network-based analysis methods, proposing a hybrid framework for capturing and modeling systemic financial risk, which is quantified using the

Δ C o V a R

algorithm. The FoPDEs model is formulated based on reaction–diffusion equations and discretized using the Caputo fractional derivative. Parameter estimation is conducted through a composite optimization strategy, and numerical simulations are carried out to investigate the underlying mechanisms and dynamic behavior encoded in the equations. For empirical evaluation, we utilize data from China’s financial and real estate sectors. The results demonstrate that our model achieves a Mean Relative Accuracy (MRA) of 95.5% for daily-frequency data, outperforming LSTM and XGBoost under the same conditions. For weekly-frequency data, the model attains an MRA of 91.7%, exceeding XGBoost’s performance of 90.25%. Further analysis of parameter dynamics and event studies reveals that the fractional-order parameter

α

, which controls the memory effect of the model, tends to remain low when

Δ C o V a R

exhibits sudden surges. This suggests that the model assigns greater importance to past data during periods of financial shocks, capturing the persistence of risk dynamics more effectively. Full article

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26 pages, 1292 KiB

Open AccessArticle

Linear Damped Oscillations Underlying the Fractional Jeffreys Equation

by Emad Awad, Alaa A. El-Bary and Weizhong Dai

Fractal Fract. 2025, 9(9), 556; https://doi.org/10.3390/fractalfract9090556 (registering DOI) - 23 Aug 2025

Abstract

In this study, we consider a fractional-order extension of the Jeffreys equation (also known as the dual-phase-lag equation) by introducing the Reimann–Liouville fractional integral, of order

0 < ν < 1

, to the Jeffreys constitutive law, where for

ν = 1

it [...] Read more.

In this study, we consider a fractional-order extension of the Jeffreys equation (also known as the dual-phase-lag equation) by introducing the Reimann–Liouville fractional integral, of order

0 < ν < 1

, to the Jeffreys constitutive law, where for

ν = 1

it corresponds to the conventional Jeffreys equation. The kinetical behaviors of the fractional equation such as non-negativity of the propagator, mean-squared displacement, and the temporal amplitude are investigated. The fractional Langevin equation, or the fractional damped oscillator, is a special case of the considered integrodifferential equation governing the temporal amplitude. When

ν = 0

and

ν = 1

, the fractional differential equation governing the temporal amplitude has the mathematical structure of the classical linear damped oscillator with different coefficients. The existence of a real solution for the new temporal amplitude is proven by deriving this solution using the complex integration method. Two forms of conditional closed-form solutions for the temporal amplitude are derived in terms of the Mittag–Leffler function. It is found that the proposed generalized fractional damped oscillator equation results in underdamped oscillations in the case of

0 < ν < 1

, under certain constraints derived from the non-fractional case. Although the nonfractional case has the form of classical linear damped oscillator, it is not necessary for its solution to have the three common types of oscillations (overdamped, underdamped, and critical damped), unless a certain condition is met on the coefficients. The obtained results could be helpful for analyzing thermal wave behavior in fractals, heterogeneous materials, or porous media since the fractional-order derivatives are related to the porosity of media. Full article

(This article belongs to the Special Issue Recent Trends in Computational Physics with Fractional Applications)

31 pages, 1723 KiB

Open AccessArticle

A Novel Nonlinear Different Fractional Discrete Grey Multivariate Model and Its Application in Energy Consumption

by Jun Zhang and Jiayi Liu

Fractal Fract. 2025, 9(9), 555; https://doi.org/10.3390/fractalfract9090555 - 22 Aug 2025

Abstract

With global energy demand escalating and climate change posing unprecedented challenges, accurate forecasting of regional energy consumption has emerged as a cornerstone for national energy planning and sustainable development strategies. This study develops a novel nonlinear different fractional discrete grey multivariate model (NDFDGM(r_i,N)). This model improves the shortcomings of the conventional GM(1,N) in handling nonlinear relationships and variable differences by introducing different fractional order accumulation and nonlinear logarithmic conditioning terms. In addition, the Firefly Algorithm (FA) was utilized to optimize the model’s hyperparameters, significantly enhancing the prediction accuracy. Through empirical analysis of energy consumption data in China’s eastern, central and western regions and across the country, it has been confirmed that the NDFDGM model outperforms others during both the simulation and forecasting phases, and its predicted MAPE values are, respectively, 1.4585%, 1.4496%, 2.0673% and significantly lower than that of compared models. The findings indicate that this model can effectively capture the complex characteristics of energy consumption, and its prediction results offer a solid scientific foundation for guiding energy strategies and shaping policy decisions. Finally, this paper conducts extrapolation and predictive analysis using the NDFDGM(r_i,N) to explore the development trends of energy consumption in the whole country in the coming three years and puts forward energy policy suggestions for different regions to promote the optimization and sustainable development of the energy structure. Full article

(This article belongs to the Special Issue Applications of Fractional-Order Grey Models, 2nd Edition)

22 pages, 424 KiB

Open AccessArticle

Solution of Time-Fractional Partial Differential Equations via the Natural Generalized Laplace Transform Decomposition Method

by Hassan Eltayeb and Shayea Aldossari

Fractal Fract. 2025, 9(9), 554; https://doi.org/10.3390/fractalfract9090554 - 22 Aug 2025

Abstract

This work presents an efficient analytical technique for obtaining approximate solutions to time-fractional system partial differential equations. The proposed method combines the Natural generalized Laplace transform with the decomposition method to construct a systematic solution framework. A general formulation of the method is developed for a broad class of time-fractional system equations. In particular, we check the validity and effectiveness of the approach by providing three illustrative examples, confirming its accuracy and applicability in solving both linear and nonlinear fractional problems. Full article

(This article belongs to the Special Issue Recent Computational Methods for Fractal and Fractional Nonlinear Partial Differential Equations)

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15 pages, 1082 KiB

Open AccessArticle

Fractal Modeling of Nonlinear Flexural Wave Propagation in Functionally Graded Beams: Solitary Wave Solutions and Fractal Dimensional Modulation Effects

by Kai Fan, Zhongqing Ma, Cunlong Zhou, Jiankang Liu and Huaying Li

Fractal Fract. 2025, 9(9), 553; https://doi.org/10.3390/fractalfract9090553 - 22 Aug 2025

Abstract

In this study, a new nonlinear dynamic model was established for functionally graded material (FGM) beams with layered/porous fractal microstructures, aiming to reveal the cross-scale propagation mechanism of flexural waves under large deflection conditions. The characteristics of layered/porous microstructures were equivalently mapped to the fractal dimension index. In the framework of the fractal derivative, a fractal nonlinear wave governing equation integrating geometric nonlinear effects and microstructure characteristics was derived, and the coupling effect of finite deformation and fractal characteristics was clarified. Four groups of deflection gradient traveling wave analytical solutions were obtained by solving the equation through the extended minimal (G′/G) expansion method. Compared with the traditional (G′/G) expansion method, the new method, which is concise and expands the solution space, generates additional csch² soliton solutions and csc² singular-wave solutions. Numerical simulations showed that the spatiotemporal fractal dimension can dynamically modulate the amplitude attenuation, waveform steepness, and phase rotation characteristics of kink solitary waves in beams. At the same time, it was found that the decrease in the spatial fractal dimension will make the deflection curve of the beam more gentle, revealing that the fractal characteristics of the microstructure have an active control effect on the geometric nonlinearity. This model provides theoretical support for the prediction and regulation of the wave behavior of fractal microstructure FGM components, and has application potential in acoustic metamaterial design and engineering vibration control. Full article

(This article belongs to the Special Issue Fractal Theory and Models in Nonlinear Dynamics and Their Applications, 2nd Edition)

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