Fractional Processes and Systems in Computer Science and Engineering

1. Introduction

Fractional calculus, with conceptual origins tracing back to the foundational dialogues between L’Hôpital and Leibniz in 1695, has undergone a remarkable transformation from abstract mathematical inquiry into a key framework for studying complex systems [1,2,3,4]. By extending the derivative to non-integer orders, fractional-order operators provide a sophisticated mathematical language for capturing the intrinsic memory and long-range dependence that define the evolution of diverse processes [5,6]. Consequently, it provides a tool for researchers to precisely quantify the intricacies of natural and engineered systems [7,8,9,10,11]. The evolution of fractional calculus has catalyzed its widespread adoption across an expansive scientific landscape, such as information science [12,13], physics [14,15,16], engineering [17,18,19,20,21], ecology [22,23], biology [24,25], social sciences [26,27], and medical sciences [28,29].

This Special Issue, entitled “Fractional Processes and Systems in Computer Science and Engineering,” was established to showcase high-quality research that pushes the boundaries of this field. While the title highlights the technical core, the scope of the contributions extends far beyond traditional computer science and engineering, encompassing diverse disciplines such as geophysics, environmental science, and economic decision-making. Out of 20 submissions received, 10 high-quality papers were accepted for publication, for an acceptance rate of 50%. The 10 articles featured in this Special Issue represent a cross-section of modern fractional research, bridging the gap between abstract mathematical constructs and high-impact implementations. Collectively, these studies demonstrate that the fractional order serves as a universal tuning knob for complexity, from the microscopic capacity decay of Li-ion batteries to the macroscopic fluctuations in global climate oscillations. By weaving these disparate threads, this Special Issue provides a comprehensive reference for experts and a source of inspiration for future explorations into the memory-dependent and persistent nature of our world.

2. An Overview of Published Articles

The papers featured in this Special Issue span a broad spectrum of disciplines, collectively advancing the frontiers of fractional calculus from pure mathematical constructs to complex real-world implementations. These contributions are categorized into the following five thematic areas:

2.1. Theoretical Development

Contribution 1, “Donsker-Type Construction for the Self-Stabilizing and Self-Scaling Process”, provided a foundational contribution to stochastic theory by establishing a Donsker-type construction for a new class of self-stabilizing processes. Their work transcends classical Lévy models by allowing local jump intensities to vary with the process value dynamically. The essence of the research is the rigorous proof of the process’s existence and its multifaceted nature as a Markovian martingale exhibiting self-regulating and self-scaling properties. The paper serves as a theoretical cornerstone, offering a sophisticated mathematical framework for capturing state-dependent feedback in complex systems, bridging pure probability theory with advanced dynamic modeling.

2.2. Earth, Ocean, and Environmental Applications

Contribution 2, ”Ensemble Mean Dynamics of the ENSO Spatiotemporal Oscillator with Fractional Stochastic Forcing”, achieved a breakthrough in stochastic dynamic modeling by replacing traditional white noise or memoryless Markov processes with a fractional Ornstein–Uhlenbeck (FOU) process characterized by power-law decay. The authors theoretically constructed a nonlinear spatiotemporal oscillator model for ENSO driven by long-memory noise. The researchers derived ensemble-mean dynamic equations and identified an anomalous growth rate triggered by the cumulative effect of long-memory forcing. The study quantitatively reveals the modulation logic of the fractional order on the system’s evolution rate; specifically, as the order decreases (signifying enhanced memory), this anomalous growth rate intensifies, thereby persistently dominating the divergence or dissipation behavior of the ensemble-mean system. This discovery not only offers a novel dynamical perspective for understanding ENSO irregularities but also establishes a new mathematical paradigm for the stability analysis of stochastic nonlinear systems.

Contribution 3, “Space–Time Variations in the Long-Range Dependence of Sea Surface Chlorophyll in the East China Sea and the South China Sea”, offerred a significant methodological advancement in environmental remote sensing and fractional statistical inference. Their theoretical breakthrough lies in the transition from static Hurst exponent analysis to a space–time-varying fractional modeling framework. By employing the multi-fractional generalized Cauchy model (mGCM), the authors successfully captured the non-stationary and heterogeneous long-range dependence (LRD) of sea surface chlorophyll (SSC) in the East and South China Seas. The key contribution of this work is the revelation that anthropogenic interventions tend to diminish the memory depth of marine ecosystems. Their findings demonstrate that fractional parameters can serve as complexity or sensitive indicators for monitoring ecosystems under human pressure, providing a robust statistical bridge between abstract fractal theory and real-world environmental monitoring.

Contribution 4, “Multifractal Structures and the Energy-Economic Efficiency of Chinese Cities: Using a Classification-Based Multifractal Method”, progressed our understanding of urban complexity and sustainability by advancing beyond monofractal constraints. They applied a classification-based multifractal method that precisely captures the heterogeneous spatial essence of urban systems. The core contribution lies in illustrating the nonlinear driving mechanisms that connect multifractal spatial organization to energy–economic efficiency. By identifying an advantageous structure following a cubic polynomial form, the authors provide a diagnostic toolkit for detecting urban inefficiencies. Representing an interdisciplinary paradigm, the authors transform fractal geometry into a strategic socio-ecological modeling tool, offering a rigorous mathematical blueprint for optimizing urban forms toward carbon neutrality and economic synergy.

2.3. Engineering Systems and Mechanical Modeling

Contribution 5, “Fractional Poisson Process for Estimation of Capacity Degradation in Li-Ion Batteries by Walk Sequences”, achieved a significant advancement in engineering reliability by modeling Li-ion battery degradation as a dual-fractional process. Transcending traditional models that overlook non-monotonicity, the authors integrated fractional Brownian motion as a diffusion term and the fractional Poisson process (fPp) as a jump term. The authors rigorously established a link between random capacity jumps and long-range dependence, capturing the memory of historical degradation more accurately. By implementing this approach on NASA datasets, the method demonstrates superior predictive performance over the fractional Brownian motion, fractional Levy stable motion, Wiener, and long short-term memory models. The research is a novel application of fractional stochastic dynamics in micro-engineering, providing a robust tool for advanced energy system health management.

Contribution 6, “Fractional Calculus to Analyze Efficiency Behavior in a Balancing Loop in a System Dynamics Environment”, advanced the field of system dynamics by integrating fractional calculus into the analysis of balancing loops. Unlike traditional simulations that rely on idealized scenarios, the authors developed a more realistic mathematical model to capture the varying efficiencies and irregularities inherent in real-world systems. The core innovation lies in using fractional operators to represent the non-local and memory-dependent nature of feedback mechanisms, providing a higher degree of precision in understanding system stabilization. Within this Special Issue, the model is a vital methodological bridge, demonstrating how fractional theory can optimize complex engineering systems by moving beyond the constraints of integer-order modeling toward more accurate, efficiency-aware dynamic analysis.

Contribution 7, “Theoretical Analysis of Viscoelastic Friction System Characteristics of Robotic Arm Brake Based on Fractional Differential Theory”, enhanced the theoretical understanding of mechanical stability by modeling viscoelastic friction in robotic brakes through fractional differential theory. Their model surpasses traditional integer-order damping models by employing Caputo fractional derivatives to capture the inherent memory effects of braking materials. The focal aspect of the study is its rigorous nonlinear analysis, which deciphers how the fractional order acts as a pivotal parameter in modulating bifurcations and chaotic transitions. By quantifying the relationship between fractional dynamics and self-excited vibrations, the study provides a robust mathematical gateway for vibration suppression. The paper reveals the power of fractional mechanics in optimizing high-performance engineering systems.

2.4. Computer Networks

Contribution 8, “A Low-Computational Burden Closed-Form Approximated Expression for MSE Applicable for PTP with gfGn Environment”, addressed a critical challenge in network synchronization by introducing a low-computational-burden closed-form approximation for the mean square error (MSE) within a generalized fractional Gaussian noise (gfGn) environment. Their method is less complex than traditional numerical methods, enabling real-time performance evaluation for the Precision Time Protocol (PTP). The key contribution of this research is a mathematical derivation that quantifies clock skew estimation accuracy under long-range-dependent jitter without excessive computational overhead, which is ideal for real-time industrial applications. By linking advanced fractional signal processing with the practical constraints of embedded communication systems, the study provides a strategic analytical tool for optimizing 5G and industrial internet synchronization. This paper highlights the transformative potential of fractional modeling in enhancing data communication reliability and efficiency.

2.5. Statistical Modeling and Decision Assessment

Contribution 9, “Fractional Time-Varying Autoregressive Modeling: Parallel GAM and PINN Approaches to Dynamic Volatility Forecasting”, introduced a sophisticated fractional time-varying autoregressive (FTVAR) model to address the non-stationarity of financial volatility. Their method provides an advancement in static modeling by integrating long-memory properties with time-evolving parameters. The core innovation lies in a parallel computational framework combining generalized additive models (GAMs) and physics-informed neural networks (PINNs) for interpreting time-varying coefficients and embedding system dynamics. This synergy allows the model to capture complex long-range dependencies while maintaining adaptability to market shifts. The model connects fractional statistics and artificial intelligence, offering a powerful mathematical blueprint for dynamic risk forecasting in volatile financial systems.

Contribution 10, “Aggregation Operator and Its Application in Assessing First-Class Discipline Construction in Industry-Characteristic Universities”, advanced decision science by introducing the complex cubic fractional orthotriple fuzzy set (CCFOFS) to manage uncertainty in institutional evaluations. The theoretical breakthrough is the leveraging of fractional orders to refine the expression of ambiguous information beyond classical fuzzy constraints. The authors developed a sophisticated hybrid aggregation operator (complex cubic fractional orthotriple fuzzy Dombi-weighted power partitioned Muirhead mean, CCFOFDWPPMM) that synthesizes Dombi operations and Muirhead mean logic. This synergy allows for a more nuanced characterization of the interdependencies among assessment criteria. The authors advanced the application of fractional logic to management engineering, offering a robust mathematical framework for objective value assessments in complex socio-academic systems.

3. Concluding Remarks

The ten papers featured in this Special Issue collectively demonstrate the power of fractional calculus in depicting the evolution of complex systems. From the foundational construction of self-stabilizing processes to empirical studies spanning marine ecology, energy engineering, and financial decision-making, these contributions illustrate how fractional theory has evolved from a purely mathematical inquiry into a multifaceted instrument for solving real-world challenges. We wish to express our sincere gratitude to all the authors who contributed to this Special Issue and to the reviewers for their rigorous evaluations. Special thanks are also extended to the editorial staff of Fractal and Fractional for their constant support and professional assistance throughout the publication process. Through these explorations across disciplinary boundaries, we gain a more profound understanding of the memory-dependent and persistent nature of our complex world. We hope this Special Issue will inspire innovative thinking in fractional dynamics and collectively drive this captivating field toward new horizons.