Vol. 32 No. 1 (2025)

Open Access

Article

Article ID: 2510

Generalized fixed-point theorem for strict almost ϕ-contractions with binary relations in b-metric spaces and its application to fractional differential equations

by Jiaojiao Wu, Fei He, Shu-fang Li

Advances in Differential Equations and Control Processes, Vol.32, No.1, 2025;

The present study is centered around establishing a generalized fixed-point theorem for strict almost ϕ-contractions in b-metric spaces in the context of binary relations. Through the introduction of an innovative lemma, we offer distinct proof methodologies that diverge from the conventional ones in metric spaces. The achieved outcomes not only fortify but also broaden the domain of prior fixed-point theorems in the pertinent literature. Moreover, as a practical exemplification, the existence and uniqueness of solutions to fractional differential equations are illustrated convincingly, thereby connecting the theoretical and applied dimensions of the research.

Open Access

Article

Article ID: 2075

Mathematical modelling and controllability analysis of fractional order coal mill pulverizer model

by Ghanshyam Malviya, Jaita Sharma, Vishant Shah, Gargi Trivedi

Advances in Differential Equations and Control Processes, Vol.32, No.1, 2025;

This paper investigates the controllability of nonlinear dynamical systems and their applications, with a focus on fractional-order systems and coal mill models. A novel theorem is proposed, providing sufficient conditions for controllability, including constraints on the steering operator and nonlinear perturbation bounds. The theorem establishes the existence of a contraction mapping for the nonlinear operator, enabling effective control strategies for fractional systems. The methodology is demonstrated through rigorous proof and supported by an iterative algorithm for controller design. Additionally, the controllability of a coal mill system represented as a nonlinear differential system, is analyzed. The findings present new insights into the interplay of fractional dynamics and nonlinear systems, offering practical solutions for real-world control problems.

Open Access

Article

Article ID: 2799

Rolling optimization control method for hydro-photovoltaic-storage microgrid based on stochastic chance constraints

by Qianjin Gui, Wenfa Xu, Xiaoyang Li, Lirong Luo, Haifeng Ye, Zhengfeng Wang

Advances in Differential Equations and Control Processes, Vol.32, No.1, 2025;

Hydro-photovoltaic-storage (HPS) microgrid has gradually become an important measure to optimize the energy structure and ensure the reliability of regional power supply. However, due to the strong randomness and spatiotemporal correlations of hydropower and photovoltaic (PV) output, traditional deterministic optimization methods are difficult to support the accurate regulation and reliable operation of microgrid with a high proportion of renewable energy integration. On this basis, a rolling optimization control method for HPS microgrid based on stochastic chance constraints is proposed. A novel multivariate scenario reduction method considering hydro-PV correlations is presented to characterize the uncertainty of renewable energy output, and a day-ahead stochastic optimal scheduling model based on chance-constrained programming is constructed. Combined with stochastic model predictive control strategies, the day-ahead scheduling plan can be adjusted at multiple time scales, both intraday power compensation and real-time adjustments, to suppress the intraday power fluctuations induced by day-ahead scenario errors and reduce the influence of the uncertainty of hydro-PV power output on microgrid operation. Experimental results show that compared with the traditional deterministic scheduling method, the proposed method can effectively improve the stability and economy of HPS microgrid operation under complex uncertain conditions.

Open Access

Article

Article ID: 2807

Performance evaluation of post-processed kinematic precise point positioning solution for environmental applications

by Ahmed Al Shouny

Advances in Differential Equations and Control Processes, Vol.32, No.1, 2025;

Precise Point Positioning (PPP) is a modern satellite-based technique known for its simplicity, efficiency, and cost-effectiveness, eliminating the need for a reference or base station. This study evaluates the accuracy of Precise Point Positioning (PPP) solutions for both static and kinematic observations using the CSRS-PPP service. To ensure a fair comparison, PPP-derived results were assessed against relative positioning techniques. Field measurements, including static and kinematic data, were collected across a 39 km² study area in northern Egypt to generate topographic contour maps. The findings indicate that PPP is a viable alternative for static positioning, achieving a 2D horizontal accuracy of ±2.54 cm, though vertical accuracy is lower at 11.3 cm. In kinematic mode, horizontal accuracy is ±5 cm, while vertical accuracy decreases to 18.4 cm. While the achieved 2D accuracy meets the needs of most environmental applications, the lower height precision may not be suitable for tasks requiring high vertical accuracy.

Open Access

Article

Article ID: 2259

Learning of a certain homogeneous reducible differential equation by means of ChatGpt in engineering students during the second semester of 2024 in Antofagasta-Chile

by Jorge Olivares, Byron Droguett, Pablo Martin

Advances in Differential Equations and Control Processes, Vol.32, No.1, 2025;

The main objective of this research work was to investigate the learning of a certain homogeneous reducible differential equation by means of ChatGpt in engineering students, during the second semester of 2024 in Antofagasta-Chile. This research followed a qualitative case study approach. Four students of the differential equations course were chosen. Personalized interviews of three questions, related to the general objective and two specific ones, were established after solving a certain exercise, through ChatGpt collaboration. It was found that the opinions expressed about the use of this artificial intelligence are very positive and valuable, evidencing what was stated by several authors. Finally, it can be concluded that the perception of ChatGpt enriches the mathematical confidence in the development of computers, which generates security in learning.

Open Access

Article

Article ID: 2489

Numerical solution of a 3D mathematical model for the progression of tumor angiogenic factor in a tissue

by Melike Keleş Duman, Serdal Pamuk

Advances in Differential Equations and Control Processes, Vol.32, No.1, 2025;

In this work, the movement of tumor angiogenic factor in a three-dimensional tissue is obtained by the Method of Lines. This method transforms a partial differential equation into a system of ordinary differential equations together with the initial and boundary conditions. The more the number of lines is increased, the more the accuracy of the method increases. This method results in very accurate numerical solutions for linear and non-linear problems in contrast with other existing methods. We present Matlab-generated figures, which are the movement of tumor angiogenic factor in porous medium and explain the biological importance of this progression. The computer codes are also provided.

Open Access

Article

Article ID: 2759

Herd immunity in a coronavirus disease 2019 epidemic model with consideration of vaccination and quarantine interventions

by Moh Hasan, Faizal Rifky Fahreza

Advances in Differential Equations and Control Processes, Vol.32, No.1, 2025;

During the pandemic of COVID-19, people had reduced contact among each other. As a result of this behavior, several factors, such as economic conditions and the teaching and learning process, have been affected. Hence, it is important to identify whether the impact of COVID-19 is no longer as severe as when it was first observed. The study aimed to analyze herd immunity against COVID-19 in Indonesia according to the bifurcations and simulations of mathematical models of COVID-19 transmission. Based on the bifurcation of the disease system, whether the current pandemic was controlled with standard interventions was evaluated. The system behavior can be compared with herd immunity that should be achieved in a specific population. Thus, whether a system has resulted in the achievement of herd immunity can be evaluated. The behavior of this system can provide information on the achievement of group immunity during disease outbreaks.

Open Access

Editorial

Article ID: 2589

Transforming frontiers: The next decade of differential equations and control processes

by Ji-Huan He

Advances in Differential Equations and Control Processes, Vol.32, No.1, 2025;

Mathematics serves as the fundamental basis for innovation, propelling technological advancement. In the forthcoming decade, the convergence of differential equations and control processes is poised to redefine the frontiers of scientific exploration. The integration of artificial intelligence and machine learning with differential equations is set to inaugurate a new era of problem-solving, enabling the extraction of latent physical insights and accelerating solution discovery. Multi-scale modeling, with its capacity to span disparate physical domains, has the potential to resolve long-standing puzzles in fields such as fluid mechanics and nanoscience. Furthermore, the integration of fractal geometry with differential equations holds the promise of novel perspectives for understanding and optimizing complex systems, ranging from urban landscapes to turbulent flows. The integration of artificial intelligence (AI) with control innovations is poised to play a pivotal role in the development of next-generation technologies, with the potential to transform diverse sectors such as medicine, communication, and autonomous systems. This paper explores these developments, highlighting their potential impacts and emphasizing the necessity for interdisciplinary collaboration to leverage their full potential.

(This article belongs to the Special Issue Nonlinear Vibration Systems for MEMS Systems and Energy Harvesting)

Open Access

Editorial

Article ID: 2989

Differential equations: a bibliometric analysis

by João Paulo Davim

Advances in Differential Equations and Control Processes, Vol.32, No.1, 2025;

N/A

Open Access

Editorial

Article ID: 3096

Differential equation-driven intelligent control: Integrating AI, Quantum computing, and adaptive strategies for next-generation industrial automation

by Yue Cheng, Cheng-Li Luo, Chen Zhong, Hong Lin, Dragan Marinkovic, Ji-Huan He

Advances in Differential Equations and Control Processes, Vol.32, No.1, 2025;

The increasing intricacy of industrial systems highlights the inadequacies of conventional control theories in the management of high-dimensional nonlinear dynamics, real-time coupling, and multi-scale modelling. This article introduces a transformative paradigm—differential equation-driven intelligent control—that synergizes artificial intelligence (AI), quantum computing, and adaptive strategies to redefine next-generation industrial automation. The following innovations are at the core of this paradigm: Physics-informed neural networks (PINNs) for solving partial differential equations (PDEs), Quantum-enhanced linear algebra for stochastic differential equation (SDE) optimization, and symbolic regression for automated discovery of fractional-order dynamic models. A case study on flexible robotic arm dynamics demonstrates the tunability of hybrid rigid-flexible systems via fractional-order parameters and adaptive Lyapunov-based control. The concept of Equations as a Service (EaaS) is proposed to democratize access to distributed computational solvers, enabling real-time optimization for applications such as drone swarm coordination and carbon-neutral manufacturing. A number of critical challenges are addressed in this text, including the interpretability of AI (for example, through the use of SHAP-based explainability tools), the reliability of hybrid quantum-classical solvers, and ethical governance frameworks. Through interdisciplinary collaboration, the vision for self-evolving factories by 2030 is outlined—where differential equations autonomously refine parameters using real-time sensor data. Examples include smart grids adapting to renewable energy fluctuations at millisecond scales and robotic assembly lines recalibrating dynamics to mitigate material defects. The overarching objective of this paradigm shift, termed EaaS, is to transition differential equations from their traditional role as static descriptors to that of self-optimizing assets. This transition is expected to lay the foundation for resilient, explainable, and sustainable ecosystems in the era of Industry 5.0.