Advances in Differential Equations and Control Processes

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

Jiaojiao Wu, Fei He, Shu-fang Li — Fri, 14 Feb 2025 02:05:35 +0000

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.

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

Gargi Trivedi, Ghanshyam Malviya, Jaita Sharma, Vishant Shah — Tue, 04 Mar 2025 08:14:53 +0000

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.

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

Ji-Huan He — Wed, 22 Jan 2025 00:00:00 +0000

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.