Vol. 32 No. 3 (2025)

Open Access

Article

Article ID: 3329

Mathematical Modeling of Criminal Activities: An Approach Based on Homotopy Perturbations

by Snežana Stojičić, Vladica Stojanović, Radovan Radovanović, Dušan Joksimović, Mihailo Jovanović

Advances in Differential Equations and Control Processes, Vol.32, No.3, 2025;

 The manuscript deals with solving a mathematical model of criminal activities (MCA model) in order to facilitate the understanding of the dynamics of crime and implement certain measures in its prevention and deterrence. For the MCA model, which consists of two coupled ordinary differential equations of the diffusion type, the stability and existence of a unique Cauchy solution are first examined. Thereafter, coupled equations of the MCA model are transformed into a dimensionless system on the unit interval and solved using the homotopy perturbation method (HPM). In this way, recursive sequences of approximate HPM solutions are obtained, and, under certain conditions, their convergence has been proven. In the following, using different initial conditions, several numerical simulations of the proposed HPM technique in solving the MCA model were conducted, as well as a practical application of the proposed procedure in modelling real-world crime rate data.

Open Access

Article

Article ID: 3050

Preconditioned Performance Trajectory Tracking Control of a Dynamic Adaptive Sliding Mode Observer for Quadrotor UAV

by Qiu Xu, Li Zhang

Advances in Differential Equations and Control Processes, Vol.32, No.3, 2025;

The trajectory tracking accuracy of quadrotor UAVs is greatly challenged by external disturbances and model uncertainties. To overcome these issues, this paper presents a prescribed performance control approach incorporating a dynamic adaptive sliding mode observer. First, external disturbances and model uncertainties are treated as lumped disturbances within the quadrotor UAV system. By designing a sliding mode observer with a nonlinear gain adjustment mechanism, these disturbances can be estimated and compensated for in real time, without requiring prior knowledge of their bounds. Building upon this, the prescribed performance function is further improved, and a new performance function independent of initial conditions is developed to find out that the system states converge within the predefined performance boundaries, achieving precise trajectory tracking. The system’s convergence is rigorously analysed with Lyapunov stability theory. And we can make sure the method using lots of simulations. The results indicate algorithm achieves fast error convergence and precise trajectory tracking, even in the presence of dynamic environmental disturbances.

Open Access

Article

Article ID: 3444

Partial differential analytical expression for the failure rate change of electrical components under multi-fault coupling

by Tiejun Cui, Pengpeng Wei, Shasha Li

Advances in Differential Equations and Control Processes, Vol.32, No.3, 2025;

With the trend of high integration and complex working conditions of electronic equipment, multi-fault coupling failures have become a key threat to operational reliability. To study the influence mechanism of multi-factor-induced multi-fault modes on the failure of electrical components and consider the time-dependent effect of component operation, an analytical expression in the form of a partial differential equation for the component failure rate is established. The time-dependent of failure rate, multi-fault coupling terms, and coupling coefficients in the expression are further determined. The research shows that constructing the partial differential expression for failure rate should consider electromigration, corrosion, hot carrier, and dielectric breakdown faults and their influencing factors. By introducing multi-fault coupling terms, the impacts of parameters such as temperature, current density, etc. on various fault modes and component failure rates are reflected. Electromigration-corrosion, heat-carrier-dielectric breakdown accelerate the occurrence of faults, and the analytical and approximate formulas for coupling coefficients are provided. Case analysis obtains the failure rates of each fault, the component failure rate, and two coupling coefficients; and it is found that the failure rate changes significantly at 100s, serving as a critical life point. This study provides a method for analyzing the failure rate of electrical components under multi-factor influences over time.

(This article belongs to the Special Issue Mathematical Analysis Advances in System Fault Analysis, Prediction and Control (Close))

Open Access

Article

Article ID: 2940

Recent advances in differential equations, control processes, and secure cryptographic networks for medical data exchange

by Chafaa Hamrouni

Advances in Differential Equations and Control Processes, Vol.32, No.3, 2025;

Advances in differential equations and control theory are reshaping how secure, efficient medical data-exchange systems are designed. In parallel, blockchain offers decentralized trust, cryptographic integrity, and auditable access control for healthcare networks. Yet the choice of storage and transmission architecture strongly affects scalability, latency, privacy, and cost. This work investigates how mathematical modeling via differential equations and modern control processes can be coupled with blockchain to strengthen security and interoperability across distributed healthcare systems. We comparatively examine three deployment models: (1) on-chain storage, (2) off-chain, cloud-backed storage with blockchain access control, and (3) local institutional storage integrated with federated learning. On-chain designs maximize transparency and tamper-resistance but incur substantial computation and storage overhead. Off-chain approaches improve scalability while retaining verifiable control through the ledger. Local storage with federated learning safeguards patient privacy by keeping raw data within institutions and sharing only encrypted updates or proofs on chain. Persistent challenges include storage bloat, network delays, heterogeneous regulations, and evolving attack surfaces. To address these issues, we outline optimization strategies grounded in system dynamics stability analysis, resource allocation, and control-oriented tuning to balance throughput, privacy, and reliability. The study synthesizes theoretical insights with implementation considerations, offering a unified perspective on building resilient, performant, and privacy-preserving medical data-exchange frameworks that leverage blockchain under mathematically principled control.

Open Access

Article

Article ID: 2859

Shaping new limits: the future evolution of mathematical models and control strategies

by Chafaa Hamrouni

Advances in Differential Equations and Control Processes, Vol.32, No.3, 2025;

Mathematics underpins scientific advancement and fuels innovation across disciplines. In the next decade, the convergence of differential equations and control theory with emerging technologies, especially artificial intelligence and machine learning will redefine modeling, prediction, and decision-making for complex systems. This integration enhances predictive fidelity, reveals latent structure in high-dimensional data, and accelerates discovery cycles. Recent work in environmental forecasting illustrates these gains, where deep learning has achieved high-accuracy prediction of atmospheric variables and pollutant concentrations. Multi-scale modeling will be central to linking phenomena across spatial and temporal ranges, enabling impactful applications from nanotechnology to fluid dynamics. Parallel progress in fractal geometry offers new tools to analyze, quantify, and optimize intricate structures and dynamics, informing studies of urban growth, heterogeneous media, and chaotic flows. AI-assisted control strategies are poised to transform healthcare, autonomous systems, and communication networks by delivering adaptive, data-driven policies that improve robustness, efficiency, and safety. This review synthesizes the state of the art at the intersection of differential equations, control processes, and AI, surveying methodological advances, benchmark applications, and emerging computational pipelines. It also identifies open challenges including model interpretability, data scarcity across scales, stability guarantees for learning-based controllers, and reproducible evaluation protocols that demand tightly coordinated, interdisciplinary research. By unifying mathematical rigor with AI-driven inference and control, the field is positioned to build more predictive, reliable, and efficient solutions for the next generation of scientific and engineering problems.

Open Access

Article

Article ID: 3151

A parameter-free series solution approach for differential equations in fluid flow

by A. Alameer

Advances in Differential Equations and Control Processes, Vol.32, No.3, 2025;

Viscous incompressible flow across a converging/diverging channel generally known as Jeffery-Hamel flow is an important form of flow in the fluid dynamics sector that exists in a variety of engineering systems, rivers, and in the biological world. This paper proposes a novel Maclaurin Series Method (MSM) to investigate Hamel’s fractal flow pattern in a wedge-shaped region. The fundamental partial differential equations are changed by suitable transformation into the dimensionless non-linear ordinary differential equation. The resulting equation is solved through MSM. The Maclaurin series method obtains the solution of the two-dimensional incompressible viscous flow in the converging / diverging channels according to initial condition. The MSM provides an efficient and accurate alternative to traditional solution techniques. To validate the Maclaurin series method, error analysis of the solution is calculated and presented in tabular form, demonstrating excellent agreement with benchmark results. Furthermore, the MSM solution is plotted for various β values. The comparison between MSM approximate and exact solutions confirms the reliability and effectiveness of the method. Overall, the results indicate that the suggested approach is an effective and reliable tool for solving fluid flow problems.

Open Access

Article

Article ID: 3103

Existence and multiplicity of solutions to N-Laplacian equation with discontinuous exponential growth in ℝN

by Mengyuan Xi

Advances in Differential Equations and Control Processes, Vol.32, No.3, 2025;

This research explores the existence and multiplicity of solutions to N-Laplacian equations with discontinuous exponential nonlinearities in the whole Euclidean space. Through combining symmetric rearrangement techniques and variational methods for non-differentiable functionals, it identifies sufficient conditions for the existence of weak solutions when perturbation parameters are small, and uncovers the rich solution structure caused by discontinuous growth and non-smooth operators.These studies connect critical Sobolev growth and exponential nonlinearities, which is an important link in phase transition models and nonlinear analysis.We have proven the existence and multiplicity of weak solutions for the N-Laplacian equation with discontinuous exponential growth . Notably, when the perturbation parameter is sufficiently small, there exist at least multiple weak solutions, which stem from the interaction between the discontinuous exponential nonlinearity and the N-Laplacian operator. Compared to previous findings, our results extend the existing literature on elliptic equations with critical growth and discontinuous nonlinearities. Additionally, the combination of priori estimates with non-differentiable variational methods constitutes a novel approach, distinct from traditional techniques in earlier studies.

Open Access

Article

Article ID: 3619

Global martingale solutions to a stochastic superquadratic Shigesada-Kawasaki-Teramoto type population system

by Xi Lin

Advances in Differential Equations and Control Processes, Vol.32, No.3, 2025;

For a stochastic cross-diffusion population system with superquadratic transition rate, we show that a global martingale solution exists. The existence of a global martingale solution proof for a stochastic population system is quite different from the existence of a weak solution proof for a deterministic population system. For deterministic population systems, we apply the entropy method to show a weak solution exists. For stochastic population systems, we rely on the Galerkin approximation scheme to derive the sequence of approximated solutions. We apply the Itbo formula to derive uniform estimates. After the tightness property be proved based on the estimation, a space changing result then be used to confirm the limit is a martingale solution of the cross-diffusion system. In the uniform estimation process, we notice that we have to estimate stochastic processes that are not in the finite dimensional space, otherwise we can not derive strong enough estimation for the tightness proof. We are not able to apply the Itbo formula to stochastic processes that are not finitely dimensional processes. In this situation, we have to introduce an auxiliary sequence. The estimation of the approximated sequence has to be derived on the estimation of an auxiliary sequence, which is the key idea of this work. The nonnegative property for approximated solutions has been shown by the standard Stampacchia method.

Open Access

Article

Article ID: 3604

Application of intelligent control to aircraft landing system

by Teng-Chieh Yang, Jih-Gau Juang

Advances in Differential Equations and Control Processes, Vol.32, No.3, 2025;

Conventional automatic landing systems (ALS) primarily utilize proportional integral-derivative (PID) controllers in combination with gain-scheduling techniques. Their designs are generally constrained to the specific flight envelopes established by Federal Aviation Administration regulations. However, when environmental disturbances exceed these operational boundaries, the ALS may be deactivated. Consequently, developing a more intelligent ALS is crucial for maintaining safety across a broader spectrum of turbulent conditions. This paper proposes an intelligent control method that integrates the Cerebellar Model Articulation Controller (CMAC) with a fuzzy logic system for the development of an advanced aircraft automatic landing system. Multiple fuzzy modules are embedded within the CMAC structure: Type-1 fuzzy CMAC, adaptive Type-1 fuzzy CMAC, Type-2 fuzzy CMAC, and adaptive Type-2 fuzzy CMAC. Flight control principles are incorporated into its design. The Lyapunov stability theory is applied to ensure system stability, and adaptive learning rules are established to maintain this stability. Simulation results verify that the proposed controller can accurately track the desired landing trajectory and effectively adapt to various environmental conditions. Therefore, even under turbulent conditions, the adaptive fuzzy CMAC achieves reliable aircraft guidance and landing performance.

Open Access

Review

Article ID: 3728

Stochastic Differential Equation Model for Detecting Digital Currency Market Manipulation: A Systematic Review

by Lei Shen, Hanqiao Tang

Advances in Differential Equations and Control Processes, Vol.32, No.3, 2025;

Digital currency markets exhibit extreme volatility, heavy tails, and bursty order-flow that complicate surveillance and heighten manipulation risk. This systematic review synthesizes the state of the art in stochastic differential equation (SDE)–based approaches for detection and monitoring of abusive practices in cryptocurrencies. Following PRISMA 2020 and PICOS, an Elsevier-indexed search (2015–2025) yielded 273 records; after screening and eligibility assessment, 20 primary studies (2018–2025) were included. The literature clusters into six methodological families: (i) stochastic volatility and jump–diffusion models for heavy-tailed returns and volatility smiles; (ii) Hawkes and related point-process models for clustered order arrivals, spoofing, pump-and-dump, and wash trading; (iii) Markov and regime-switching diffusions that delineate latent “fair” versus manipulated regimes; (iv) hybrid SDE–machine learning frameworks for high-frequency prediction; (v) change-point and sequential detection methods grounded in likelihood ratios and optimal stopping; and (vi) meta-studies consolidating performance trends. Across studies, Hawkes-type intensities consistently outperform Poisson and threshold baselines in event detection; regime-switching models align with known market breaks; and hybrid neural–SDE systems achieve the strongest forecasting but at reduced interpretability. We formalize a taxonomy linking model structure to surveillance objective, and we delineate a practical trade-off between transparency (classical SDEs) and predictive accuracy (hybrid models). The review highlights open needs in explainable hybrid designs, reproducible datasets, and real-time deploy ability for exchanges and regulators. By connecting applied probability, financial engineering, and control, the paper clarifies how SDE frameworks can underpin robust, auditable market-integrity tools for digital assets.