Vol. 33 No. 1 (2026)

Open Access

Articles

Article ID: 3815

Unraveling structural equation modeling: Key assumptions, model fit, and trends

by Jack Ng Kok Wah

Advances in Differential Equations and Control Processes, Vol.33, No.1, 2026;

Structural equation modeling (SEM) serves as a cornerstone analytical tool across disciplines, enabling robust tests of complex relationships. Yet, core assumptions, model fit, measurement invariance, missing data handling, and causal inference validity spark ongoing debates. Despite methodological progress, challenges linger in parameter estimate reliability, sensitivity to model changes, and integration with alternative approaches. This study critically synthesizes recent empirical and theoretical insights to scrutinize these assumptions. A review of contemporary studies spotlights trends like refined fit evaluation, SEM-fsQCA synergies, and machine learning incorporation. Findings expose inconsistencies in missing data treatment and model respecification, varying by discipline. Quantitative focus sharpens model fit indices, while qualitative views stress theoretical justification hurdles. Cross-disciplinary analysis (psychology, finance, education, healthcare, marketing) reveals uneven assumption adherence, questioning generalizability, especially for cross-cultural data and ordinal variables. Hybrid integrations, such as SEM with system dynamics or network analysis, boost predictive accuracy and curb violations. SEM endures as powerful but demands nuanced assumption testing for theoretical and empirical soundness. Implications urge interdisciplinary collaboration on validation. Limitations encompass publication bias and the omission of unpublished advances. Future work should probe alternative fit techniques, violation impacts, and AI-driven diagnostics, fostering reliable, replicable SEM applications.

Open Access

Articles

Article ID: 3725

Unified framework of four Caputo fractional differences for initial and final value problems in discrete fractional calculus with variable bounds

by Xiaomin Li, Huaigu Tian, Peijun Zhang, Jun Ma

Advances in Differential Equations and Control Processes, Vol.33, No.1, 2026;

Fractional calculus has emerged as a powerful tool for characterizing non-classical dynamic phenomena, yet its discretization remains fragmented, with existing studies primarily focusing on single combinations of variable bounds and difference directions. To address this gap, this paper proposes a unified theoretical framework for discrete fractional calculus by systematically introducing four novel Caputo fractional difference definitions, which integrate variable upper/lower-limit sums with forward/backward difference operations. First, we rigorously derive the fundamental properties of these four definitions, including the commutativity of fractional sums and differences, and their consistency with integer-order difference operations. Second, we construct fractional difference equations for each definition, establish their equivalence to Volterra sum equations, and provide explicit solutions and strict proofs for their corresponding initial and final value problems. To validate the theoretical results, we design four targeted computational cases and numerical simulations, confirm the consistency between theoretical solutions and numerical results, and intuitively demonstrate the long-memory effect of fractional-order discrete systems. Furthermore, we present a concise comparison of the four definitions, clarifying their suitability for discrete systems with distinct boundary conditions and dynamic characteristics. This work not only completes the theoretical system of discrete fractional calculus with variable bounds but also provides standardized and targeted mathematical tools for modeling complex discrete dynamic processes, laying a solid foundation for the practical application of discrete fractional calculus in fields such as engineering control, infectious disease modeling, and economic dynamics.

Open Access

Articles

Article ID: 3774

Data-driven hierarchical decision support for civil aviation maintenance safety risk: A fusion of Bayesian network and system dynamics

by Jirong Duan, Ming Cheng

Advances in Differential Equations and Control Processes, Vol.33, No.1, 2026;

To enhance the scientificity and precision of risk analysis and management decision-making in aircraft maintenance operations, this study proposes a risk analysis-decision model tailored to maintenance events. Based on actual civil aviation maintenance scenarios, the model employs real data to conduct data-driven analysis and precisely calculates the occurrence probabilities of various risk factors by constructing a Bayesian risk probability network. Meanwhile, it selects three categories of key risk factors: personnel (A), management (B), and organization (C), to build a system dynamics scenario, thereby simulating the long-term implementation effects of different management strategies. The research findings indicate that the existing maintenance management system demonstrates a certain level of risk buffering efficacy under normal operating conditions, effectively preventing risks from evolving into higher severity levels. The combinations of key risk factors at different severity levels exhibit a hierarchical characteristic, specifically manifesting as three tiers dominated by organization and safety barriers, personnel capabilities and behaviors, and daily operations and slow-variable risks, respectively. It is proposed that maintenance safety risk governance should adopt a graded and differentiated management strategy. At the decision-making level, the model is capable of simulating the long-term impacts of different management strategies. The study reveals that increasing management investment can significantly reduce process risks, whereas systemic risks and frontline operational errors require sustained, long-term resource allocation for improvement.

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

Open Access

Articles

Article ID: 3861

A multi-stage decision-making model for urban fire emergency with multi-granularity uncertain linguistic information and prospect theory

by Xuemei Zhou, Nady Slam

Advances in Differential Equations and Control Processes, Vol.33, No.1, 2026;

Existing fire emergency decision-making models often struggle with accurately handling multi-granularity uncertain linguistic information, loss aversion, and a lack of adaptability to dynamic fire evolution. To address these gaps, this study adopts two-tuple linguistic representation (2TLR) for quantifying multi-granularity linguistic information and combines the Analytic Hierarchy Process (AHP) with the entropy weight method (EWM) to determine the ability weights of the experts. Furthermore, a six-dimensional dynamic reference point is generated via the random forest algorithm, and the integration of prospect theory (PT) with a sequential decision-making framework (SDF) is implemented for the dynamic optimization of response plans. Validation through real-world cases demonstrates that the proposed Multi-stage Prospect Selection (M-PS) model outperforms both the TOPSIS method and the single PT model, compared with these two methods, the proposed M-PS model can effectively prioritize the avoidance of high-risk scenarios, accurately reflect decision-makers’ loss aversion tendency, and realize dynamic decision-making through updating the decision plan sequentially, thereby providing reliable support for urban fire emergency management. At the same time, in this study we conduct a comparative analysis of core metrics between existing methods and the proposed M-PS model. The evaluation across five dimensions demonstrates that the proposed M-PS model delivers superior performance.

Open Access

Article

Article ID: 3719

Stability of totally-positive switched linear systems with mode-dependent average dwell time switching

by Yanping Guo, Yijing Li, Lei Tai, Qiang Yu

Advances in Differential Equations and Control Processes, Vol.33, No.1, 2026;

Totally-positive switched linear systems (TSLSs), as one of the special switched system classes, have both the complex dynamic behavior of switched systems and their special dynamic properties of totally positive dynamical systems. Recently, TSLSs have attracted scientists’ extensive attention, due to their wide applications, such as economics, biology, communication, and electronic information engineering. The research focuses on the stability issue of TSLSs. Several new exponential stability criteria of TSLSs in both continuous-time and discrete-time cases are obtained by combining the strategy of mode-dependent average dwell time (MDADT) and the multiple linear co-positive Lyapunov function approach. Those stability criteria obtained are presented in the form of linear constraints, making them easy to verify and apply through tools such as linear programming (LP). Since the MDADT framework only limits the average dwell time (ADT) of each subsystem and does not impose restrictions on the switching order or subsystem activation frequency, the conclusion of this paper is robust for switching sequences. The corresponding ADT stability criteria have also been inferred. Furthermore, it is pointed out that the stability issue under arbitrary switching can be solved by the common linear co-positive Lyapunov function (CLCLF) method. Finally, the efficiency of the results is verified by two numerical examples. One of them is from the epidemiological models, which provides a practically motivated TSLSs to make the validation more convincing.

Open Access

Article

Article ID: 3821

Adaptive Enriched Rational Spectral Methods with sinh transformations and asymptotic correctors for variable-coefficient singular perturbation problems

by Lufeng Yang

Advances in Differential Equations and Control Processes, Vol.33, No.1, 2026;

This paper introduces and analyzes novel Enriched Rational Spectral Methods for efficiently solving singular perturbation problems exhibiting sharp boundary layers. While spectral methods are known for their ‘spectral accuracy’ in solving smooth problems, their performance deteriorates for stiff differential equations because they fail to resolve rapid transitions in the solution. To overcome this limitation, we propose a rational spectral collocation framework enriched with asymptotic corrector functions. These correctors are derived directly from a boundary layer analysis of the variable-coefficient operator itself, enabling them to accurately capture the solution's singular behavior. Two specific schemes are proposed: the Enriched Spectral Method (ESM) and the Enriched Rational Spectral Method combined with a sinh transformation (ERSM-sinh). In ERSM-sinh, the corrector functions are integrated with a sinh transformation whose parameters—layer location and width—are determined from asymptotic estimates. The correction parameters are obtained implicitly by solving the discrete algebraic system arising from the original problem. Extensive numerical experiments on convection-diffusion and reaction-diffusion problems with variable coefficients demonstrate the superior performance of our methods. Results show that ERSM-sinh maintains robust spectral accuracy, significantly outperforms existing approaches such as RSC-SSM and RSCAT for variable-coefficient problems, and achieves high precision with minimal computational cost—even for very small perturbation parameters (e.g., ε = 10⁻¹⁰). This work provides a high-resolution, efficient, and generalizable framework for singularly perturbed boundary value problems.

Open Access

Article

Article ID: 3811

A unified multi-physics model for co-design: Enhancing efficiency and enabling compact thermal management in vanadium redox flow battery stacks

by Jacer Hamrouni, Leila Abdelgader, Chafaa Hamrouni, Abdennaceur Kachouri Kachouri, Mounir Baccar

Advances in Differential Equations and Control Processes, Vol.33, No.1, 2026;

This work develops a control-oriented, lumped-parameter model for vanadium redox flow battery (VRFB) stacks. The framework integrates mass, charge, energy, and momentum transport with electrochemical kinetics via a coupled system of ordinary differential equations (ODEs) and algebraic constraints, bridging system dynamics and electrochemical engineering. A key methodological advancement is the application of a hydraulic-electrical network analogy, utilizing Kirchhoff's laws to simulate electrolyte flow and shunt current pathways across a 20-cell stack, thereby transforming complex three-dimensional physics into a tractable, control-oriented formulation. The model directly links physical fidelity to actionable performance insights. Simulations identify that non-uniform flow distribution induces significant local state-of-charge gradients, exacerbating shunt currents. This parasitic effect can reduce effective charging current by up to 2.1% and increase discharge overpotentials. Through analysis of these coupled interactions, the study demonstrates that optimized flow management and thermal control can mitigate losses. Specifically, regulating stack temperature below 40 °C via a novel targeted tank-cooling strategy rather than full-system cooling prevents vanadium precipitation while improving round-trip efficiency, achieving a 27.2% reduction in cooling energy consumption. Furthermore, the model reveals that tank-based heat rejection dominates convective heat transfer (85.8%), enabling a transformative redesign where thermal management is consolidated at the tanks. This permits a more compact stack enclosure and reduces balance-of-plant complexity. The work establishes a validated mathematical framework that advances the fundamental understanding of coupled transport in VRFBs and provides a direct pathway to designing more efficient, compact, and cost-effective systems.

Open Access

Article

Article ID: 3793

Hybrid calculation-estimation modeling for flow field optimization: Enhancing efficiency of biomimetic vanadium redox flow batteries

by Jacer Hamrouni, Kabashi Khatir Kabashi, Chafaa Hamrouni, Abdennaceur Kachouri, Mounir  Baccar

Advances in Differential Equations and Control Processes, Vol.33, No.1, 2026;

This study introduces a hybrid calculation-estimation framework to optimize flow field designs for vanadium redox flow batteries (VRFBs), prioritizing directly calculable geometric and physical parameters over empirically fitted coefficients to enhance model fidelity and predictive accuracy. A fully validated three-dimensional multi-physics model, coupling fluid dynamics with electrochemical kinetics, is developed to systematically evaluate three distinct flow field architectures: a conventional serpentine design, a nature-inspired biomimetic leaf-venation network, and a modified serpentine channel featuring embedded micro-pillar perturbators. Comparative analysis reveals that biomimetic design achieves the most favorable trade-off between hydraulic and electrochemical performance. Its low-resistance, hierarchically branched architecture facilitates uniform electrolyte distribution across the porous electrode, resulting in a 35% reduction in pressure drop and a corresponding 3.2% increase in net system efficiency relative to the conventional baseline. In contrast, while the perturbator-enhanced design achieves the highest limiting current density (190 mA cm⁻²) by inducing localized vortex mixing to enhance mass transport, this gain is offset by a significant increase in pumping losses. The findings underscore that directly calculated parameters such as branching geometry and flow path length are critical drivers of performance. This work provides a principled modeling strategy and offers generalizable design guidelines, demonstrating that nature-inspired engineering is a key pathway toward developing next-generation, high-efficiency VRFB systems.

Open Access

Article

Article ID: 3584

Physics-informed neural networks for solving steady-state heat conduction inverse problems

by Tangwei Liu, Qiong Zou, Xiaoqing Ruan, Dingding Yan, Jeevan Kafle, Zhongzhou Lan

Advances in Differential Equations and Control Processes, Vol.33, No.1, 2026;

This paper investigates numerical methods for a class of three-dimensional steady-state heat conduction inverse problems. By employing Physics-Informed Neural Networks (PINNs), the three-dimensional heat conduction inverse problems are reformulated as optimization problems with respect to a properly defined loss function. Two cases with different additional conditions are considered: one incorporates an additional boundary temperature gradient condition, while the other involves an additional partial internal temperature measurement. Corresponding efficient algorithms are developed to solve the resulting optimization problems. To optimize the performance of the proposed numerical framework, systematic sensitivity analyses are performed to rigorously justify the selection of key hyperparameters (e.g., activation functions and network architecture). Additionally, to validate the mathematical effectiveness and noise robustness of the algorithm, this study primarily employs synthetic data with controlled noise levels for quantitative evaluation. Numerical results demonstrate that the proposed method can efficiently and accurately approximate the solutions to the three-dimensional (3D) heat conduction inverse problems for both polynomial and non-polynomial cases. In addition, a theoretical analysis is provided to interpret the method's stability against noise amplification. Future work will focus on applying the proposed framework to real-world field data to further validate its practical engineering value.

Open Access

Article

Article ID: 3876

Existence of periodic solutions for a nonlinear plate coupling system with thermal memory and external forces

by Xia Li, Jiang-Long Shen, Hang-Jing Xiong, Run-Fa Zhang

Advances in Differential Equations and Control Processes, Vol.33, No.1, 2026;

This paper is devoted to investigating the existence and uniqueness of T-periodic solutions for a nonlinear thermoelastic plate coupling system with thermal memory effects and time-periodic external forces, derived from the non-Fourier heat flux law—a model more physically realistic for characterizing the thermal response of materials with transient heat conduction behavior. To address the mathematical challenges of this coupled system, we first transform the original high-order system into an equivalent first-order evolution system via auxiliary and memory variable substitutions. Using the Galerkin method to construct finite-dimensional approximate solutions, we then apply the Leray-Schauder fixed-point theorem to prove the existence of approximate periodic solutions, deriving uniform a priori estimates for their derivatives in Hilbert space V 0 via Hölder’s, Poincaré’s, and Gronwall’s inequalities. The Sobolev compact embedding theorem verifies the convergence of approximate solutions, establishing the existence of T-periodic solutions for the original system; uniqueness is further proven via an energy difference functional and Gronwall’s lemma under a smallness condition on external forces. This work enriches the theoretical framework for periodic solutions of memory-type thermoelastic coupling systems and provides a foundation for engineering dynamic analysis of plate structures.

Open Access

Article

Article ID: 3873

Time-optimal control with bang-bang property for strongly coupled nonlinear microwave heating systems

by Dongsheng Luo, Lianying Zhang, Peiyong Zhang, Ailiang Zhao

Advances in Differential Equations and Control Processes, Vol.33, No.1, 2026;

This paper makes a rigorous mathematical analysis of the time-optimal control problem for a nonlinear microwave heating system described by coupled partial differential equations. The work extends the established theory for linear models to a physically realistic nonlinear regime, where magnetic field permeability exhibits a nonlinear dependence on the material’s evolving temperature field. The first result establishes the exact controllability of this nonlinear distributed parameter system. This is achieved by applying the Kakutani Fixed-Point Theorem to an appropriately defined solution operator, proving that the system state can be driven from any admissible initial temperature distribution to a specified target profile within a finite time horizon using suitable control inputs. Leveraging this controllability foundation and employing crucial a priori energy estimates, we subsequently prove the existence of at least one time-optimal control via minimizing sequences and weak compactness arguments. The central contribution is the rigorous analytic proof of the bang-bang property for these time-optimal controls. This structural property is demonstrated by contradiction, using a pivotal quantitative relation—derived from the controllability analysis—that links the minimum achievable control time to the L 2 -norm of the control force. The proof conclusively shows that any time-optimal control must saturate the prescribed control constraints almost everywhere in the time-space domain, taking values only at the extremes of the admissible set. These results lay a firm theoretical foundation for optimal control protocol design in nonlinear microwave heating, confirming that efficient strategies are inherently of switching type and offering a benchmark for future numerical and experimental work as well.

Open Access

Article

Article ID: 3902

A multimodal deep learning-based dynamic prediction model for colorectal cancer liver metastasis

by Haitao Zheng, Dehui Wen, Liwei Zhang, Haiyong Lu, Xiaoyu Li, Yongxin Li

Advances in Differential Equations and Control Processes, Vol.33, No.1, 2026;

Colorectal cancer liver metastasis (CRLM) remains a major determinant of long-term outcomes. Existing clinical models are typically static and single-modality, limiting early warning and individualized follow-up. We prospectively enrolled 300 treatment-naïve colorectal cancer patients. We collected preoperative three-phase contrast-enhanced ultrasound (CEUS) dynamic sequences, longitudinal serum marker measurements (EZH2/CD10) from preoperation through 12 months, and 35 clinical–pathological variables. The proposed Dynamic Modality Alignment Network (DMA-Net) includes (i) an imaging encoder based on an enhanced 3D-ResNet18 to extract perfusion kinetics, (ii) a molecular encoder using BiLSTM with temporal attention to model serial biomarkers, and (iii) a clinical encoder (MLP) for structured variables. A dynamic alignment module and cross-modal attention fuse modalities, followed by a discrete-time survival head that outputs month-specific conditional hazards and cumulative risks. On the held-out test set, the tri-modal model achieved an area under the curve (AUC) of 0.918 at 12 months with favorable calibration (Brier score 0.123), outperforming a traditional Cox model built from clinical variables (AUC 0.782, Brier score 0.177). Time-dependent evaluation showed stable AUCs from 3 to 12 months (0.904–0.919). Ablation experiments indicated that imaging and molecular branches contributed most to discrimination, whereas clinical variables improved calibration. Multimodal dynamic modeling integrating CEUS perfusion, longitudinal biomarkers, and clinical variables improves early warning and risk stratification for CRLM, and provides a practical framework to support personalized surveillance.

Open Access

Article

Article ID: 3966

SI/SIS/SIR models for malware propagation in P2P networks: Numerical analysis and perspectives for fractional-order extensions

by Dušan Džamić, Aleksa Marković

Advances in Differential Equations and Control Processes, Vol.33, No.1, 2026;

In this paper, we examined the suitability of epidemic models on networks to assess their potential application for detecting malware propagation patterns in peer-to-peer (P2P) computer networks. We analyzed how the Susceptible-Infected (SI), Susceptible-Infected-Susceptible (SIS), and Susceptible-Infected-Recovered (SIR) models, which were originally developed for biological viruses, can be applied to digital viruses. Using the Gnutella network dataset as a representative topology of P2P networks, we simulated infection scenarios to evaluate how scale-free network properties and the presence of high-degree nodes acting as super-spreaders influence the propagation speed and network saturation. The obtained results show that the examined models can be used and provide valuable insight into epidemic dynamics. However, the existing models are not perfect, and the introduction of additional states, such as L for latency and Q for quarantine, is proposed, since these are relevant for digital devices and digital viruses. More precisely, the absence of latent (L) and quarantine (Q) components leads to an overestimation of infection speed and an inability to model strategic isolation. Accordingly, this study provides empirical evidence that standard biological models are not sufficient for accurate predictions in the field of cybersecurity in P2P environments, and that future modeling efforts should move from basic compartmental models toward more advanced frameworks, such as SEIR and SIQR, to realistically capture malware activation delays and the impact of active defense strategies.

Open Access

Article

Article ID: 3938

A modified ancient Babylonian algorithm for nonlinear oscillators

by Kaipeng Xu, Fang Yang, Xingwei Zhou

Advances in Differential Equations and Control Processes, Vol.33, No.1, 2026;

This paper focuses on the frequency-amplitude relationship of nonlinear oscillators and proposes an improved ancient Babylonian algorithm. This algorithm constructs a solution framework based on linear and nonlinear operators in a unique iterative form, and cleverly selects the initial guess value and determines the frequency equation. Through in-depth exploration of several representative nonlinear oscillator examples (covering different forms of nonlinear terms and parameter settings), it fully demonstrates its specific operation and effectiveness verification process in the solution process. The results show that this algorithm performs well in weakly nonlinear oscillator problems, and the obtained results are highly consistent with existing methods or exact solutions. Moreover, it is equivalent to He's frequency formula under specific conditions, strongly supporting the effectiveness of the latter. At the same time, it clearly reveals the influence of the law of the nonlinear term coefficient and amplitude on the accuracy of the algorithm. However, in the case of strongly nonlinear systems, the algorithm has certain limitations. This study combines ancient numerical wisdom with modern nonlinear dynamics, providing a computationally simple and effective tool for oscillator engineering, while also indicating directions for improvement to enhance strong nonlinear performance.

Open Access

Article

Article ID: 3910

Reinforcement learning from demonstration for robust control of superconducting qubits: Decoherence suppression via environmental engineering

by Chafaa Hamrouni, Leila Abdelgader

Advances in Differential Equations and Control Processes, Vol.33, No.1, 2026;

This work develops a control-oriented theoretical framework for the manipulation of transient emission spectra and field-state purity in a qubit interacting with a broadband squeezed reservoir. We model the joint qubit-field dynamics using a time-resolved open quantum system formalism. This establishes a direct, quantitative connection between transient spectral characteristics, quantum purity evolution, and controllable system parameters. The analysis reveals that nonclassical reservoir properties, particularly squeezing-induced correlations, strongly influence both spectral deformation and coherence redistribution during the early-time dynamics. Building on this foundation, the work explores practical control strategies aimed at engineering the system’s dynamical response. Pulse shaping and time-dependent modulation of qubit-reservoir coupling offer effective control tools. These regulate spectral broadening, suppress unwanted decoherence, and accelerate convergence toward stabilized operating regimes. In addition, feedback-assisted qubit probing protocols are investigated as a means of monitoring and controlling field purity while minimizing measurement back-action on the squeezed mode. Our results show that well-designed control loops can balance information extraction and disturbance. This enables efficient regulation of nonclassical field properties. The proposed approach highlights transient spectral measurements as a powerful diagnostic and control resource, linking observable emission features to underlying quantum correlations and purity dynamics. From an engineering perspective, the framework offers practical guidelines. These guide the design of feedback-stabilized quantum emitters and reservoir-engineered coherence control. These findings are directly relevant to emerging quantum technologies, including cavity and circuit quantum electrodynamics, quantum sensing, and quantum communication systems operating in nonclassical environments.