Open Access

Article

Article ID: 3929

Quantum purity exchange dynamics in a qubit–resonator system subject to squeezed-vacuum driving

by Leila Abdelgader, Chafaa Hamrouni

Advances in Differential Equations and Control Processes, Vol.33, No.2, 2026;

This work presents a theoretical framework to study the non-Markovian dynamics of a two-level quantum emitter interacting with a broadband squeezed electromagnetic reservoir, and both one- and two-photon interaction processes are incorporated. Mathematical modeling uses a time-convolution less projection operator technique. This yields a time-local master equation. The coefficients of this equation are derived from integrals over the reservoir's squeezed correlation functions:  and . The model is validated through rigorous numerical simulation of the resulting dynamical equations. Testing involves computing key physical observables: the transient emission spectrum  and the field linear entropy . These predictions are systematically analyzed against variations in squeezing parameters , coupling strengths , and detector bandwidth . The results confirm that the model successfully captures phase-dependent decoherence, spectral modulation, and purity oscillations. Notably, two-photon processes suppress decoherence under strong squeezing. The consistency between analytical derivations and numerical outcomes validates the framework. It is established as a predictive tool for quantum optics in engineered nonclassical environments. This study directly connects engineered reservoir properties specifically its nonclassical photon statistics to observable, time-dependent quantum phenomena. The findings offer fundamental insights and a predictive tool for quantum control, sensing, and information processing in tailored electromagnetic environments.

Open Access

Article

Article ID: 4147

Effect of heater location and wall waviness on buoyant convection in a porous wavy cavity using heat function approach

by Huey Tyng Cheong, Sivasankaran Sivanandam

Advances in Differential Equations and Control Processes, Vol.33, No.2, 2026;

Natural convection and thermal transport in a porous square cavity with a wavy cold wall and a localized heat source on the left sidewall are numerically examined in this work. The cavity is filled with a fluid-saturated porous medium and is governed by the Darcy model under steady, laminar flow conditions with the Boussinesq approximation. A heater of fixed length is mounted on the left sidewall at three different points, namely the lower, center, and upper positions, while the right sidewall is maintained at a constant cold temperature and modeled with varying waviness in terms of amplitude and number of undulations. The remaining walls are considered adiabatic. The governing dimensionless equations for energy and stream functions are discretized using the finite difference technique and solved iteratively for various heater positions, right sidewall waviness, and Darcy–Rayleigh values after transforming the physical wavy domain into a rectangular computational domain. Results are presented in the form of Nusselt numbers, isotherms, streamlines, and heatlines. The findings indicate that the heater position has a significant influence on the convection flow, and heat transfer performance. The averaged heat transmission rate is improved by the right sidewall’s increased waviness. Among the heater placements, lower heating produces the highest averaged heat transfer for higher Darcy–Rayleigh numbers, whereas center heating is more effective under weak convection conditions. This study provides useful insight into the thermal design of porous systems involving non-uniform heating, such as solar air conditioning, ventilation, and heating systems.

Open Access

Article

Article ID: 4021

Within-host dynamics of ZIKV-CHIKV co-infection: Stability analysis and effective therapeutic strategies

by Ahmed Elaiw, Zainab Alkhudhari , Aatef Hobiny

Advances in Differential Equations and Control Processes, Vol.33, No.2, 2026;

In this paper, we develop a mathematical model that describes the within-host co-dynamics of two arboviruses, Zika virus (ZIKV) and Chikungunya virus (CHIKV). The model is also modified to investigate the impact of various treatment strategies. The model incorporates four cell types: uninfected target cells, latently infected cells, actively infected cells, and antibodies. The analysis establishes that all solutions remain nonnegative and bounded over time. It further reveals the presence of four distinct steady states: the disease-free steady state, the ZIKV-only steady state, the CHIKV-only steady state, and the coexistence steady state representing co-infection. The next-generation matrix technique was applied to determine the reproduction numbers for the ZIKV-only model, the CHIKV-only model, and the ZIKV-CHIKV co-infection model (denoted by R_Z^L, R_C^L and R₀^L = max{R_Z^L, R_C^L }, respectively) as well as the invasion reproduction numbers R_Z^L,^inv and R_C^L,^inv which determine whether a virus can successfully invade an existing infection state. We conducted a mathematical analysis to determine the existence of equilibrium points and to establish the criteria for their global stability. Global stability is verified through the application of suitably constructed Lyapunov functions. The effects of four therapeutic strategies are included: (i) antiviral therapy that prevents viral infection of target cells, (ii) antiviral therapy that suppresses viral production, (iii) immune-stimulating treatment, and (iv) therapy that increases the rate of antibody circulation. Simulations show antivirals outperform immune-boosting strategies in clearing co-infection, while combining both offers synergy by suppressing replication and enhancing host defenses. The proposed model, along with the theoretical analysis, is new and offers a useful framework for studying viral co-infections.  

Open Access

Article

Article ID: 3353

Bielecki–Hyers–Ulam stability of non-linear fractional Volterra Fredholm Hammerstein integro-delay dynamic systems with instantaneous impulses on time scale

by Syed Omar Shah, Jun Zheng

Advances in Differential Equations and Control Processes, Vol.33, No.2, 2026;

In this paper, we study the existence and uniqueness of solutions, Bielecki–Hyers–Ulam stability, and Bielecki–Hyers–Ulam–Rassias stability for non-linear fractional Volterra Fredholm Hammerstein integro-delay dynamic systems with instantaneous impulses on time scale. Such systems provide a unified framework that encompasses both continuous and discrete models, making them highly appropriate for describing complex real-world phenomena involving memory effects, hereditary properties, and sudden perturbations. Sufficient conditions are established for the existence and uniqueness of solutions to the considered systems. In particular, the Picard operator and the Banach fixed point theorem are utilized to prove the existence and uniqueness of solutions. Moreover, we analyze the qualitative behavior of solutions by proving Bielecki–Hyers–Ulam stability and Bielecki–Hyers–Ulam–Rassias stability. To obtain these stability results, Grönwall’s inequality on time scales is used as the main analytical tool. For our results, some suitable assumptions are imposed along with appropriate Lipschitz conditions on the nonlinear terms. By constructing appropriate contractive mappings in a suitably defined Bielecki-type normed space, we develop a unified and systematic framework to handle the combined effects of integral operators, fractional dynamics, delay arguments, and impulsive perturbations. Finally, an illustrative example is provided to demonstrate the effectiveness and applicability of the theoretical findings.

Open Access

Perspective

Article ID: 4007

From linear w-γ correlation to redshift-dependent dynamics: A complete phenomenological framework and testing roadmap for dark energy

by Tongfeng Zhao

Advances in Differential Equations and Control Processes, Vol.33, No.2, 2026;

Based on an adaptive universe model, this perspective article presents a phenomenological framework that correlates the dark energy equation of state w with the cosmic growth index γ via the linear relation w(a) = −1 + η(γ(a) − 0.55). Recognizing that the coupling between dark energy and structure formation may evolve with cosmic time, the framework is extended to a redshift-dependent formulation: w(z) = −1 + η(z)[γ(z) − 0.55] + Δw_bg(z), where η(z) captures the structure-dependent coupling and Δw_bg(z) accounts for intrinsic background evolution. Several physically motivated parameterizations of η(z) are proposed, including continuous forms (smooth transition and oscillatory) and a phenomenological piecewise model aligned with distinct phases of structure formation history. Built upon an interacting dark sector model that strictly conserves energy and momentum, the framework maintains the spacetime geometry of General Relativity while naturally addressing both the Hubble tension (via enhanced late-time expansion) and the S₈ tension (via suppressed structure growth). A hierarchical Bayesian testing roadmap with Fisher forecasts demonstrates that upcoming surveys (DESI, Euclid, Roman) can decisively detect couplings of magnitude |η| ≳ 0.05 at high significance. The framework yields distinctive, testable predictions—including correlated w(z) and fσ₈(z) evolution, a gravitational slip parameter η_slip = 1 that distinguishes it from modified gravity theories, and scale-dependent signatures in the nonlinear regime—providing a comprehensive path to either validate or falsify the hypothesized dark energy–structure growth connection.