Search Results (56)

Weakly nonlinear free-surface flows past disturbances are traditionally modeled using the forced Korteweg–de Vries (fKdV) equation with a prescribed instantaneous pressure field. However, physical wake responses possess finite relaxation times and advection scales that diagnostic algebraic closures fail to capture. This paper introduces a novel coupled system in which the surface pressure is a dynamical field governed by an advection–reaction–diffusion equation driven by band-limited curvature. Using linear spectral theory and numerical validation, we derive a phase-speed criterion demonstrating that energy transfer is determined by the comparison between the pressure drift speed and the surface phase speed. A sharp stability theorem proves that, to leading order in the coupling strength and for a non-negative even response transfer function whose drift speed exceeds the Froude detuning, the system is spectrally stable if and only if the response is band-limited below a critical wavenumber $k_c$. Furthermore, an exact energy identity establishes that passivity and linear stability are equivalent. Finally, we demonstrate resonance steering: while coupling typically increases the wave resistance for monotone spectra, tuning the response to a spectral zero of a multi-lobe footprint reduces the drag significantly relative to its classical value. This result identifies an explicit performance–strongness trade-off, providing a mathematically strong structure for wave drag minimization through dynamic pressure control. Full article

The management of estuarine ecosystem sustainability is a complex problem that requires models that are physically sound, socially meaningful, and interpretable from a mechanistic standpoint. Even though classical AI has demonstrated promise in environmental forecasting, black-box models typically fall short of meeting basic conservation requirements or accounting for anthropogenic stresses that alter water quality. This work introduces a novel framework based on Deep Physics-Informed Neural Networks (Deep PINNs) to predict the dynamics of dissolved oxygen (DO) in the Bouregreg Estuary (Morocco). We advance baseline standards by directly integrating the non-linear advection–diffusion–reaction (ADR) transport equations into the loss function of a deep residual architecture (ResNet with 12–20 layers). This integration ensures that the model takes into account two important aspects of estuarine hydrodynamics: gravitational circulation and the salt wedge effect. The incorporation of a socio–hydro–physical nexus, which uses regional water-pricing indices and urban wastewater discharge volumes from the Rabat-Salé municipal area $(\sim \mathrm{120,000} {\mathrm{m}}^3/\mathrm{d}\mathrm{a}\mathrm{y})$ as proxy variables for anthropogenic pressure, is a unique aspect of this work. The Deep PINN achieves a better coefficient of determination ($R^2=0.998$) and a Nash–Sutcliffe efficiency ($\mathrm{N}\mathrm{S}\mathrm{E}=0.997$), outperforming the traditional ANFIS and ANN baselines by $89.1\%$ in terms of predictive error reduction ($\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=0.041\pm 0.002 \mathrm{m}\mathrm{g}/\mathrm{L}$). In situations where unconstrained data-driven models fall short, the framework exhibits physical robustness in capturing vertical DO stratification in addition to numerical accuracy. Urban effluent volumes have a significant impact on predictive variance, accounting for 28% of the model internal attribution—more than the relative influence of thermal solubility, according to mechanistic feature attribution analysis using SHAP (Shapley Additive exPlanations). Finally, exploratory management scenarios suggest that summer hypoxia could hypothetically be mitigated through a $20\%$ reduction in discharge volumes. This study bridges the gap between scientific modeling and policy implementation by providing a physics-consistent digital twin framework for environmental stewardship in support of UN SDG 6 and Morocco’s National Water Plan. Full article

Accurate network traffic prediction is important for the autonomy, resilience and resource orchestration of 6G and AI-native communication infrastructures, while also supporting green networking and digital twin network applications. However, existing data-driven prediction models face several limitations: over-reliance on massive labeled data, physically implausible predictions, black-box non-interpretability and over-parameterization that impairs edge deployment. To address these issues, this paper proposes a Physics-Informed Network Traffic Prediction (PINTP) framework for graph topology network traffic prediction, which formalizes network traffic evolution as Graph-based Advection–Diffusion–Reaction (ADR) equations and embeds physical regularization into the neural architecture. The framework adopts a hybrid differentiation paradigm unifying automatic differentiation for temporal dynamics and spectral graph theory-derived operators for discrete spatial topologies, and designs a physics-constrained composite loss function with data-driven collocation to balance data fidelity and physical consistency. Experiments are conducted in two complementary settings: a 100-node synthetic random-graph benchmark that evaluates the full graph-topological formulation, and a topology-unavailable real-world telemetry proxy based on Alibaba Cluster Trace v2018 for evaluating sparse-label physics-informed temporal regularization. Comparative analysis with mainstream baselines, including Multilayer Perceptron (MLP), Spatio-Temporal Graph Convolutional Network (STGCN), Graph WaveNet, Transformer, Temporal Convolutional Network (TCN), and XGBoost, shows that the proposed PINTP/PINN implementation achieves a test $R^2$ of 0.898 and MSE of 0.000723 on the 100-node synthetic graph benchmark, close to the strongest Transformer result ($R^2=0.900$, MSE = 0.000710), while using substantially fewer trainable parameters. PINTP/PINN also outperforms Graph WaveNet, STGCN and TCN in this setting, indicating that physics-informed regularization can remain competitive as graph size increases. On the Alibaba proxy task, PINTP/PINN achieves the strongest result among the evaluated models with a test $R^2$ of 0.963. In an independent Alibaba ablation protocol, physical regularization (e.g., $\lambda =10.0$) reduces the mean squared error by 89.15% compared with pure data-driven models and helps mitigate overfitting. This work presents a systematic PINTP framework for graph topology network traffic prediction, achieving competitive prediction accuracy with high parameter efficiency and a degree of physical interpretability. It helps address several limitations of traditional data-driven models, indicates potential for future deployment-oriented studies on real-time network management and resource-constrained edge analytics, and provides an interpretable modeling route for physics-informed network analytics in next-generation communication systems. Full article

Accurate prediction of contaminant transport and self-purification processes in rivers remains challenging because pollutant dispersion, biochemical reactions, and hydrodynamic conditions interact across multiple spatial scales. This study aims to develop and compare mathematical models for soluble contaminant transport and biodegradable organic matter removal in channels and rivers. Unsteady advection–diffusion–reaction equations were formulated for one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) transport scenarios and solved through numerical techniques based on the transformation of partial differential equations into systems of ordinary differential or algebraic equations. In parallel, the classical Streeter–Phelps model and an extended formulation incorporating turbulent diffusion were implemented to evaluate organic load degradation and oxygen deficit dynamics. Simulations were performed using a Matlab R2019a-based computational framework under representative hydraulic and reaction conditions obtained from literature data and empirical correlations. The results showed that, under specific conditions, the 3D model reproduced trends comparable to those predicted by the 2D model, while the latter approached the behavior of the 1D formulation. The Streeter–Phelps model predicted an organic load removal efficiency of 97.74%, a purification index of 1.9564, a critical time of 18.43 h, and a critical distance of 6.93 km. These findings provide a useful framework for river water-quality assessment and support future applications involving complex hydrodynamic and pollutant-loading scenarios. Full article

This paper investigates numerical modeling of transport and diffusion processes of harmful impurities governed by the advection–diffusion–reaction equation, along with the solution of corresponding direct and inverse problems. Particular emphasis is placed on identifying pollution source parameters and reconstructing spatiotemporal concentration distributions from limited and noisy observational data. Classical numerical methods, including stable finite-difference schemes, are employed for solving direct problems. Inverse problems are tackled using modern approaches such as regularization techniques, global optimization, and machine learning methods. In particular, evolutionary optimization algorithms and physics-informed neural networks (PINNs) are considered, enabling the integration of physical laws, observational data, and prior information within a unified computational framework. Computational experiments demonstrate that hybrid approaches combining classical numerical methods with machine learning significantly enhance the accuracy and stability of inverse problem solutions, especially under incomplete or noisy data conditions. Neural network-based methods exhibit strong approximation capabilities and effectively recover unknown model parameters. The results highlight the potential of integrating numerical and intelligent methods for environmental monitoring and pollutant dispersion forecasting, and can be applied in the development of operational analysis and environmental management systems. Full article

This paper is concerned with the numerical solution of the variable-order time fractional advection–diffusion–reaction equation (VO-TFADRE) in two space dimensions. We first propose a Crank–Nicolson (C-N) discretization scheme based on central difference operators and L1 formula for space and time variables, respectively. Then, we apply the C-N scheme to construct a new algorithm, namely the explicit group (EG) method, for the model problem under consideration. The EG method utilizes the idea of small fixed-size groups of mesh points and comes with computational merits as compared with the C-N scheme. Stability and convergence analyses are given in this work. The resulting discretization leads to large sparse linear systems, which are solved using the Bi-CGSTAB iterative method. Numerical experiments demonstrate that both the C–N and EG schemes achieve accurate approximations, while the EG method significantly reduces computational time. To economize further on the computational cost, we propose a parallelized version of the EG method for solving the VO-TFADRE. Carried out numerical simulations reveal that the parallel algorithm is more efficient than the serial algorithm for solving the problem under consideration. Full article

This study focuses on the use of Physics-Informed Neural Networks (PINNs) to solve the 1D Advection–Diffusion–Reaction (ADR) equation. The performance of the PINN model is evaluated in comparison with the classical Crank–Nicolson Finite Difference Method (CNFDM) and validated against analytical solutions to assess improvements in accuracy, robustness, and flexibility. Quantitative analysis reveals that the PINN achieved a high level of accuracy with absolute errors ranging from approximately $2.13\times {10}^{-4}$ to $1.17\times {10}^{-3}$ across the spatial domain. The study utilizes a neural network architecture with two hidden layers of 80 neurons each, optimized through a two-stage training process involving Adam and L-BFGS optimizers. This work contributes to the growing field of physics-informed machine learning by demonstrating the strengths and quantitative reliability of the PINN technique for solving complex partial differential equations in transport phenomena. Full article

We revisit a hyperbolic wildfire model based on reaction–diffusion dynamics with relaxation effects and extend it by incorporating an advection transport term that accounts for wind-driven fire spread. After a planar two-dimensional reformulation and non-dimensionalization of the model, the analysis is restricted to the minimal ignition regime characterized by the presence of a logistic reaction term governing the evolution of the fire-affected tree fraction. The focus of the study is to assess the influence of the effective wind velocity on the propagation dynamics of the fire-affected tree fraction. For this purpose, analytical solutions of the extended wildfire model are derived by applying the Simple Equations Method (SEsM) in its (1,1) variant using a Riccati-type ordinary differential equation as a simple equation. The obtained families of exact solutions describe physically relevant transition fronts connecting fire-unaffected and fully fire-affected states, or vice versa. Numerical simulations of the derived analytical solutions are performed to demonstrate how the internal front thickness and the profile morphology depend on the specific variant of the Riccati-type solution and on the magnitude of the effective wind velocity. A phase-plane stability and bifurcation analysis of the reduced traveling wave system is carried out. Hopf bifurcation thresholds with respect to the effective wind velocity parameter are identified, revealing transitions between monotone front propagation and oscillatory regimes. A regime map is constructed in the parameter plane spanned by the effective wind velocity and the traveling wave speed. This regime diagram delineates regions of qualitatively different propagation behavior, including monotone advancing fronts, possible oscillatory regimes, and regimes in which traveling wave fronts cease to exist. Full article

The present work is focused on a predator–prey model with the Holling type II interaction function, which is influenced by diffusion, advection and nonlinear reaction effects. Firstly, we show that the solutions of this dynamical model are bounded and unique. Secondly we use the Lyapunov function and then show that the equilibrium points are globally stable. Thirdly, we obtain the solution profile when the diffusion coefficient is small. For this purpose we introduce self-similar structures to convert the nonlinear partial differential equations into nonlinear ordinary differential equations and then use the singular perturbation technique to solve these equations. Fourthly, we use the Hamiltonian and Lighthill’s technique to obtain upper stationary solutions for a small coefficient of the advection term. Lastly, we consider a large diffusion coefficient and obtain the asymptotic profiles of nonstationary solutions with the help of nonlinear point scaling. Full article

This paper investigates a slow–fast stochastic reaction–diffusion–advection equation with Hölder-continuous coefficients, where the irregularity of the coefficients presents significant analytical challenges. Our approach fundamentally relies on techniques from Poisson equations in Hilbert spaces, through which we establish optimal strong convergence rates for the approximation of the averaged solution by the slow component. The key advantage that this paper presents is that the coefficients are merely Hölder continuous yet the optimal rate can still be obtained, which is crucial for subsequent central limit theorems and numerical approximations. Full article

Robust epidemiological models are essential for managing COVID-19, especially in diverse urban settings. In this study, we present a fractional advection–diffusion–reaction model to analyze COVID-19 spread in three major Turkish cities: Ankara, Istanbul, and Izmir. The model employs a Caputo-type time-fractional derivative, with its order dynamically determined by the Hurst exponent, capturing the memory effects of disease transmission. A nonlinear reaction term models self-reinforcing viral spread, while a Gaussian forcing term simulates public health interventions with adjustable spatial and temporal parameters. We solve the resulting fractional PDE using an implicit finite difference scheme that ensures numerical stability. Calibration with weekly case data from February 2021 to March 2022 reveals that Ankara has a Hurst exponent of 0.4222, Istanbul 0.1932, and Izmir 0.6085, indicating varied persistence characteristics. Distribution fitting shows that a Weibull model best represents the data for Ankara and Istanbul, whereas a two-component normal mixture suits Izmir. Sensitivity analysis confirms that key parameters, including the fractional order and forcing duration, critically influence outcomes. These findings provide valuable insights for public health policy and urban planning, offering a tailored forecasting tool for epidemic management. Full article

Four formulations based on the Kirchhoff transformation and time linearization for the numerical study of one-dimensional reaction–diffusion equations, whose heat capacity, thermal inertia and reaction rate are only functions of the temperature, are presented. The formulations result in linear, two-point boundary-value problems for the temperature, energy or heat potential, and may be solved by either discretizing the second-order spatial derivative or piecewise analytical integration. In both cases, linear systems of algebraic equations are obtained. The formulation for the temperature is extended to two-dimensional, nonlinear reaction–diffusion equations where the resulting linear two-dimensional operator is factorized into a sequence of one-dimensional ones that may be solved by means of any of the four formulations developed for one-dimensional problems. The multidimensional formulation is applied to a two-dimensional, two-equation system of nonlinearly coupled advection–reaction–diffusion equations, and the effects of the velocity and the parameters that characterize the nonlinear heat capacities and thermal conductivity are studied. It is shown that clockwise-rotating velocity fields result in wave stretching for small vortex radii, and wave deceleration and thickening for counter-clockwise-rotating velocity fields. It is also shown that large-core, clockwise-rotating velocity fields may result in large transient periods, followed by time intervals of apparent little activity which, in turn, are followed by the propagation of long-period waves. Full article

The effects of relaxation, convection, and anisotropy on a two-dimensional, two-equation system of nonlinearly coupled, second-order hyperbolic, advection–reaction–diffusion equations are studied numerically by means of a three-time-level linearized finite difference method. The formulation utilizes a frame-indifferent constitutive equation for the heat and mass diffusion fluxes, taking into account the tensorial character of the thermal diffusivity of heat and mass diffusion. This approach results in a large system of linear algebraic equations at each time level. It is shown that the effects of relaxation are small although they may be noticeable initially if the relaxation times are smaller than the characteristic residence, diffusion, and reaction times. It is also shown that the anisotropy associated with one of the dependent variables does not have an important role in the reaction wave dynamics, whereas the anisotropy of the other dependent variable results in transitions from spiral waves to either large or small curvature reaction fronts. Convection is found to play an important role in the reaction front dynamics depending on the vortex circulation and radius and the anisotropy of the two dependent variables. For clockwise-rotating vortices of large diameter, patterns similar to those observed in planar mixing layers have been found for anisotropic diffusion tensors. Full article

This study introduces the higher-order unconditionally positive finite difference (HUPFD) methods to solve the linear, nonlinear, and system of advection–diffusion–reaction (ADR) equations. The stability and consistency of the developed methods are analyzed, which are necessary and sufficient for the numerical approach to converge to the exact solution. The problem under consideration is of the Cauchy type, and hence, Von Neumann stability analysis is used to analyze the stability of the proposed schemes. The HUPFD’s efficacy and efficiency are investigated by calculating the error, convergence rate, and computing time. For validation purposes, the higher-order unconditionally positive finite difference solutions are compared to analytical calculations. The numerical results demonstrate that the proposed methods produce accurate solutions to solve the advection diffusion reaction equations. The results also show that increasing the order of the unconditionally positive finite difference leads an implicit scheme that is conditionally stable and has a higher order of accuracy with respect to time and space. Full article

In this paper, the enhanced higher-order unconditionally positive finite difference method is developed to solve the linear, non-linear and system advection diffusion reaction equations. Investigation into the effectiveness and efficiency of the proposed method is carried out by calculating the convergence rate, error and computational time. A comparison of the solutions obtained by the enhanced higher-order unconditionally positive finite difference and exact solution is conducted for validation purposes. The numerical results show that the developed method reduced the time taken to solve the linear and non-linear advection diffusion reaction equations as compared to the results obtained by the higher-order unconditionally positive finite difference method. Full article