This study presents an extended numerical investigation of the unsteady magnetohydrodynamic (MHD) flow of a Jeffrey nanofluid containing motile microorganisms past a vertically stretching cylinder. The model incorporates nonlinear Forchheimer drag to simulate non-Darcian porous resistance, variable thermal conductivity and viscosity for realistic material behavior, and the Dufour—Soret effects to account for double-diffusive convection. Buongiorno’s two-component model is used to characterize nanoparticle dynamics, including Brownian motion and thermophoresis. The nonlinear partial differential equations system receives solution through an implicit finite difference method which achieves second-order spatial accuracy. The simulation investigates how different parameters including Hartmann number and Deborah number and Forchheimer number and Eckert number and thermophoresis and Brownian motion and bioconvection-related parameters affect the velocity and temperature and concentration and microorganism distributions. The research findings provide essential knowledge about how to enhance transport and control flows in systems used for thermal management and bio-reactors and porous media processing.

Objective.Convection-reaction equation based magnetic resonance electrical properties imaging (cr-MREPT), especially in phase-based form, is one of the most commonly used MREPT methods since it can successfully overcome internal boundary artifacts, does not rely on transceive phase approximation and is relatively easy to implement. While the partial differential equation system solved in this method introduces regularization compared to Helmholtz MREPT, it also increases the solution time especially when the method is applied in large 3D volumes and reduces the practicality of the applications.Approach.We divide the large 3D volume of interest into smaller regions (local ROIs) for which the solutions can be obtained faster, we then parallelize the solutions of the individual regions and finally combine the results to obtain the whole conductivity distribution. Sensitivities of the reconstructed conductivity at a certain voxel, to theB1phase data of nearby voxels and to the Dirichlet Boundary condition imposed at the boundary of the solution region are investigated to determine the optimum size for the small regions.Main results.Conductivity distributions for various phantoms are successfully reconstructed with significantly reduced computational times, and up to approximately 100 times acceleration of the solution is achieved using a 72-core server. This can be further increased with a high-performance computer, where theoretical limit is the solution time for a single local ROI. Significance.Multislice reconstruction of large phantoms, such as a human head, is possible within a reasonable solution time with the proposed method. These findings enhance the potential for the practical applications of phase-based cr-MREPT.

This paper develops a stochastic optimal control framework for modeling age-structured population dynamics under climate-induced migration, with application to Nigeria’s Niger Delta region. Climate-related slow-onset and extreme hazards, including flooding, sea-level rise, and environmental degradation, drive internal displacement that disproportionately affects younger working-age groups and intensifies urban demographic pressure and infrastructure strain. The proposed model extends the deterministic McKendrick–von Foerster equation into a stochastic partial integro-differential system by incorporating a climate-sensitive migration kernel with multiplicative Wiener noise to represent persistent uncertainty and optional Lévy jumps to capture abrupt extreme events. Policy interventions, including relocation incentives, infrastructure capacity enhancements, and adaptive zoning, are formulated as controls to minimize an expected long-term cost functional that penalizes demographic imbalances, intervention effort, and migration-related disruptions. Optimality conditions are derived from an adapted stochastic Pontryagin maximum principle in infinite-dimensional spaces, resulting in a forward–backward stochastic partial differential equation system. The well-posedness of the state dynamics is proven using semigroup theory and fixed-point methods, the existence of optimal controls is established through compactness and continuity arguments, and long-term ergodic behavior under persistent noise is analyzed using Lyapunov functionals. Numerical solutions combine finite-difference discretization of the age variable, Euler–Maruyama time-stepping, and Monte Carlo integration for stochastic terms, with convergence demonstrated under Lipschitz and stability assumptions. A case study in Rivers State, centered on Port Harcourt and involving an estimated population of approximately 7 million, is calibrated using UN World Population Prospects age distributions, World Bank Groundswell Africa internal climate migration projections, and regional flood probability estimates. Simulations indicate that stochastic optimal policies reduce expected urban demographic overload variance by 20–35% relative to deterministic baselines under representative flood scenarios, while promoting more balanced age structures and supporting resilient urban planning. The study contributes to environmetrics by advancing uncertainty quantification for climate-induced migration modeling and provides a reproducible Python-based decision-support framework for evidence-based policy in climate-vulnerable coastal developing regions.

ABSTRACT In this work, the unsteady heat and mass transport properties of a Casson fluid passing over an oscillating vertical cylinder embedded in a Darcy–Forchheimer porous medium are examined. The growing industrial application of oscillatory cylindrical systems in drilling operations, increased oil recovery, biochemical reactors, and polymer processing, where non‐Newtonian fluids interact with porous materials under periodic motion is the motivation behind this work. In order to effectively represent transport phenomena found in petroleum reservoirs, chemical mixing towers, food processing facilities, and heat exchange devices, this model integrates Soret and Dufour effects, viscous dissipation, chemical reaction, and heat generation/absorption. A dimensional partial differential system of equations is created by formulating the governing equations of momentum, energy, and concentration. The suitable transformations are then applied in the governing model to obtain the dimensionless form in terms of partial differential equations. To solve the equations numerically, a reliable and effective Crank‐Nicolson finite difference technique is implemented. The understanding of how to regulate heat and mass flow in porous geometries is made easier by this work. The effects of significant parameters on velocity, temperature, and concentration are investigated numerically and graphically.

Mean-square exponential stability and stabilization for linear MIMO stochastic parabolic partial difference systems with time delay

ABSTRACT In this paper, we investigate the existence of global nontrivial weak solutions to certain partial differential systems defined in the exterior domain of hyperbolic space. We establish sufficient conditions for the nonexistence of such global solutions, providing insights into the behavior of these solutions in the context of hyperbolic geometry. Our results highlight the challenges in the existence of global solutions in non‐Euclidean settings.

The Calogero–Bogoyavlenskii–Schiff (CBS) equation serves as a cornerstone for nonlinear integrable partial differential systems that govern complex wave phenomena. This study investigates the [Formula: see text]-dimensional CBS-negative-order CBS (CBS–nCBS) model, unveiling novel soliton solutions. Using the bilinear neural network method, a bilinear representation of the model is derived, and multi-soliton solutions are identified through various neural network architectures. The bilinear neural framework employs single-layer architectures (4–3–1, 4–4–1) combined with diverse test functions to capture kink–breather waves, M-lumps, and lump–kink interaction solitons. A novel Jacobi elliptic activation function is applied to explore rogue wave soliton solutions. Spatiotemporal visualizations, such as 3D with contour, density, and 2D plots, generated using MATLAB for selected parameters, offer valuable insights into the structural evolution of these solutions. Moreover, the stability analysis of the governing equation is investigated using linear stability analysis under small perturbations, and the results are illustrated with dispersion analysis graphs. These visualizations offer a quantitative representation of the intricate dynamics at play. The bilinear neural methodology, utilizing established test functions, provides a systematic and robust approach for analyzing complex nonlinear phenomena. The findings enhance the understanding of soliton structures and expand the applicability of the CBS–nCBS equation in nonlinear wave theory.

This study handles the robust sampled-data H_infty fuzzy control analysis for a category of nonlinear partial differential systems (NPDSs) holding disturbances. As for now, the Takagi-Sugeno (T–S) fuzzy model serves superior by describing a broad category of nonlinear systems, and therefore, originally, a T–S fuzzy model is employed to illustrate the nonlinear parabolic partial differential systems. Here, the primary focus of this research is on designing a resilient sampled-data H_infty fuzzy estimator-based controller which is competent in stabilizing the T–S fuzzy closed-loop partial differential systems (PDSs) and to tolerate the disruption under a specified level. By the virtue of Lyapunov stability theory, Green’s formula and several inequality techniques, the robust stabilization design problem based on a sampled-data fuzzy H_infty estimator is effectively addressed using a set of linear matrix inequalities (LMIs). Moreover, the impacts of the diffusion phenomenon and the designed controller are clearly reflected in the derived criteria. Further, the acquired criteria can be checked for their practicability by the virtue of MATLAB LMI control toolbox. Finally, simulation results are presented to demonstrate the effectiveness of the proposed criteria.

This paper focuses on analyzing how initial stress influences wave propagation phenomena in a microelongated thermoelastic medium described within the framework of fractional conformable derivative, considering both the dual phase lag (DPL) and refined dual phase lag (RDPL) theories. The fundamental governing equations for heat transfer, mechanical motion, and microelongation are established to incorporate finite thermal wave speed and microelongation effects. Through an appropriate non-dimensionalization procedure and the application of the normal mode analysis technique, the coupled partial differential system is transformed into a form that admits explicit analytical solutions. These solutions provide expressions for displacement, microelongation, temperature distribution, and stress components, allowing a comprehensive examination of the thermomechanical wave behavior within the medium. To better comprehend the theoretical results, numerical evaluations are performed to emphasize the comparison of DPL and RDPL in the presence and absence of initial stress, as well as the influence of the fractional-order parameter and different times on wave properties. The results show that initial stress has a considerable effect on wave propagation characteristics such as amplitude modulation, propagation speed, and attenuation rate. Furthermore, the use of fractional conformable derivatives and the RDPL formulation allows for more precise modeling and control of the thermal relaxation dynamics. The current study contributes to a better understanding of the linked microelongated and thermal effects in thermoelastic media, as well as significant insights for designing and modeling advanced microscale thermoelastic systems.

This article studies a global and local piecewise fuzzy control for semilinear time-fractional parabolic partial differential equation (PDE) systems. First, the Takagi‒Sugeno (T-S) fuzzy model is used to characterize a semilinear time-fractional parabolic PDE system. Then, on the basis of the T-S fuzzy model, a fuzzy controller is designed to make the resulting closed-loop semilinear time-fractional parabolic PDE system Mittag‒Leffler stable. Meanwhile, the existence condition of the stabilizing fuzzy controller is given in the form of a linear matrix inequality (LMI). Finally, the effectiveness of the proposed scheme is verified by experiment simulation.

The practically fixed-time leaderless cluster synchronization is addressed for uncertain spatiotemporal networks (USTNs) with coopetition interactions, actuator faults and external disturbances. Firstly, by introducing sign-based coupling, a class of USTN is formulated to capture the dynamics of coopetition interactions among different clusters, which provides a more accurate representation compared to dynamical networks with unsigned coupling. Secondly, a practical fixed-time (PFT) convergence theorem is developed for a general partial differential system, which relaxes the constraints on the derivative of the Lyapunov function and provides a less conservative method for estimating the settling time. Subsequently, a distributed fault-tolerant control algorithm is designed to drive the cluster synchronization error to an adjustable attraction region in a fixed time. By exploring specific properties of the intra-cluster Laplacian matrix and proposing a new inter-degree balanced condition, several flexible synchronization criteria are derived and a quantitative relationship among control parameters, the settling time and the size of the attraction region is presented. Finally, the effectiveness of the developed controllers and criteria is validated through a coupled reaction-diffusion neural network.

A class of partial differential equations with random noise is employed to model the pipe acoustic system. A high-precision compact differential scheme is constructed for its solution. To ensure numerical stability, a buffer layer technique is applied to absorb outgoing waves. The propagation of acoustic waves under different modes is simulated. Furthermore, a specific numerical example is provided, and the results show good agreement with theoretical analysis.

This work develops and analyzes an incommensurate fractional FitzHugh–Nagumo (FHN) reaction–diffusion system in which each state variable evolves with a distinct fractional order. The formulation extends the classical and commensurate fractional models by incorporating heterogeneous memory effects that break temporal symmetry between the activator and inhibitor variables. After establishing the mathematical framework, the equilibrium states of the system are derived and subjected to a detailed local stability analysis in both diffusion-free and diffusion-driven regimes. Explicit stability criteria are obtained by examining the spectral properties of the linearized operator under incommensurate fractional dynamics. Numerical simulations based on a Caputo L1 discretization scheme corroborate the theoretical results and demonstrate how asymmetric memory orders influence transient behavior, convergence rates, and the qualitative structure of the solutions. The study provides the first systematic stability characterization of an incommensurate fractional FitzHugh–Nagumo reaction–diffusion model, highlighting the role of fractional-order asymmetry in shaping the system’s dynamical response.

Crohn's disease (CD) is a recurrent chronic autoimmune disease, which is an inflammatory disease of the intestine with epithelial granulomas. The number of patients has been increasing significantly, and its pathogenesis and treatments are arousing hot discussions in the academic community. Taking into account the spatial heterogeneity of lesion distribution and the periodic recurrence, this paper uses a partial functional differential system with the free diffusion of bacteria and immunocytes and immune response latency to model the process of CD, based on the Lauffenburger-Kennedy bacterial infection model. In order to describe the spatial distribution and recurrence, we analyze the stability of the inflammation equilibrium state, and deduce the diffusion-driven Turing bifurcations and delay-driven Hopf bifurcations, drive the critical conditions for occurrence. Furthermore, through the analysis of Turing-Hopf bifurcations, the coupling effect of two factors is explored to obtain spatiotemporal patterns that better reflect clinical manifestations of CD. In addition, both theoretical and numerical results reveal that the motility is a necessary factor in the production of intestinal epithelial granulomas, while the immune response latency is an important factor in the recurrence. A small effective diffusion rate and a large time delay would lead to two spatially non-homogeneous steady states and a stable periodic solution, ultimately giving rise to a pair of stable spatially non-homogeneous periodic solutions through Turing-Hopf bifurcations. Our conclusions may provide some insights into the control mechanisms for Crohn's disease.

The MHD (magnetic hydrodynamics) boundary problem in three-dimensional domains of certain types is considered within the framework of discrete potential theory. The discrete character of the information obtained from remote sensing of the Earth and planets of the Solar System can be taken into account when using the basic principles of this theory. This approach makes it possible to reconstruct the spatial distribution of magnetic fields and the velocity field with relatively high accuracy using the heterogeneous data in some network points. In order to restore the magnetic image of a planet with a so-called dynamo, the subsequent approximations approach is implemented. The unknown physical field is represented as a sum of terms of different magnitudes. Such an approach allows us to simplify the nonlinear partial differential equation system of magnetic hydrodynamics and extend it to discrete magnetic field and velocity vectors. The solution of the simplified MHD equation system is constructed for some classes of bounded domains in Cartesian coordinates in three-dimensional space.

To study the blood flow circulation dynamics, the present work thoroughly investigates the one-dimensional blood flow and transport in arteries and veins. We construct the one-dimensional optimum subalgebra and provide exact solutions of the governing equations using Lie symmetry analysis. A systematic framework is developed to generate a network of nonlocally related partial differential equation systems. These systems, constructed from local conservation laws and symmetry-based approaches, yield nonlocal symmetries that are classified and applied to obtain further exact solutions. One such solution is used to study the evolution of weak discontinuity waves. The findings shed light on blood circulation dynamics and suggest potential uses in hemodynamic modeling and analysis.

This research aims to present a comprehensive investigation of the self-organization of two-dimensional patterns composed of numerous dots and lines using numerical techniques, specifically focusing on the Gray–Scott (GS) model, a well-established framework for studying reaction–diffusion systems. The staggered grid finite volume approach is utilized and the Crank–Nicholson method (θ=12⋅) is used for time discretization in our simulations, as it is known to yield a reliable and accurate solution. The present study involves an investigation into the stability of the temporal semi-discrete numerical scheme, along with a comprehensive analysis of the errors through the utilization of standard solutions that demonstrate second-order convergence in both temporal and spatial domains. The utilization of pattern formulations proves to be extremely useful in the analysis of the interaction between diffusion and reactions. This is accomplished by solving the coupled partial differential system using finite difference techniques, which serve to enhance accuracy while simultaneously preserving the stability of the system. To further examine the system’s behavior, the finite volume methodology (FVM) was utilized to examine the dynamics of the GS model. The assessment of stability is performed on fixed points, wherein analytical solutions are compared to computational techniques such as the Alternate Direction Implicit methodology and the FVM. Furthermore, we employ the FVM in spatial discretization and the Crank–Nicolson scheme in temporal discretization to solve the GS model. This study represents a novel application of two distinct methodologies to analyze error co-variances and system efficiency in a particular system. The utilization of these approaches in combination is unprecedented in the existing literature and aligns with our current comprehension of the subject matter. The results demonstrate the accuracy and reliability of the proposed numerical schemes, highlighting their potential applicability to a wide range of reaction–diffusion systems. The findings of this research have significant implications for several disciplines, including material science, chemical engineering, and biophysics, where self-organization patterns are essential to the creation of innovative materials, novel devices, and biological systems. The results of our research provide vital insight into the fundamental mechanics of pattern formation. In addition, these discoveries have the potential to be utilized in the advancement and enhancement of a wide variety of technologies and applications.

The purpose of this paper is to investigate a class of novel symmetric coupled fuzzy fractional partial differential equation system involving the Caputo–Katugampola (C-K) generalized Hukuhara (gH) derivative. Within the framework of C-K gH differentiability, two types of gH weak solutions are defined, and their existence is rigorously established through explicit constructions via employing Schauder fixed point theorem, overcoming the limitations of traditional Lipschitz conditions and thereby extending applicability to non-smooth and nonlinear systems commonly encountered in practice. A typical numerical example with potential applications is proposed to verify the existence results of the solutions for the symmetric coupled system. Furthermore, we introduce Ulam–Hyers stability (U-HS) theory into the analysis of such symmetric coupled systems and establish explicit stability criteria. U-HS ensures the existence of approximate solutions close to the exact solution under small perturbations, and thereby guarantees the reliability and robustness of the systems, while it prevents significant deviations in system dynamics caused by minor disturbances. We not only enrich the theoretical framework of fuzzy fractional calculus by extending the class of solvable systems and supplementing stability analysis, but also provide a practical mathematical tool for investigating complex interconnected systems characterized by uncertainty, memory effects, and spatial dynamics.

Purpose Despite considerable progress in fluid dynamics, the intricate interplay between buoyancy-induced forces and thermal dispersion in nanofluid flows over exponentially stretching/shrinking surfaces remains limited, particularly when multiple solutions emerge due to the nonlinearity of governing equations. This study aims to bridge that gap by examining the heat and momentum transport in AA7075-water nanofluid. Design/methodology/approach Using similarity transformations, the complex partial differential system is reduced to coupled nonlinear ordinary differential equations, which are solved numerically using MATLAB. The analysis reveals dual solutions within certain parameter ranges, accompanied by a saddle-node bifurcation and flow separation, indicating critical transitions in flow structure. A comprehensive temporal stability analysis is conducted to identify the stable solution. To gain deeper insight into the thermal performance, response surface methodology is used to construct a predictive quadratic regression model for the Nusselt number. Findings Sensitivity analysis shows a strong positive correlation between the thermal dispersion parameter and Nusselt number, emphasizing the role of dispersion in enhancing convective transport. Physically, increasing the dispersion parameter significantly improves heat transfer in both stretching and shrinking cases. However, in the shrinking regime, higher Darcy numbers and nanoparticle volume fractions reduce thermal efficiency due to increased resistance. Conversely, stretching flows benefit from increased permeability, which enhances convective heat transfer. Originality/value The research offers valuable insights into optimizing nanofluid-based thermal systems, particularly in microchannel flows, heat exchangers and material processing applications requiring precise thermal regulation. By identifying multiple solutions and analyzing their stability, the study advances understanding and improves predictive capabilities for controlling flow transitions in practical nanofluid environments.

Boundary control of stochastic partial differential systems with delays and Lévy noise