Partial Differential System Research Articles

Extension of Symmetrized Neural Network Operators with Fractional and Mixed Activation Functions

We propose a novel extension to symmetrized neural network operators by incorporating fractional and mixed activation functions. This study addresses the limitations of existing models in approximating higher-order smooth functions, particularly in complex and high-dimensional spaces. Our framework introduces a fractional exponent in the activation functions, allowing adaptive nonlinear approximations with improved accuracy. We define new density functions based on q-deformed and ?-parametrized logistic models and derive advanced Jackson-type inequalities that guarantee uniform convergence rates. Additionally, we provide a rigorous mathematical foundation for the proposed operators, supported by numerical validations demonstrating their efficiency in handling oscillatory and fractional components. The results extend the applicability of neural network approximation theory to broader functional spaces, paving the way for applications in solving partial differential equations and modeling complex systems.

Computing and Communication Structure Design for Fast Mass-Parallel Numerical Solving PDE

Partial differential equations and systems with certain boundary conditions specify continuous processes significant for both large-scale simulations in computer-aided design using HPC and subsequent real-time control of embedded applications using dedicated hardware. The paper develops a spectrum of techniques based on a family of place-transition nets aimed at the computing and communication structure design for fast mass-parallel numerical solving of PDEs. For the HPC domain, we develop models of interconnects in the form of infinite nets and graphical programs in the form of Sleptsov nets. For the embedded control domain, we develop specialized lattices for fast numerical solving PDE based on integer number approximation specified with Sleptsov-Salwicki nets to be implemented on dedicated hardware, which we prototype on FPGAs. For mass-parallel solving of PDEs, we employ ad-hoc finite-difference schemes and iteration methods that allow us to recalculate the lattice values in a single time cycle suitable for control of hypersonic objects and thermonuclear reactions.

Solutions for Modelling the Marine Oil Spill Drift

Oil spills represent a critical environmental hazard with far-reaching ecological and economic consequences, necessitating the development of sophisticated modelling approaches to predict, monitor, and mitigate their impacts. This study presents a computationally efficient and physically grounded modelling framework for simulating oil spill drift in marine environments, developed using Python coding. The proposed model integrates core physical processes—advection, diffusion, and degradation—within a simplified partial differential equation system, employing an integrator for numerical simulation. Building on recent advances in marine pollution modelling, the study incorporates real-time oceanographic data, satellite-based remote sensing, and subsurface dispersion dynamics into an enriched version of the simulation. The research is structured in two phases: (1) the development of a minimalist Python model to validate fundamental oil transport behaviours, and (2) the implementation of a comprehensive, multi-layered simulation that includes NOAA ocean currents, 3D vertical mixing, and support for inland and chemical spill modelling. The results confirm the model’s ability to reproduce realistic oil spill trajectories, diffusion patterns, and biodegradation effects under variable environmental conditions. The proposed framework demonstrates strong potential for real-time decision support in oil spill response, coastal protection, and environmental policy-making. This paperwork contributes to the field by bridging theoretical modelling with practical response needs, offering a scalable and adaptable tool for marine pollution forecasting. Future extensions may incorporate deep learning algorithms and high-resolution sensor data to further enhance predictive accuracy and operational readiness.

Mean Square Stability and Stabilization for Linear Parabolic Stochastic Partial Difference Systems

This paper studies the mean square stability and stabilization of linear parabolic stochastic partial difference systems, which contain space–time characteristics and stochastic noise. A definition of the mean square stability for this system is proposed. Using stochastic analysis and some mathematical analysis methods, a strict decreasing sequence is constructed to represent the expectation of the sum of squares of the state variable along the spatial dimension. The sufficient conditions of the mean square stability are established based on system parameters, and then the convergence along the time axis is rigorously proven by the Squeeze criterion. Moreover, some stabilization criteria are derived by designing a linear feedback controller. Finally, two examples are given to illustrate the effectiveness of the results.

Symbolic Neural Ordinary Differential Equations

Differential equations are widely used to describe complex dynamical systems with evolving parameters in nature and engineering. Effectively learning a family of maps from the parameter function to the system dynamics is of great significance. In this study, we propose a novel learning framework of symbolic continuous-depth neural networks, termed Symbolic Neural Ordinary Differential Equations (SNODEs), to effectively and accurately learn the underlying dynamics of complex systems. Specifically, our learning framework comprises three stages: initially, pre-training a predefined symbolic neural network via a gradient flow matching strategy; subsequently, fine-tuning this network using Neural ODEs; and finally, constructing a general neural network to capture residuals. In this process, we apply the SNODEs framework to partial differential equation systems through Fourier analysis, achieving resolution-invariant modeling. Moreover, this framework integrates the strengths of symbolism and connectionism, boasting a universal approximation theorem while significantly enhancing interpretability and extrapolation capabilities relative to state-of-the-art baseline methods. We demonstrate this through experiments on several representative complex systems. Therefore, our framework can be further applied to a wide range of scientific problems, such as system bifurcation and control, reconstruction and forecasting, as well as the discovery of new equations.

Hybrid model of hydrogen thermal desorption from structural materials

To solve problems of hydrogen power engineering, there is an intensive search for materials for hydrogen storage. Thermal desorption spectrometry (TDS) is one of the effective experimental methods for studying the interaction of structural materials with hydrogen isotopes. A sample (we consider a thin plate made of a material with metallic properties) pre-saturated with dissolved atomic hydrogen is heated relatively slow in a vacuum chamber. The degassing flux is registered using a mass spectrometer. The spectrum is the dependence of the desorption flux density from a two-sided surface of the sample on the current temperature. Quite often, several local peaks are registered on the spectrum. Traditionally, this is associated with the reversible capture of various kinds of traps (inhomogeneities of the material) with different binding energies. However, numerical experiments on models with dynamic boundary conditions describing the dynamics of surface concentrations show the possibility of a different scenario. The following scheme is possible: The first peak occurs when hydrogen leaves the surface and the subsurface volume. Then, a large concentration gradient is formed at the surface. For this reason, and during continued heating, diffusion influx from the volume is significantly activated, which leads to the next peak of desorption. Recommendations on how to distinguish degassing scenarios corresponding to these essentially different physicochemical reasons are given. This is fundamentally important for the correct recalculation of modeling results from laboratory samples to real constructions. The hybrid thermal desorption model can be considered as a computational algorithm for solving a partial differential system using an approximation by an ODE system (but this is not a straight-line method).

INFLUENCES OF VISCOSITY AND TEMPERATURE VARIATIONS ACROSS THE THIN LUBRICANT LAYER ON THE CYLINDRICAL SLIDE BEARING PARAMETERS

The article analyses changes in apparent viscosity and temperature across the thickness of the lubricating non-Newtonian liquid layer and determines the influence of these variations on the hydrodynamic pressure, temperature, and load-carrying capacity of monotone rotational cylindrical bearing surfaces. To this end the authors identify particular semi-analytical recurrent solutions of a strongly non-linear, secondorder partial differential system of five equations with variable coefficients in cylindrical coordinates and impose proper boundary conditions on it. Scientific calculations for cylindrical bearings referring to changes in temperature and viscosity across the thickness of the lubricating film imply significant average ten and fifty percent variations in the load-carrying capacity compared to the results obtained from classical calculations in contemporary scientific literature, where the temperature and oil dynamic viscosity are assumed constant across the film thickness. Moreover, the load-carrying capacity values obtained based on semi-analytical calculations for temperature and viscosity gradient variations across the thin film layer differ by only two or three percent, compared to the corresponding experimental values measured at several points of measurement in real cylindrical journal bearings. This fact confirms the correctness of the new calculations performed for cylindrical journal bearings.

Nanofluid magnetoconvection and entropy generation: a computational study for water treatment and resource management

This research exploration emerged from the critical need to revolutionize heat transfer techniques, particularly in pivotal domains like nuclear technologies, electronics and energy-efficient systems. The motivation for this study endeavour stemmed from the complex interrelation among nanofluids, magnetic fields and their potential for enhancing heat exchange. A pragmatic numerical approach is utilized to examine the Cu–H2O nanofluid flow situation within an enclosure featuring cooled vertical walls and a heat-generating source, while ensuring insulation for the remaining edges. The evaluation analyses the contribution of entropy, including total, viscous and thermal entropies, establishing a connection to real-world heat transfer challenges. The Galerkin finite element algorithm is utilized to solve the partial differential system of the modelled problem. The phenomena of entropy generation, fluid flow and heat transfer are studied under the influence of parameters such as the Hartmann number, Rayleigh number, magnetic field inclination angle and nanoparticle volume fraction. The study reveals that irreversibility increases with the magnetic field inclination angle, while entropy generation decreases with an increase in the Hartmann number. The primary innovation of this study is uncovering new dimensions with widespread practical implications by deciphering the complex dynamics of nanofluid convection with entropy generation and inclined magnetic influence. This research holds significant potential for advancing heat transfer applications in water treatment and resource management, aligning with the journal’s focus on sustainable and innovative water solutions.

Dual Neural Network (DuNN) method for elliptic partial differential equations and systems

Dual Neural Network (DuNN) method for elliptic partial differential equations and systems

Retraction notice to "Exponential stability analysis of delayed partial differential equation systems: Applying the Lyapunov method and delay-dependent techniques" [Heliyon 10 (2024) e32650

Retraction notice to "Exponential stability analysis of delayed partial differential equation systems: Applying the Lyapunov method and delay-dependent techniques" [Heliyon 10 (2024) e32650

NEW SOLUTIONS FOR LUBRICANT VISCOSITYAND TEMPERATURE VARIATIONS ACROSS THE THIN FILM ONARBITRARY SLIDING BEARING SURFACES

The aim of this paper is to analyse the changes in apparent dynamic viscosity and temperature across any thinnon-Newtonian lubricating liquid layer, and determine the influence of such variations on the hydrodynamicpressure and load-carrying capacity for arbitrary curvilinear monotone or non-monotone rotational and nonrotationalsliding bearing surfaces.This requires determining particular semi-analytical solutions of a strongly non-linear, second-order partialdifferential system of five equations with variable coefficients in curvilinear coordinates, and imposingproper curvilinear boundary conditions on it. After initial numerical calculations for any bearing surface,especially with a conical or spherical shape, the changes in temperature and viscosity across the thickness ofthe lubricating film change the load-carrying capacity by nearly 20 per cent compared to the results obtainedfrom classic calculations in the contemporary scientific literature, where the temperature and oil dynamicviscosity are assumed constant across the film thickness.

Mathematical Aspects of General Relativity

General relativity is an area at the interface of partial differential equations, differential geometry, global analysis, mathematical physics and dynamical systems. It interacts with astrophysics, cosmology, high energy physics, and numerical analysis. The field is rapidly expanding and has witnessed remarkable developments and interconnections with other fields in recent years.The workshop Mathematical Aspects of General Relativity was organised by Carla Cederbaum (Tübingen), Mihalis Dafermos (Cambridge/Princeton), Jim Isenberg (Eugene) and Hans Ringström (KTH Stockholm). There were 48 on-site and 4 online participants. There were 16 one hour talks, nine 30 minute talks and four 10 minute talks.

Stochastic parabolic system: homogenization in time-dependent potential and random fluctuations

Abstract We study the convergence of the solution to the parabolic system with time-dependent, zero-mean Gaussian potential. Depending on the decorrelation properties of the Gaussian potential, the spatial dimension, and the order of the elliptic operator, we prove that the solution to the parabolic system strongly converges to the deterministic solution of the homogenized system. Additionally, we also get
that the random fluctuation converges in distribution, after a certain scaling transformation, to the solution of the stochastic partial differential system with additive noise. Unlike the error term, the random fluctuation can asymptotically capture the stochasticity in the solution.

Multidimensional integrable systems from contact geometry

Upon having presented a bird’s eye view of history of integrable systems, we give a brief review of certain recent advances in the longstanding problem of search for partial differential systems in four independent variables, often referred to as (3+1)-dimensional or 4D systems, that are integrable in the sense of soliton theory. Namely, we review a recent construction for a large new class of (3+1)-dimensional integrable systems with Lax pairs involving contact vector fields. This class contains inter alia two infinite families of such systems, thus establishing that there is significantly more integrable (3+1)-dimensional systems than it was believed for a long time. In fact, the construction under study yields (3+1)-dimensional integrable generalizations of many well-known dispersionless integrable (2+1)-dimensional systems like the dispersionless KP equation, as well as a first example of a (3+1)-dimensional integrable system with an algebraic, rather than rational, nonisospectral Lax pair. To demonstrate the versatility of the construction in question, we employ it here to produce novel integrable (3+1)-dimensional generalizations for the following (2+1)-dimensional integrable systems: dispersionless BKP, dispersionless asymmetric Nizhnik–Veselov–Novikov, dispersionless Gardner, and dispersionless modified KP equations, and the generalized Benney system.

Blow-up criterion for the 3-D inhomogeneous incompressible viscoelastic rate-type fluids with stress-diffusion

<p>In this paper, we investigate the blow-up criterion of an evolutionary partial differential equation system controlling the flows of incompressible viscoelastic rate-type fluids with stress-diffusion, where the extra stress tensor describing the elastic response of the fluid is purely spherical. By utilizing this criterion, the global well-posedness of the system can be readily obtained. Despite being a physical simplification, this model exhibits features that necessitate novel mathematical approaches to tackle the technically complex structure of the associated internal energy, as well as the more complicated forms of the corresponding entropy and energy fluxes. The paper provides the first rigorous proof of the existence of a global solution to the model under small initial data.</p>

Analytical approach to control the irreversibilities in a chemically reactive nanofluid flow over a Darcy–Forchheimer medium with nonuniform heat source/sink

AbstractIn real‐world processes, such as heat transfer, fluid flow, and chemical reactions, irreversibility occurs frequently, which induces an augment in entropy. The optimization of the entropy generation in a mechanical system is one of the fundamental areas of research to channel the energy of the system for utilization. This research exhibits an entropy generation in a nanofluid flow drenched in a fluid‐saturated non‐Darcy porous medium toward a stagnation point. The flow line also experiences the existence of nonuniform heat generation/absorption following the first‐order chemical reaction, which enhances the applicability of this model in different domains of science and engineering. A special form of the Lie group approach is applied to revert the governing partial differential system into its self‐similar ordinary differential form. The solutions of the transformed nonlinear ODEs are acquired analytically through the DTM ‐Padé as well as numerically by the Runge–Kutta–Fehlberg (RKF‐45) method coupled with the shooting process. The fallouts are vividly sketched graphically. One of the upshots reveals that the escalation of space and temperature‐reliant heat absorption parameters can optimize the thermal irreversibility of the system and so as the total entropy generation. Meanwhile, by incrementing the Darcy parameter from 0 to 0.8, the wall shear stress decays by 23.7%, whereas with the same hike in the Forchheimer resistance factor, declines the rate of heat and mass transfer approximately by 2.6% and 3.6%, respectively. Another upshot reveals that with the escalation of the space‐dependent heat source parameter from 0.1 to 0.5, the Nusselt number significantly decreased by 23.9%. Meanwhile boosting the temperature‐reliant heat sink parameter from ‐0.1 to ‐0.5 causes an inclination in the rate of heat transportation by 10.5%. Moreover, it is found that mass transport irreversibility can be controlled by enhancing the constructive chemical reaction rate. We hope that this study will be beneficial for many engineering and industrial processes, particularly in geothermal energy extraction, nuclear waste disposal management, groundwater filtration machinery and food processing equipment.

Numerical exploration of bioconvection in optimizing nanofluid flow through heated stretched cylinder in existence of magnetic field

PurposeNanofluids are used in technology, engineering processes and thermal exchanges. In thermal transfer processing, these are used for the smooth transportation of heat and mass through various mechanisms. In the current investigation, we have examined multiple effects like activation energy thermal radiation, magnetic field, external heat source and especially slippery effects on a bioconvective Casson nanofluid flow through a stretching cylinder.Design/methodology/approachSeveral studies used non-Newtonian fluid models to study blood flow in the cardiovascular system. In our research, Lewis numbers for bioconvection and the influence of important parameters, such as Brownian diffusion and thermophoresis effects, are also considered. This system is developed as a partial differential equation for the mathematical treatment. Well-defined similarity transformations convert partial differential equation systems into ordinary differential equations. The resultant system is then numerically solved using the bvp4c built-in function of MATLAB.FindingsAfter utilizing the numerical approach to the system of ordinary differential equations (ODEs), the results are generated in the form of graphs and tables. These generated results show a suitable accuracy rate compared to the previous results. The consequence of various parameters under the assumed boundary conditions on the temperature, motile microorganisms, concentration and velocity profiles are discussed in detail. The velocity profile decreases as the Magnetic and Reynolds number increases. The temperature profile exhibits increasing behavior for the Brownian motion and thermal radiation count augmentation. The concentration profile decreased on greater inputs of the Schmidt number and magnetic effect. The density of motile microorganisms decreases for the increased value of the bio-convective Lewis number.Originality/valueThe numerical analysis of the flow problem is addressed using graphical results and tabular data; our reported results are refined and novel based on available literature. This method is useful for addressing such fluidic flow efficiently.

CMMSE: Solutions in a Broad Sense to the Boundary Value Problem for First‐Order Partial Differential Systems

ABSTRACTThis article examines the initial‐boundary value problem for a system of first‐order partial differential equations. Issues of existence and uniqueness of the solution in a broad sense are considered, while taking into account both periodic and multipoint conditions. The definition of a solution in a broad sense is introduced; the initial problem is reduced to the initial‐boundary value problem for ordinary differential equations. The two‐point boundary value problem for ordinary differential equations systems is studied by the Dzhumabaev method (parameterization method), which allows us to move on to the equivalent multipoint boundary value problem with functional parameters. An algorithm to find an approximate solution to the problem in a broad sense has been developed.

On a Certain Subclass of Multivalent Function Defined by Generalized Ruscheweyh Derivative

Fractional calculus is the prominent branch of applied mathematics, it deals with a lot of diverse possibility of finding differentiation as well as integration of function <i>f</i>(<i>z</i>) when the order of differentiation operator ‘D’ and integration operator ‘J’ is a real number or a complex number. The combination of fractional calculus with geometric function theory is the dynamic field of the current research scenario. It has many applications not only in the field of mathematics but also in the different fields like modern mathematical physics, electrochemistry, viscoelasticity, fluid dynamics, electromagnetic, the theory of partial differential equations systems, Mathematical modeling. Various new subclasses of univalent and multivalent functions defined by using different operators. In this research paper, we work on the formation of new subclass of analytic and multivalent functions defined under the open unit disk. By using Generalized Ruscheweyh derivative operator we define a new subclass of analytic and multivalent functions. The main aim of this research article is to derive interesting characteristics of new subclass of multivalent functions, which mainly include coefficient bound, growth and distortion bounds for function and its first derivative, extreme point and obtain unidirectional results for the multivalent functions which are belonging to this new subclass.

Case study of radiation absorptive and chemical reactive impacts on unsteady MHD natural convection flowing of Cu-Al2O3/H2O hybridized nanofluid

Case study of radiation absorptive and chemical reactive impacts on unsteady MHD natural convection flowing of Cu-Al2O3/H2O hybridized nanofluid