Partial Differential Equations Research Articles

Regional control strategies for a spatiotemporal SQEIAR epidemic model: application to COVID-19

In this work, we develop a spatial SEIAR-type epidemic model considering a quarantined population (denoted as Q), which we call the SQEIAR model. The dynamics of the SQEIAR model are described by six Partial Differential Equations (PDEs) that represent the changes in the susceptible, quarantined, exposed, asymptomatic, infected, and recovered populations. Our goal is to reduce the number of susceptible, exposed, asymptomatic, and infected individuals while accounting for the environment, which plays a critical role in the spread of epidemics. We then propose a novel strategy for epidemic control, incorporating two key control measures: regional quarantine for the susceptible population and treatment for the infected. This approach serves as an alternative to widespread quarantine, minimising the economic, social, and other potential impacts. Additionally, we consider the possibility of reinfection among recovered individuals, a common occurrence in many diseases. To demonstrate the practical utility of our results, a numerical example centred on COVID-19 is presented.

Partial Differential Equations in Module of Copolynomials over a Commutative Ring

Partial Differential Equations in Module of Copolynomials over a Commutative Ring

Cervical cancer incidence rates considering migration status in mainland China using Bayesian model-Estimation based on 2016 cancer registry data.

In mainland China, cancer registration relies on household-registered populations, overlooking migrant populations. Estimating cervical cancer incidence among permanent residents, including migrants, offers a more accurate representation of the true burden. The data from 487 cancer registries across China in 2016 were analyzed using a Bayesian spatial regression model with the integrated nested Laplace approximation-stochastic partial differential equation method. The study estimated cervical cancer incidence among household-registered populations and adjusted for migrant populations using a weighting method based on interprovincial distribution and age stratification to derive the incidence of cervical cancer in the permanent residents. Data from the China Population Census, the China Migrants Dynamic Survey, and the Urban Statistical Yearbook were incorporated. The estimated crude incidence rate of cervical cancer among permanent residents was 17.4/100,000 in mainland China, with an age-standardized incidence rate (ASIR) of 17.2/100,000. The largest disparities in cervical cancer crude incidence rate between permanent residents and household-registered populations were observed in Guizhou (2.4/100,000, 95% CI 1.9-2.9/100,000), Zhejiang (-1.2/100,000, 95% CI -1.8 to -0.6/100,000) and Tianjin (-1.1/100,000, 95% CI -1.5 to -0.7/100,000). The number of the estimated cervical cancer incident cases was 8948. Guangdong saw an increase of 887 cases, while Henan had a decrease of 1430 cases. Guizhou had the highest ASIR (28.1/100,000), and Beijing had the lowest ASIR (11.0/100,000). The significance of this study is that it improves the accuracy of cervical cancer data in China. These findings provide evidence for developing cervical cancer prevention and control strategies, and offer insights for other countries and regions facing migration challenges.

Optimal stochastic control for carbon emission with allowances trading

The main focus of this paper is the optimal control strategy for minimizing carbon emission reduction cost in a company. Due to the existence of the carbon market, the company can purchase carbon allowances in the carbon market to increase its own emission limit, and it will adopt measures in order to decrease its emissions. We build a model with the goal of minimizing the total emissions reduction cost for the company, which involves two random processes. This model can then be transformed into a Hamilton–Jacobi–Bellman (HJB) equation, which is a semi-linear partial differential equation that is spatially two-dimensional. We prove the existence and uniqueness of the classical solution. Finally, some numerical results are shown.

From disorganized data to emergent dynamic models: Questionnaires to partial differential equations

From disorganized data to emergent dynamic models: Questionnaires to partial differential equations

Radial limits of solutions to elliptic partial differential equations

Radial limits of solutions to elliptic partial differential equations

A Rigorous Integrator and Global Existence for Higher-Dimensional Semilinear Parabolic PDEs via Semigroup Theory

In this paper, we introduce a general constructive method to compute solutions of initial value problems of semilinear parabolic partial differential equations on hyper-rectangular domains via semigroup theory and computer-assisted proofs. Once a numerical candidate for the solution is obtained via a finite dimensional projection, Chebyshev series expansions are used to solve the linearized equations about the approximation from which a solution map operator is constructed. Using the solution operator (which exists from semigroup theory), we define an infinite dimensional contraction operator whose unique fixed point together with its rigorous bounds provide the local inclusion of the solution. Applying this technique for multiple time steps leads to constructive proofs of existence of solutions over long time intervals. As applications, we study the 3D/2D Swift–Hohenberg, where we combine our method with explicit constructions of trapping regions to prove global existence of solutions of initial value problems converging asymptotically to nontrivial equilibria. A second application consists of the 2D Ohta–Kawasaki equation, providing a framework for handling derivatives in nonlinear terms.

Exponential almost Sure Stabilization of Nonlinear Delay Differential Systems under Stochastic Optimal Control Driven by Ito Brownian Noise

This study investigates the role of Brownian white noise in stabilizing nonlinear optimal control delay differential equations (OCDDES) that are typically unstable in their deterministic form. The technique applied involves the use of Lyapunov sample exponent and a specialized partial differential equation suggested by Mao, (1997). It is demonstrated that if the noise scaling parameters of the stochastically perturbed equation is finite, then the new stochastic optimal control delay differential equation (SOCDDES ) is self - stabilized in an almost sure exponential sense. This phenomenon does not occur in deterministic optimal control delay differential equations where noise is absent.

Artificial neural networks computing system for Ree-Eyring hybrid nanofluid flow over a solar panel sheet: Hamilton–Crosser model

Background: The utilisation of solar radiation for generating thermal energy has garnered a significant amount of attention, especially with the rising demand for sustainable power and heating sources. Nanofluids appeared as promising options for enhancing the efficiency of solar-thermal systems owing to their superior heat transmission properties. In this aspect, the present research deals with Ree-Eyring hybrid nanofluid ( SWCNT − MWCNT / H 2 O ) flow in the presence of thermal radiation effect and shape factor to examine the thermal behaviour in solar panels. Additionally, the innovation lies in the field of artificial neural networks and fluid flow analysis as an integrated approach. Methodology: The considered physical interpretation is stated in the form of nonlinear partial differential equations. To facilitate the computation process, the governing equations turned into nondimensional ordinary differential equations with the help of invariant transformations. The converted equations are treated scientifically using the NDSolve methodology, which automatically adjusts the step size. In further, an artificial neural network-based multi-layer perceptron deploying the Levenberg–Marquardt model is employed to forecast the measurements of physical quantities. Core Findings: The markable outcomes from this advanced work imply that the stretching surface has an important role in analysing the thickness of the boundary layer. The improving Weissenberg number enhances the boundary layer thickness, leading to a magnification in the momentum field of the non-Newtonian fluid. In addition, thermal radiation is captured as a second significant influence on temperature distribution. It follows that enhancing the Radiation parameter upsurges the thermal layer. The variation of the volume fraction of nanoparticles resulted in higher temperatures due to the Hamilton–Crosser model, which involves the cylindrical shape of the nanoparticles. Validation: The validation of the current work is explored by comparing it with the existing literature in certain instances. The incorporation of multi-layer perceptron for hybrid nanofluid delivers the performance capacity of high prediction with the error 4.83E−06, 1.11E−10 and 2.4E−08. Applications: The existing optimisation approach provides a new perspective on solar-powered offshore, solar vehicles, solar-powered charging stations, solar power greenhouse and solar water pump implementation.

Spectral framework using modified shifted Chebyshev polynomials of the third-kind for numerical solutions of one- and two-dimensional hyperbolic telegraph equations

This investigation discusses a numerical approach to solving the hyperbolic telegraph equation in both one and two dimensions. Applying the Galerkin method is the basis of this approach. We use appropriate combinations of third-kind modified shifted Chebyshev polynomials (3KMSCPs) as basis functions to transform the governing partial differential equations into a collection of algebraic equations. Through spectral Galerkin techniques, we establish the convergence error to demonstrate that our algorithm is more effective and efficient. Five examples are examined to verify the effectiveness and resilience of the applied method by comparing errors and illustrating the results. Our results show that the current numerical solutions align closely with exact solutions. The current algorithm is simple to set up and is better suited to solving certain difficult partial differential equations.

Thermo-Magnetic Bioconvection Flow in A Semi-Trapezoidal Enclosure Filled With A Porous Medium Containing Oxytactic Micro-Organisms: Modelling Hybrid Magnetic Bio-Fuel Cells

Abstract Hybrid fuel cells are becoming increasingly popular in 21st century energy systems engineering. These systems combine multiple features including various geometries, electromagnetic fluids, bacteria (micro-organisms), thermo-solutal convection and porous media. Motivated by these developments in the present work we simulate the two-dimensional magnetohydrodynamic (MHD) natural triple convection flow in a semi-trapezoidal enclosure saturated with electrically conducting water containing oxytactic microorganisms and oxygen species. Darcy?s model is deployed for porous media drag effects. The primitive governing partial differential conservation equations for mass, momentum, energy, oxygen species and motile micro-organism species density are transformed using a vorticity-stream function formulation and non-dimensional variables into a nonlinear boundary value problem. A numerical solution is obtained using a finite difference method with incremental time steps. The mathematical model features a number of controlling parameters i.e. Prandtl number, Rayleigh number, Bioconvective Rayleigh number, Darcy parameter, Hartmann (magnetic body force) number, Lewis number, Péclet number, Oxygen diffusion ratio, fraction of consumption oxygen to diffusion of oxygen parameter. Transport characteristics (streamlines, isotherms, oxygen iso-concentration and motile micro-organism concentration) are computed for several of these parameters. Microorganisms? impact on the rate of heat transfer at the boundaries is found to be beneficial or destructive, depending on combination of other parameters in the simulations. Additionally, Nusselt number and oxygen species Sherwood number are computed at the hot vertical wall. The simulations are relevant to hybrid electromagnetic microbial fuel cells.

Exploration of the effect of fractional elements in nonlinear transmission lines

Purpose The purpose of this paper is to explore how fractional derivatives affect the transient and steady-state behaviour of nonlinear transmission lines. This problem is of significance for high-frequency design of systems such as high-speed sampling systems and radar systems. Design/methodology/approach This paper shall consider the transient and steady-state responses of nonlinear transmission lines when fractional derivatives are considered. A lumped-parameter model is considered and the product-integration implicit trapezoidal rule shall be used for simulations. Findings The important observation is that small deviations of the order of the derivative from an integer order can have a significant effect on the transient and steady-state behaviour. This includes a change in the speed of the wave on the transmission line and on its damping. Originality/value The work is novel as it uses a lumped-parameter model with nonlinear capacitors and explores the effect on the dynamical behaviour when fractional derivatives are present. This is in contrast to the typical approach of using a partial differential equation derived under certain assumptions such as the nature of the nonlinear capacitor.

Numerical Study of Williamson Fluid Flow over a Stretching Sheet with Newtonian Heating Embedded in a Porous Medium in Presence of Thermal Radiation and Heat Source/Sink

Williamson fluid has been utilized to explore the aligned magnetic field in the presence of thermal radiation and heat source/sink utilizing Newtonian heating on a stretched sheet soaked in porous medium. The governing partial differential equations are transformed into nonlinear ordinary differential equations by applying similarity transformations. These equations are then solved using MATLAB and the Runge-Kutta fourth-order technique with a shooting technique. Graphs were used to investigate the effects of different factors on dimensionless velocity and temperature. A decrease in the velocity field and an improvement in the distribution of temperature are caused by an enhancement in the Non-Newtonian Williamson fluid parameter and permeability restriction K, whereas the opposite effect is evident when the Aligned angle parameter is increased. When the conjugate factor for Newtonian heating parameter is increased, temperature rises, even as the reverse effect is seen in Prandtl number The coefficient of skin friction is an decreasing function and the Nusselt number is increasing function of Non-Newtonian Williamson fluid parameter and permeability restriction K. while the reverse effect is seen in Aligned angle parameter . The Nusselt number is increased function of heat source/sink parameter , and Newtonian heating parameter . While the reverse effect is seen in Prandtl number .

Output tracking problem for a class of 3-D hyperbolic PDEs

The output tracking problem (OTP) refers to the procedure of determining an input control that ensures the system output follows a specified trajectory. In this paper, we investigate the OTP for controlled systems governed by a class of 3-D hyperbolic partial differential equations (PDEs), aiming for an exact alignment between the system output and the desired trajectory. Utilizing backstepping techniques, semigroup theory, the Hahn-Banach theorem, Laplace transforms, and Green's functions, our approach begins by establishing the OTP solution as a bounded feedback law, with Green's functions allowing the solution of a Cauchy-Euler equation. This is followed by deriving a detailed formulation for the complete state of the corresponding control problem. Finally, the approach provides an explicit formula for the OTP solution, using the desired trajectory and the solutions of specific kernel PDEs.

A new approach for solving the multidimensional control optimization problems with first‐order partial differential equations constraints

AbstractThe purpose of this paper is to provide the linearization technique to solve the multidimensional control optimization problem (MCOP) involving first‐order partial differential equation (PDEs) constraints. Firstly, we use the modified objective function approach for simplifying the aforesaid extremum problem (MCOP) and show that the solution sets of the original control optimization problem and its modified control optimization problem (MCOP) are equivalent under convexity assumptions. Further, we use the absolute value exact penalty function method to transform (MCOP) into a penalized control problem (MCOP) . Then, we establish the equivalence between a minimizer of the modified penalized optimization problem (MCOP) and a saddle point of the Lagrangian defined for the modified optimization problem (MCOP) under appropriate convexity hypotheses. Moreover, the results established in the paper are illustrated by some examples of MCOPs involving first‐order PDEs constraints.

Quasi-rectifiable Lie algebras for partial differential equations

Abstract We introduce families of quasi-rectifiable vector fields and study their geometric and algebraic aspects. Then, we analyse their applications to systems of partial differential equations. Our results explain, in a simple manner, the properties of families of vector fields describing hydrodynamic-type equations by means of k-waves. Facts concerning families of quasi-rectifiable vector fields, their relation to Hamiltonian systems, and practical procedures for studying such families are developed. We introduce and analyse quasi-rectifiable Lie algebras, which are motivated by geometric and practical reasons. We classify different types of quasi-rectifiable Lie algebras, e.g. indecomposable ones up to dimension five. New methods for solving systems of hydrodynamic-type equations are established to illustrate our results. In particular, we study hydrodynamic-type systems admitting Riemann k-wave solutions through quasi-rectifiable Lie algebras of vector fields. We develop techniques for obtaining the submanifolds related to quasi-rectifiable Lie algebras of vector fields and systems of partial differential equations admitting a nonlinear superposition rule: the PDE Lie systems.

An Upscaling Machine Learning Method Based On Physical Information

Objective: The objective of this study is to investigate a machine learning methodology based on the physics of dynamic fluid flow, with the aim of incorporating the knowledge of physical laws into learning algorithms to estimate precise results with lower computational cost. Theoretical Framework: This section presents the main concepts and theories that underpin the research. Key theories include nonlinear partial differential equations (PDEs), Darcy's Law, and the conservation of mass and energy, providing a solid foundation for understanding the research context. Method: The methodology adopted for this research involves modeling a one-dimensional two-phase fluid flow problem in a porous medium, governed by a first-order hyperbolic nonlinear equation. Data collection was conducted through numerical simulations, using micro and macro grids to evaluate permeability and boundary conditions. Results and Discussion: The results revealed that considering learning only in absolute permeability yielded good estimates of reservoir pressures. However, small discrepancies were observed in the estimation of saturations. The discussion contextualizes these results in light of the theoretical framework, highlighting the identified implications and relationships, as well as the study's limitations. Research Implications: The practical and theoretical implications of this research are discussed, providing insights into how the results can be applied or influence practices in the field of reservoir modeling and fluid flow simulations. These implications may include optimizing oil extraction processes and improving predictive models in reservoir engineering. Originality/Value: This study contributes to literature by presenting an innovative approach that integrates physical laws into machine learning algorithms, resulting in more accurate and efficient estimates. The relevance and value of this research are evidenced by the potential application of the results in optimizing industrial processes and improving computational simulations.

Application of physics-informed neural networks (PINNs) solution to coupled thermal and hydraulic processes in silty sands

The accurate modeling of water and heat transport in soils is crucial for both geo-environmental and geothermal engineering. Traditional modeling methods are problematic because they require well-defined boundaries and initial conditions. Recently, physics-informed neural networks (PINNs), which incorporate partial differential equations (PDEs) to solve forward and inverse problems, have attracted increasing attention in machine learning research. In this study, we applied PINNs to tackle hydraulic and thermal transport coupling forward problems in silty sands. A fully connected deep neural network was utilized for training. This neural network model leverages automatic differentiation to apply the governing equations as constraints, based on the mathematical approximations established by the neural network itself. We conducted forward problems and compared the solutions derived from PINNs with those from Finite Element Method (FEM) simulations. The forward problem results demonstrate the PINNs model’s capability in predicting hydraulic transport, heat transport, and thermal–hydraulic coupling in silty sands under various boundary conditions. The PINNs exhibited great performance in simulating the thermal–hydraulic coupling problem. The accuracy of the PINNs solutions shows its potential for simulation in geotechnical engineering.

Mathematical model and stability of SWCNT- and MWCNT-based nanofluid flow with thermal and chemically reactive effects inside a porous vertical cone

This study investigates the significance of single-walled (SWCNTs) and multi-walled (MWCNTs) carbon nanotubes with a convectional fluid (water) over a vertical cone under the influences of chemical reaction, magnetic field, thermal radiation and saturated porous media. The impact of heat sources is also examined. Based on the flow assumptions, the fundamental flow equations are modeled as partial differential equations (PDEs). Using the appropriate transformation, the PDEs are converted to ordinary differential equations and then solved via RK4 in MATLAB. To confirm the results, a comparison is made with a previously investigated problem, finding good agreement. The emerging dimensionless physical parameters impact on the flow problem is determined through graphs and tables. Analysis reveals a dual solution for the suction and injection parameter. Therefore, stability examination is implemented to confirm a stable solution. The aim of the study is to analyze SWCTs and MWCTs in a vertical cone with stability to establish that only the first solution is reliable. The analysis here signifies that large volume fractions can be substituted to increase the nanofluid movement. The mathematical model and graphical demonstration indicate that the velocity of MWCNTs is higher than SWCNTs. Moreover, the local skin friction, rate of heat transfer, and Nusselt and Sherwood numbers improve with Biot number.

Dynamics of Photoinduced Charge Carrier and Photothermal Effect in Pulse-Illuminated Narrow Gap and Moderate Doped Semiconductors

When a sample of semiconducting material is illuminated by monochromatic light, in which the photon energy is higher than the energy gap of the semiconductor, part of the absorbed electromagnetic energy is spent on the generation of pairs of quasi-free charge carriers that are bound by Coulomb attraction. Photo-generated pairs diffuse through the material as a whole according to the density gradients established, carrying part of the excitation energy and charge through the semiconducting sample. This energy is indirectly transformed into heat, where the excess negatively charged electron recombines with a positively charged hole and causes additional local heating of the lattice. The dynamic of the photoexcited charge carrier is described by a non-linear partial differential equation of ambipolar diffusion. In moderate doped semiconductors with a low-level injection of charge carriers, ambipolar transport can be reduced to the linear parabolic partial differential equation for the transport of minority carriers. In this paper, we calculated the spectral function of the photoinduced charge carrier distribution based on an approximation of low-level injection. Using the calculated distribution and inverse Laplace transform, the dynamics of recombination photoinduced heat sources at the surfaces of semiconducting samples were studied for pulse optical excitations of very short and very long durations. It was shown that the photoexcited charge carriers affect semiconductor heating depending on the pulse duration, velocity of surface recombination, lifetime of charge carriers, and their diffusion coefficient.