Use Of Partial Differential Equations Research Articles

Fractional Analysis of Magnetohydrodynamics Maxwell Flow Over an Inclined Plate with the Effect of Thermal Radiation

This current study provides a comprehensive examination of a novel method for studying the dynamics of a fractionalized Maxwell flow near an inclined plate, considering non-uniform mass transfer through a permeable media. Through the use of partial differential equations, incorporating heat and mass movement effects, the study employs a combination of generalized Fick’s and Fourier’s law with the Caputo operator. Transforming the fractionalized model into dimensionless form using appropriate dimensionless values, semi-analytical solutions for the non-dimensional transmitted fractional model are obtained via the Laplace transformation method. Through graphical analysis, the precise contributions of key parameters such as heat generation, radiation, and chemical reactions are elucidated, including their impacts on the calculated heat generation parameter (Qo), radiation parameter (Nr), and others. The study’s significance lies in its implications for the design of efficient heat exchangers, fluid flow systems, and cooling components in complex engineering systems, including nuclear reactors and power generation plants. Furthermore, the fractional derivative approach offers a more accurate representation of the viscoelastic behavior of materials like polymers, crucial for optimizing fabrication processes such as extrusion and molding. The insights gained from this study extend to the realm of miniaturized fluidic devices, including bio-analysis tools, lab-on-a-chip technology, and microfluidic drug delivery systems, where improved performance and control need a grasp of Maxwell fluid dynamics. The physical outcome of this research lays the groundwork for future investigations that will maximize heat transfer efficiency in real-world systems and give insightful information on the behavior of complicated fluids. We compute and display the skin friction, mass and heat transfer rate in tabular form.

The Modeling and Control of (Renewable) Energy Systems by Partial Differential Equations—An Overview

Mathematical models of energy systems have been mostly represented by either linear or nonlinear ordinary differential equations. This is consistent with lumped-parameter dynamic system modeling, where dynamics of system state variables can be fully described only in the time domain. However, when dynamic processes of energy systems display both temporal and spatial evolutions (as is the case of distributed-parameter systems), the use of partial differential equations is necessary. Distributed-parameter systems, being described by partial differential equations, are mathematically (and computationally) much more difficult for modeling, analysis, simulation, and control. Despite these difficulties in recent years, quite a significant number of papers that use partial differential equations to model and control energy processes and systems have appeared in journal and conference publications and in some books. As a matter of fact, distributed-parameter systems are a modern trend in the areas of control systems engineering and some energy systems. In this overview, we will limit our attention mostly to renewable energy systems, particularly to partial differential equation modeling, simulation, analysis, and control papers published on fuel cells, wind turbines, solar energy, batteries, and wave energy. In addition, we will indicate the state of some papers published on tidal energy systems that can be modelled, analyzed, simulated, and controlled using either lumped or distributed-parameter models. This paper will first of all provide a review of several important research topics and results obtained for several classes of renewable energy systems using partial differential equations. Due to a substantial number of papers published on these topics in the past decade, the time has come for an overview paper that will help researchers in these areas to develop a systematic approach to modeling, analysis, simulation, and control of energy processes and systems whose time–space evolutions are described by partial differential equations. The presented overview was written after the authors surveyed more than five hundred publications available in well-known databases such as IEEE, ASME, Wiley, Google, Scopus, and Web of Science. To the authors’ best knowledge, no such overview on PDEs for energy systems is available in the scientific and engineering literature. Throughout the paper, the authors emphasize novelties, originalities, and new ideas, and identify open problems for future research. To achieve this goal, the authors reviewed more than five hundred journal articles and conference papers.

Examining share repurchase executions: insights and synthesis from the existing literature

This literature review aims to address the critical knowledge gap in the field of share repurchase executions, a financial activity involving companies repurchasing trillions of dollars' worth of their own shares. The significance of understanding these mechanisms and their impact is underscored by their potential influence on the global economy. The paper employs a comprehensive analysis of existing literature, focusing on share repurchase mechanisms and motivations. It scrutinizes both open-market repurchases and Accelerated Share Repurchase contracts. Methodological approaches in current research, such as the use of partial differential equations and tree methods, are also evaluated. The review reveals that the execution phase of share repurchases remains largely unexplored. Unanswered questions persist about trading schedules, implications, costs, broker and corporate performance, and psychological effects of beating a buyback benchmark. Additionally, the review identifies significant limitations in current research methodologies. The paper advocates for the application and development of more advanced tools like machine learning and artificial intelligence to address these gaps. It also suggests potential areas for future research, including the role of technology in share repurchase execution, psychological factors influencing corporate buybacks, and the development of performance metrics for brokers and corporations. The review serves not only to highlight existing gaps in literature but also to suggest avenues for future research that could fundamentally enhance our understanding of share repurchase executions.JEL classificationG1, G12, G14, G02, G4.

Exact solutions for magnetohydrodynamic nanofluids flow and heat transfer over a permeable axisymmetric radially stretching/shrinking sheet

We report on the magnetohydrodynamic impact on the axisymmetric flow of Al2O3/Cu nanoparticles suspended in H2O past a stretched/shrinked sheet. With the use of partial differential equations and the corresponding thermophysical characteristics of nanoparticles, the physical flow process is illustrated. The resultant nonlinear system of partial differential equations is converted into a system of ordinary differential equations using the suitable similarity transformations. The transformed differential equations are solved analytically. Impacts of the magnetic parameter, solid volume fraction and stretching/shrinking parameter on momentum and temperature distribution have been analyzed and interpreted graphically. The skin friction and Nusselt number were also evaluated. In addition, existence of dual solution was deduced for the shrinking sheet and unique solution for the stretching one. Further, Al2O3/H2O nanofluid flow has better thermal conductivity on comparing with Cu/H2O nanofluid. Furthermore, it was found that the first solutions of the stream are stable and physically realizable, whereas those of the second ones are unstable.

11-th order of accuracy for numerical solution of 3-D Poisson equation with irregular interfaces on unfitted Cartesian meshes

For the first time the optimal local truncation error method (OLTEM) with 125-point stencils and unfitted Cartesian meshes has been developed in the general 3-D case for the Poisson equation for heterogeneous materials with smooth irregular interfaces. The 125-point stencils equations that are similar to those for quadratic finite elements are used for OLTEM. The interface conditions for OLTEM are imposed as constraints at a small number of interface points and do not require the introduction of additional unknowns, i.e., the sparse structure of global discrete equations of OLTEM is the same for homogeneous and heterogeneous materials. The stencils coefficients of OLTEM are calculated by the minimization of the local truncation error of the stencil equations. These derivations include the use of the Poisson equation for the relationship between the different spatial derivatives. Such a procedure provides the maximum possible accuracy of the discrete equations of OLTEM. In contrast to known numerical techniques with quadratic elements and third order of accuracy on conforming and unfitted meshes, OLTEM with the 125-point stencils provides 11-th order of accuracy, i.e., an extremely large increase in accuracy by 8 orders for similar stencils. The numerical results show that OLTEM yields much more accurate results than high-order finite elements with much wider stencils. The increased numerical accuracy of OLTEM leads to an extremely large increase in computational efficiency.Also, a new post-processing procedure with the 125-point stencil has been developed for the calculation of the spatial derivatives of the primary function. The post-processing procedure includes the minimization of the local truncation error and the use of the Poisson equation. It is demonstrated that the use of the partial differential equation (PDE) for the 125-point stencils improves the accuracy of the spatial derivatives by 6 orders compared to post-processing without the use of PDE as in existing numerical techniques. At an accuracy of 0.1% for the spatial derivatives, OLTEM reduces the number of degrees of freedom by 900−4⋅106 times compared to quadratic finite elements. The developed post-processing procedure can be easily extended to unstructured meshes and can be independently used with existing post-processing techniques (e.g., with finite elements).

Влияние верхнего граничного условия на гидротермическое поле земляного полотна железных дорог в районах вечной мерзлоты

Purpose: Based on numerical model that takes into account the hydrothermal interrelation of soil humidity and temperature, to carry out numerical modeling of hydrothermal field of railway subgrade in the area of permafrost spread. To study the influence degree of global warming and temperature annual amplitude on hydrothermal field of subgrade. To determine the need for reasonable and economical engineering measures to ensure the stability of subgrade. Method: Calculation of hydrothermal bonding of railroad subgrade has been performed in COMSOL Multiphysics software with the use of partial differential equation (PDE) module with the methods of finite elements. Different values of the parameters of subgrade upper boundary temperature function are specified and the effect of these parameters on subgrade hydrothermal field is studied. Results: Based on measured data temperature function, the influence of global warming effect and temperature annual amplitude on subgrade hydrothermal field was determined. Change law and characteristics of temperature distribution, ice content and unfrozen water in railway subgrade in permafrost areas after hydrothermal field stabilization has been obtained. The periods of maxima and minima onset for temperature as well as for ice content and unfrozen water are given. Practical significance: The use of given numerical model allows to predict effectively the tendency of subgrade hydrothermal field change in permafrost areas under global warming influence. Based on the numerical calculation results, relevant technical recommendations are shown for elimination or reduction of engineering accidents caused by temperature changes in permafrost regions.

Half-Space Model Problem for Navier-Lamé Equations with Surface Tension

Recently, we have seen the phenomena in use of partial differential equations (PDEs) especially in fluid dynamic area. The classical approach of the analysis of PDEs were dominated in early nineteenth century. As we know that for PDEs the fundamental theoretical question is whether the model problem consists of equation and its associated side condition is well-posed. There are many ways to investigate that the model problems are well-posed. Because of that reason, in this paper we consider the <img src=image/13426244_01.gif>-boundedness of the solution operator families for Navier-Lamé equation by taking into account the surface tension in a bounded domain of <img src=image/13426244_02.gif>-dimensional Euclidean space (<img src=image/13426244_02.gif>≥ 2) as one way to study the well-posedess. We investigate the <img src=image/13426244_01.gif>-boundedness in half-space domain case. The <img src=image/13426244_01.gif>-boundedness implies not only the generation of analytic semigroup but also the maximal <img src=image/13426244_03.gif> regularity for the initial boundary value problem by using Weis's operator valued Fourier multiplier theorem for time dependent problem. It was known that the maximal <img src=image/13426244_03.gif> regularity class is the powerful tool to prove the well-posesness of the model problem. This result can be used for further research for example to analyze the boundedness of the solution operators of the model problem in bent-half space or general domain case.

Early Detection and Classification Approach for Plant Diseases based on MultiScale Image Decomposition

This paper presents a new and powerful approach for detecting and classifying leaf diseases for plant diagnosis with high accuracy. The main contribution of this paper is that a hybrid approach is proposed by using the combination of Partial Differential Equations (PDE) based image decomposition, segmentation, feature extraction, features selection and classification aiming to improve the classification accuracy and provide an excellent diagnosis. The TV-L1 Total variation model is adopted to separate the original image into texture and object components. Segmentation will be done only on the object component. Then texture, color, vein and shape features are extracted and merged in a feature vector using the codebook method. Moreover, features are selected by the RelieF feature selection algorithm to keep only relevant ones. In the classification, only selected features will be used and passed to the Multiclass Support Vector Machine algorithm SVM. The proposed approach is implemented and tested on the PV Plant Village dataset and provided a good and greater classification accuracy compared with the existing approaches from the literature. The obtained results proved that the use of PDE influences on the segmentation, which in turn, allowed us to identify correctly the leaves and provide new and optimal features, those features improves the classification accuracy rate to 95.9%.

Coupling kinetic models and advection–diffusion equations. 1. Framework development and application to sucrose translocation and metabolism in sugarcane

AbstractThe sugarcane stalk, besides being the main structural component of the plant, is also the major storage organ for carbohydrates. Previous studies have modelled the sucrose accumulation pathway in the internodal storage parenchyma of sugarcane using kinetic models cast as systems of ordinary differential equations. To address the shortcomings of these models, which did not include subcellular compartmentation or spatial information, the present study extends the original models within an advection–diffusion–reaction framework, requiring the use of partial differential equations to model sucrose metabolism coupled to phloem translocation. We propose a kinetic model of a coupled reaction network where species can be involved in chemical reactions and/or be transported over long distances in a fluid medium by advection or diffusion. Darcy’s law is used to model fluid flow and allows a simplified, phenomenological approach to be applied to translocation in the phloem. Similarly, generic reversible Hill equations are used to model biochemical reaction rates. Numerical solutions to this formulation are demonstrated with time-course analysis of a simplified model of sucrose accumulation. The model shows sucrose accumulation in the vacuoles of stalk parenchyma cells, and is moreover able to demonstrate the upregulation of photosynthesis in response to a change in sink demand. The model presented is able to capture the spatio-temporal evolution of the system from a set of initial conditions by combining phloem flow, diffusion, transport of metabolites between compartments and biochemical enzyme-catalysed reactions in a rigorous, quantitative framework that can form the basis for future modelling and experimental design.

A 2D peridynamic model for fatigue crack initiation of railheads

Due to the use of PDEs (Partial Differential Equations), classical continuum mechanics is difficult to solve the problems of discontinuous deformation. Using integral equations, peridynamic theory can overcome this difficulty. A 2D method for predicting fatigue crack initiation of railheads was presented based on the peridynamic theory. First, the reasonable values of the model parameters were analyzed. Second, the accuracy of the method was verified through comparisons with the results of classical continuum mechanics and existing conclusions in the literature. Finally, the effects of wheel-rail rolling-sliding status, friction coefficient, and wheel load on fatigue crack initiation were analyzed and discussed.

An effective modification of finite element method for heat and mass transfer of chemically reactive unsteady flow

A modified technique of Galerkin finite element method is applied to investigate the boundary layer flow of mixed convection with effects of exothermic/endothermic constructive/destructive chemical reaction and heat source. Model of the present flow problem is constructed with the use of partial differential equations and the transformed nonlinear boundary value problems are worked out numerically by the modified technique of FEM. In order to overcome the deficiency of accuracy in standard finite element method with polynomial interpolation for computing derivative numerically, finite difference formulas are coupled with FEM for computing derivatives accurately. Feature of the modified technique is tabulated for findings of the errors in derivatives and numerical values of skin friction coefficient, Nusselt number, and Sherwood number. Present coupling of finite element method with finite difference formulas can be implemented to find continuous derivatives more accurately than the standard FEM with polynomial interpolation which can also balance one of the extra advantages in applying FEM with B-splines. Comparison is made for the obtained results from present coupled technique of FEM with Matlab built in solver “bvp4c.” It is also shown that the temperature profile behaved reversely and the concentration profile did not affect exothermic/endothermic reactions but has opposite behavior for constructive and destructive chemical reactions. It can also be deducted that the velocity profile decreases for the cooled plate (for positive Grashof number) and behaves reversely for the heated plate (for negative Grashof number).

3D scene reconstruction of landslide topography based on data fusion between laser point cloud and UAV image

The main topic of this research was to propose an optimizing modeling method of landslide topography. This method fuses high-resolution imagery from an unmanned aerial vehicle (UAV) with laser point cloud. Due to the relatively high efficiency and large coverage area, the UAV system is suitable for emergency investigation of landslide. Based on the UAV images and laser point cloud, the use of partial differential equation (PDE) combines the hole repair technology with non-uniform rational B-splines (NURBS) method to rapidly reconstruct a three-dimensional (3D) scene. The mean square error (MSE) determined by comparison with measured data from the laser scanner technology is 16.1274 mm. And the error of plane calculated by the NURBS surface fitting is smallest by comparing with the least-squares plane fitting, Delaunay triangulation and polygon mesh method. In addition, the realistic 3D landslide model is obtained by texture mapping. The XiaPa landslide in Liangshan District, Xichang City, Sichuan Province China, was selected as a case study, in which the accuracy is evaluated to prove that the model has higher precision than traditional methods. The experiment results show that the integrated approach in this paper could be implemented in qualitative and quantitative risk assessment procedures pertaining to landslides.

Automated Calculation of Higher Order Partial Differential Equation Constrained Derivative Information

Developments in automated code generation have allowed extremely compact representations of numerical models and for associated adjoint models to be derived automatically via high level algorithmic differentiation. In this article these principles are extended to enable the calculation of higher order derivative information. The higher order derivative information is computed through the automated derivation of tangent-linear equations, which are then treated as new forward equations, and from which higher order tangent-linear and adjoint information can be derived. The principal emphasis is on the calculation of partial differential equation constrained Hessian actions, but the approach generalizes for derivative information at arbitrary order. The derivative calculations are further combined with an advanced data checkpointing strategy. Applications which make use of partial differential equation constrained Hessian actions are presented.

Optimization of Complex Geometry Using Tenth Order Partial Differential Equation

This paper presents an efficient and intuitive technique of shape parameterization for design optimization using a partial differential equation (PDE) of order ten. It shows how the choice of two introduced parameters can enable one to parameterize complex geometries. With the use of PDE based formulation, it is shown in this paper how the shape can be defined and manipulated on the basis of parameterization and the boundary value approaches by which complex shapes can be created. Further the boundary conditions which are used in it are a boundary and an intermediate curves for defining the shape. This technique allows complex shapes to be parameterized intuitively using a very small set of design parameters. Hence, Practical design optimization of problems becomes more achievable by applying PDE based approach of shape parameterization by incorporating standard numerical optimization techniques.

Thermoelectric converters with optimum graded materials and current distribution in one dimension and three dimensions

The description of TE‐converter performance with inhomogeneous material distribution is based on the classical Onsager–deGroot–Callen transport theory. A variational approach for performance maximization is generalized from previous 1D heat‐ and electric current distribution to general 3D devices by use of partial differential equations. The concept of multi‐stage (cascaded) TE‐converters in 1D is generalized to arbitrarily distributed electric current sources in the 3D case by introducing an extended version of the Onsager–deGroot–Callen theory. Originally insufficient material properties together with optimized current distribution allow for maximum performance with respect to a general distribution function of the thermoelectric figure of merit Z(x, T) depending simultaneously on 3D position x and local temperature T. Thermally isolating as well as non‐isolating boundary‐conditions are treated. Investigation of hitherto unsolved problems of the 1D theory leads to a new analytical solution of maximized generated electric power for optimized grading. In previous literature analytical formulas had been established for 1D TE‐efficiencies. For the outstanding maximization problem of cooling power improved results are obtained by fast inversion of tridiagonal matrices compared to previous findings with differential equation initial value algorithms. An analytic calculation is possible by optimizing the temperature extremum location in a piecewise linear profile. By its construction a close lower bound to the exact cooling power is formed, which is possibly identical to the exact value.

An irreversible investment problem with maintenance expenditure on a finite horizon: Free boundary analysis

This paper concerns a continuous-time, finite horizon, optimal irreversible investment problem with maintenance expenditure of a firm under uncertainty. We assume that the firm can make irreversible investments to expand its production capacity and spend maintenance expenditure to achieve better performance of the productivity. The objective of the firm is to construct optimal investment and maintenance policies to maximize its expected total profit over a finite horizon. Mathematically, it is a singular stochastic control problem whose value function satisfies a parabolic variational inequality with gradient constraint. The problem gives rise to two free boundaries which stand for the optimal investment and maintenance strategies, respectively. We investigate behaviors of free boundaries, study regularities of the value function, and give optimal investment and maintenance policies. As we know, this is a first integral result for an investment–maintenance problem with a finite time horizon due to use of partial differential equation (PDE) technique.

Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models 2014

Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models 2014

Chimera: a hybrid approach to numerical loop quantum cosmology

The existence of a quantum bounce in isotropic spacetimes is a key result in loop quantum cosmology (LQC), which has been demonstrated to arise in all the models studied so far. In most of the models, the bounce has been studied using numerical simulations involving states which are sharply peaked and which bounce at volumes much larger than the Planck volume. An important issue is to confirm the existence of the bounce for states which have a wide spread, or which bounce closer to the Planck volume. Numerical simulations with such states demand large computational domains, making them very expensive and practically infeasible with the techniques which have been implemented so far. To overcome these difficulties, we present an efficient hybrid numerical scheme using the property that at the small spacetime curvature, the quantum Hamiltonian constraint in LQC, which is a difference equation with uniform discretization in volume, can be approximated by a Wheeler–DeWitt differential equation. By carefully choosing a hybrid spatial grid allowing the use of partial differential equations at large volumes, and with a simple change of geometrical coordinate, we obtain a surprising reduction in the computational cost. This scheme enables us to explore regimes which were so far unachievable for the isotropic model in LQC. Our approach also promises to significantly reduce the computational cost for numerical simulations in anisotropic LQC using high performance computing.

Automatic shape optimisation of pharmaceutical tablets using Partial Differential Equations

Pharmaceutical tablets have been the most dominant form for drug delivery and most of them are used in the oral administration of drugs. These tablets need to be strong enough so that they can tolerate external stresses. Hence, during the design process, it is important to produce tablets with maximum mechanical strength while conserving the properties of powder form components. The mechanical properties of these tablets are assessed by measuring the tensile strength, which is commonly measured using diametrical or axial compression tests.This work describes the parametric design and optimisation of solid pharmaceutical tablets in cylindrical and spherical shapes, which are obtained using a formulation based on the use of Partial Differential Equations (PDEs) for shape design. The PDE-based formulation is capable of parameterised complex shapes using the information from some boundary curves that describe the shape. It is shown that the optimal design of pharmaceutical tablets with a particular volume and maximum strength can be obtained using an automatic design optimisation which is performed by combining the PDE method and a standard method for numerical optimisation.

The holonomic gradient method for the distribution function of the largest root of a Wishart matrix

We apply the holonomic gradient method introduced by Nakayama et al. (2011) [23] to the evaluation of the exact distribution function of the largest root of a Wishart matrix, which involves a hypergeometric function 1F1 of a matrix argument. Numerical evaluation of the hypergeometric function has been one of the longstanding problems in multivariate distribution theory. The holonomic gradient method offers a totally new approach, which is complementary to the infinite series expansion around the origin in terms of zonal polynomials. It allows us to move away from the origin by the use of partial differential equations satisfied by the hypergeometric function. From the numerical viewpoint we show that the method works well up to dimension 10. From the theoretical viewpoint the method offers many challenging problems both to statistics and D-module theory.