Form Of Differential Equations Research Articles

Система моніторингу забруднення повітря автотранспортом на базі автономного аеромобільного вимірювального комплексу

The article proposes an approach to constructing a system of complex and uninterrupted monitoring of harmful emissions of motor vehicles into the air. The architecture of the environmental monitoring system for measuring and forecasting the distribution of pollutant concentrations in motor vehicle exhaust gases, among which mainly CO, SO₂, NO₂, and СО₂, is presented. The mobile information and measurement complex Sniffer4D Hyper-local Air Quality Analyzer and a charging station based on solar batteries are used as the hardware. For modeling and forecasting the distribution of concentrations of harmful emissions, mathematical models of the dynamics of the distribution of concentrations of pollutants due to harmful emissions in the exhaust gases of motor vehicles are proposed in the form of differential equations that are analogs of differential equations in partial derivatives, as models of turbulent diffusion and interval models of the distribution of the background level of pollution concentration in the form of nonlinear algebraic equations. Implemented software for data collection, processing (model learning and prediction), and visualization.

Система моніторингу забруднення повітря автотранспортом на базі автономного аеромобільного вимірювального комплексу

The article proposes an approach to constructing a system of complex and uninterrupted monitoring of harmful emissions of motor vehicles into the air. The architecture of the environmental monitoring system for measuring and forecasting the distribution of pollutant concentrations in motor vehicle exhaust gases, among which mainly CO, SO₂, NO₂, and СО₂, is presented. The mobile information and measurement complex Sniffer4D Hyper-local Air Quality Analyzer and a charging station based on solar batteries are used as the hardware. For modeling and forecasting the distribution of concentrations of harmful emissions, mathematical models of the dynamics of the distribution of concentrations of pollutants due to harmful emissions in the exhaust gases of motor vehicles are proposed in the form of differential equations that are analogs of differential equations in partial derivatives, as models of turbulent diffusion and interval models of the distribution of the background level of pollution concentration in the form of nonlinear algebraic equations. Implemented software for data collection, processing (model learning and prediction), and visualization.

Energy transport features of Oldroyd-B nanofluid flow over bidirectional stretching surface subject to Cattaneo–Christov heat and mass fluxes

By considering the Cattaneo–Christov approach and Buongiorno’s model, the thermal transport model is formulated for the flow of Oldroyd-B nanofluid over a bidirectional stretching surface. The flow profile of Oldroyd-B nanofluid is examined for various physical parameters, and the effects of heat source/sink are also utilized to explore the thermal transport properties subject to thermal relaxation time. Governing mathematical models are developed on the basis of basic laws and presented in the form of Partial differential equations (PDEs). The governing partial differential equations are transformed into ordinary differential equations considering suitable dimensionless transformations. The homotopic method is applied to study the feature of heat and velocity components in fluid flow. The influence of each physical parameter over the thermal and concentration profile is displayed graphemically. It is noticed that thermal transport is decreasing with increment in thermal relaxation time. The mass transfer becomes weak with magnifying values of the stretching strength parameter. Moreover, the larger thermophoretic parameter regulates the heat transfer during fluid flow.

Formula omitted]-Laplace transforms and [formula omitted]-differential equations of the arbitrary-order, theory and applications

This paper provides a framework to study a class of arbitrary-order uncertain differential equations known as arbitrary-order Z+-differential equations. For this purpose, we first present the parametric form of Z+-numbers. Then, we introduce the basic algebraic operations on Z+-numbers, including addition, scalar multiplication, and Hukuhara difference. Definitely, these operations lead us to define the Z+-valued function. Afterward, the limit and continuity concepts of a Z+-valued function are provided under the definition of a metric on the space of Z+-numbers. Furthermore, the concepts of Z+-differentiability, Z+-integral, and Z+-Laplace transform with the convergence theorem for the Z+-valued function and its nth-order derivatives are introduced in detail. Considering all these concepts, a Z+-differential equation (Z+DE) can be expressed in the form of a bimodal differential equation combining a fuzzy initial value problem (FIVP) and a random differential equation (RDE). To this end, we use a combination of “FIVP under strongly generalized Hukuhara differentiability (SGH-differentiability)” and “random differential equation under mean-square differentiability (ms-differentiability)” to define the nth-order differential equations with Z+-number initial values. Further, the existence and uniqueness of the Z+-differential equations are examined by presenting several theorems. Finally, the effectiveness of the approaches is illustrated by solving two examples.

Electromagnetic Trihybrid Ellis Nanofluid Flow Influenced with a Magnetic Dipole and Chemical Reaction Across a Vertical Surface.

The purpose of this study is to evaluate the augmentation of thermal energy transfer in trihybrid Ellis nanofluid flow in the occurrence of magnetic dipole passes over a vertical surface. The ternary hybrid nanofluid is prepared by the dispersion of ternary nanoparticles (Al2O3, SiO2, and TiO2) in the Carreau Yasuda fluid. The velocity and heat transportation has been examined in the existence of the Darcy Forchhemier influence and heat source/sink. The phenomena of fluid flow have been mathematically designed for energy and fluid velocity in the form of a nonlinear partial differential equation (PDE)-based system. The system of PDEs is further refined to the set of ordinary differential equations via suitable similarity substitutions. The acquired dimensionless equations are numerically solved with the help of the HAM. It has been noticed that the energy contour is enhanced versus the variation of viscous dissipation and heat generation. A significant contribution of a magnetic dipole is observed to elevate the production of the thermal energy field, and an opposite trend is noticed versus the flow profile. The accumulation of Al2O3, SiO2, and TiO2 nanomaterials in the base fluid “engine oil” improves the velocity and energy profiles.

ON RITZ APPROXIMATION FOR A CLASS OF FRACTIONAL OPTIMAL CONTROL PROBLEMS

We apply the Ritz method to approximate the solution of optimal control problems through the use of polynomials. The constraints of the problem take the form of differential equations of fractional order accompanied by the boundary and initial conditions. The ultimate goal of the algorithm is to set up a system of equations whose number matches the unknowns. Computing the unknowns enables us to approximate the solution of the objective function in the form of polynomials.

Equilibrium Manipulation Planning for a Soft Elastic Rod Considering an External Distributed Force and Intrinsic Curvature

Previous work has shown that the set of equilibrium shapes of a rod is a smooth 6-dimensional manifold. We develop the Kirchhoff rod model in a differential equation form using Darboux vector that easily adds distributed force and intrinsic curvature. It can be seen that in this letter, the manifold of the equilibrium set is the same. Furthermore, we address the rod manipulation planning problem which has the certain point on rod following a predefined spatial track. We propose a novel rod manipulation planning algorithm that considers the spatial configuration distance between the current and the goal, and follows its quickest descent direction in each planning step. The experiments show that it has a much higher successful rate than straight-line path planning in the Euclidean space of the gripper and significantly less planning time than the commonly used sampling method. The hardware experiments are also conducted on a platform that consists of a manipulator, an F/T sensor and two cameras. We choose soft rubber rods in the experiments, and the results validate the proposed model and methods.

A fast solver framework for acoustic hybrid integral equations

Reliable modeling of the scattering by acoustically large and geometrically complex objects can be achieved by means of subdomain-dependent problem formulation and a numerically rigorous solution. While the objects’ inhomogeneity has driven the development of differential equation formulations and solvers, integral equation formulations, where the object’s background is modeled via a Green’s function, are advantageous for unbounded domains. In the hybrid integral equations approach (Usner et al., 2006), the interaction of separate subdomains with external fields is described by pertinent integral equations. Their Galerkin discretization leads to a dense blocked stiffness matrix. The development of compressed representations of the matrix, which are necessary for the treatment of large systems, becomes non-trivial due to the multitude of integral equation kernels and the different geometrical and physical characteristic of the subdomains. As part of the development of a fast hybrid integral equation solver framework, we consider the case of objects composed of large inhomogeneous volumes, modeled as incompressible fluids, and of simplified solids, modeled via surface integral equations. A hybrid integral equation formulation is derived and solved numerically. The iterative solution is accelerated by employing the butterfly-compressed hierarchical representation of the stiffness matrix, recently used for acoustic volume integral equations (Kaplan, 2022).

A data-driven digital twin for water ultrafiltration

Membrane-based separations are proven and useful industrial-scale technologies, suitable for automation. Digital twins are models of physical dynamical systems which continuously couple with data from a real world system to help understand and control performance. However, ultrafiltration and microfiltration membrane separation techniques lack a rigorous theoretical description due to the complex interactions and associated uncertainties. Here we report a digital-twin methodology called the Stochastic Greybox Modelling and Control (SGMC) that can account for random changes that occur during the separation processes and apply it to water ultrafiltration. In contrast to recent probabilistic approaches to digital twins, we use a physically intuitive formalism of stochastic differential equations to assess uncertainties and implement updates. We demonstrate the application of our digital twin model to control the filtration process and minimize the energy use under a fixed water volume in a membrane ultrafiltration of artificially simulated lakewater. The explicit modelling of uncertainties and the adaptable real-time control of stochastic physical states are particular strengths of SGMC, which makes it suited to real-world problems with inherent unknowns.

Hydrothermal features of the magnetite nanoparticles on natural convection flow through a square conduit by using the finite element method

Ferrofluids are made through the suspension of magnetic nanoparticles which are commonly used for the treatment of hyperthermia, malignant tumor treatment, magnetic cell separation, etc. These nanoparticles provide satisfactory results for the heat transport phenomena. Motivated by the applications of these nanoparticles, this study is performed for water-based nanofluid with a different type of magnetic nanoparticles for renewable energy and the development of the advanced cooling process of the radiator. Further, this study also talks about the impact of magnetized nanoparticles on natural convection flow occupied in a square cavity. The nanosized magnetic particles are mixed up in water to make a more convective flow. In this computational study, the momentum equation is updated with magnetohydrodynamics terms. The mathematical problem is achieved in the form of nonlinear complex partial differential equations which are simulated by using the renowned Galerkin finite element technique. The numerical code is validated with the previous study on the natural convection flow of viscous fluid in a square cavity and the verification procedure verified the good accuracy of the applied developed numerical code. The impact of the Hartmann number, Rayleigh number and the volume friction coefficient is discussed through contours and graphs. It is observed that nanofluids have more capacity to store energy as compared to regular fluids due to superior thermal transport properties. Moreover, the cobalt oxide (Co3[Formula: see text] nanoparticles provide a greater heat transfer rate due to greater thermal conductivity as compared to other nanoparticles cobalt ferrite (CoFe2[Formula: see text], magnetite (Fe3[Formula: see text] and manganese–zinc–ferrite (Mn–Zn–Fe2[Formula: see text]. The heat transfer rate is increased by 30% for cobalt oxide, 18% for magnetite, 15% for manganese–zinc–ferrite and 12% for cobalt ferrite, respectively. Hence, the cobalt oxide nanoparticles which have a greater heat transfer rate can contribute to solar energy engineering and the advanced cooling process of the radiator.

Evolutionary Approaches to the Identification of Dynamic Processes in the Form of Differential Equations and Their Systems

Evolutionary approaches are widely applied in solving various types of problems. The paper considers the application of EvolODE and EvolODES approaches to the identification of dynamic systems. EvolODE helps to obtain a model in the form of an ordinary differential equation without restrictions on the type of the equation. EvolODES searches for a model in the form of an ordinary differential equation system. The algorithmic basis of these approaches is a modified genetic programming algorithm for finding the structure of ordinary differential equations and differential evolution to optimize the values of numerical constants used in the equation. Testing for these approaches on problems in the form of ordinary differential equations and their systems was conducted. The influence of noise present in the data and the sample size on the model error was considered for each of the approaches. The symbolic accuracy of the resulting equations was studied. The proposed approaches make it possible to obtain models in symbolic form. They will provide opportunities for further interpretation and application.

A stochastic mathematical model for coupling the cytosolic and sarcoplasmic calcium movements in diseased cardiac myocytes

Several computational studies have been undertaken to explore the Ca2+-induced Ca2+ release (CICR) events in cardiac myocytes and along with experimental studies it has given us invaluable insight into the mechanism of CICR from spark/blink initiation to termination and regulation, and their interplay under normal and pathological conditions. The computational modelling of this mechanism has mainly been investigated using coupled differential equations (DEs). However, there is a lack of computational investigation into (1) how the different formulation of coupled DEs capture the Ca2+ movement in the cytosol and sarcoplasmic reticulum (SR), (2) the buffer and dye inclusion in both compartments, and (3) the effect of buffer and dye properties on the calcium behaviour. This work is set out to explore (1) the effect of different coupled formulation of DEs on spark/blink occurrence, (2) the inclusion of improved sarcoplasmic buffering properties, and (3) the effects of cytosolic and sarcoplasmic dye and buffer properties on Ca2+ movement. The simulation results show large discrepancies between different formulations of the governing equations. Additionally, extension of the model to include sarcoplasmic buffering properties show normalised fluorescent dye profiles to be in good agreement with experimental and amongst its one- and two-dimensional representations.

Flow and sensitivity analysis of viscoelastic material containing nanoparticles over a stretching sheet with Ohmic and viscous dissipation

The effects of Joule heating and viscous dissipation in the MHD flow of a viscoelastic (second-grade) fluid past a stretching sheet in the presence of metallic nanoparticles are studied. Copper nanoparticles are suspended in second-grade fluid with kerosene, water and polyvinly alcohol (PVA) that are supposed to be the base fluids. The mathematical equations that oversee the flow and heat transfer fields have been developed in the form of partial differential equations that are subsequently converted into a system of non-linear ordinary differential equations using the appropriate similarity transformation. The obtained equations are numerically solved and the impacts of different types of base fluids, solid volume fraction, magnetic number and Eckert number on the flow, temperature field, skin friction and Nusselt number are graphically analyzed. A sensitivity analysis is also utilized to identify the impact of involved parameters on skin friction and local Nusselt number. To predict the uncertainty caused by the involved parameters on skin friction and local Nusselt number, the Response Surface Methodology (RSM) is used. It is found that the volume fraction of nanoparticles does not cause uncertainty on skin friction, while the viscoelastic and magnetic number shows positive sensitivity at various levels. Also, Eckert number causes uncertainty on Nusselt number for lower values of the magnetic parameter.

Theoretical analysis of the MHD flow of a tangent hyperbolic hybrid nanofluid over a stretching sheet with convective conditions: A nonlinear thermal radiation case

The objective of this research is to examine the two-dimensional flow of a non-Newtonian water-based hybrid nanofluid containing silver and alumina nanoparticles over a stretching surface with convective conditions. The flow is considered to be steady, laminar and incompressible. The mathematical modeling of the flow analysis is considered in the form of partial differential equations which are then transformed into ordinary differential equations by mean of suitable similarity transformations. The present problem is solved with the help of semi-analytical technique called HAM. Convergence of HAM is shown with the help of Figure. The impacts of embedded parameters like magnetic parameter, Joule heating, activation energy and nonlinear thermal radiation on the flow profiles are displayed with the help of Figures and Tables. Our results showed that the greater magnetic parameter reduced the velocity profile, while a reverse impact magnetic parameter is found against temperature profile, skin frication and heat transfer rate of the hybrid nanofluid. Concentration profiles of the hybrid nanofluid flow are augmented with the increasing activation energy while reduced against chemical reaction parameter. Sherwood number has shown increasing behavior due to larger value of Schmidt number and decreasing on larger value of temperature relative parameter.

Memory response in dual-phase-lag thermoelastic medium due to instantaneous heat source

The present work deals with thermoelastic interactions in an isotropic infinite medium with memory-dependent derivative under a dual-phase-lag model. The governing equations are expressed in vector–matrix differential equation (VMDE) form in the Laplace–Fourier transform domain and solved using the eigenvalue approach. Numerical estimations of different thermophysical quantities such as displacement, temperature, and stresses are presented graphically for a coper material by employing the Gaussian quadrature formula and Honig–Hirdes method. Several remarkable points have been mentioned in the graphical representation of the field variables for various kernel functions, time-delay parameters, and magnitude of instantaneous heat source.

New Numerical Approach of Solving Highly Nonlinear Fractional Partial Differential Equations via Fractional Novel Analytical Method

In this work, the fractional novel analytic method (FNAM) is successfully implemented on some well-known, strongly nonlinear fractional partial differential equations (NFPDEs), and the results show the approach’s efficiency. The main purpose is to show the method’s strength on FPDEs by minimizing the calculation effort. The novel numerical approach has shown to be the simplest technique for obtaining the numerical solution to any form of the fractional partial differential equation (FPDE).

A Comprehensive Review on Several Techniques for Solving 2D and 3D Partial Differential Equations

Many Physical phenomenon in Sciences and engineering are modeled in the form of Partial differential equations. Partial differential equations arise in formulation of many mathematical problems may be of one dimensional, two dimensional, three dimensional etc. In this paper, various types of Partial differential equations and methods available in literature for solving two dimensional and three dimensional Partial differential equations are discussed.

Data-driven discovery of reduced plasma physics models from fully kinetic simulations

At the core of some of the most important problems in plasma physics---from controlled nuclear fusion to the acceleration of cosmic rays---is the challenge to describe nonlinear, multiscale plasma dynamics. The development of reduced plasma models that balance between accuracy and complexity is critical to advancing theoretical comprehension and enabling holistic computational descriptions of these problems. Here we report the data-driven discovery of accurate reduced plasma models, in the form of partial differential equations, directly from first-principles particle-in-cell simulations. We achieve this by using an integral formulation of sparsity-based model-discovery techniques and show that this is crucial to robustly identify the governing equations in the presence of discrete particle noise. We demonstrate the potential of this approach by recovering the fundamental hierarchy of plasma physics models---from the Vlasov equation to magnetohydrodynamics. Our findings show that this data-driven methodology offers a promising route to accelerate the development of reduced theoretical models of complex nonlinear plasma phenomena and to design computationally efficient algorithms for multiscale plasma simulations.

Explicit Calculation of Structural Commutation Relations for Stochastic and Dynamical Graph Grammar Rule Operators in Biological Morphodynamics.

Many emergent, non-fundamental models of complex systems can be described naturally by the temporal evolution of spatial structures with some nontrivial discretized topology, such as a graph with suitable parameter vectors labeling its vertices. For example, the cytoskeleton of a single cell, such as the cortical microtubule network in a plant cell or the actin filaments in a synapse, comprises many interconnected polymers whose topology is naturally graph-like and dynamic. The same can be said for cells connected dynamically in a developing tissue. There is a mathematical framework suitable for expressing such emergent dynamics, "stochastic parameterized graph grammars," composed of a collection of the graph- and parameter-altering rules, each of which has a time-evolution operator that suitably moves probability. These rule-level operators form an operator algebra, much like particle creation/annihilation operators or Lie group generators. Here, we present an explicit and constructive calculation, in terms of elementary basis operators and standard component notation, of what turns out to be a general combinatorial expression for the operator algebra that reduces products and, therefore, commutators of graph grammar rule operators to equivalent integer-weighted sums of such operators. We show how these results extend to "dynamical graph grammars," which include rules that bear local differential equation dynamics for some continuous-valued parameters. Commutators of such time-evolution operators have analytic uses, including deriving efficient simulation algorithms and approximations and estimating their errors. The resulting formalism is complementary to spatial models in the form of partial differential equations or stochastic reaction-diffusion processes. We discuss the potential application of this framework to the remodeling dynamics of the microtubule cytoskeleton in cortical microtubule networks relevant to plant development and of the actin cytoskeleton in, for example, a growing or shrinking synaptic spine head. Both cytoskeletal systems underlie biological morphodynamics.

Strength Analysis of Two-Dimensional Ceramic-Matrix-Composite Guide Vane Based on Bimodulus Constitutive Model

The challenges related to the structural design of the silicon carbide fiber-reinforced/ silicon carbide ceramic-matrix-composite (CMC) aeroengine turbine guide vanes have been analyzed in this study. This work carries complete novelty research on the strength analysis of the CMC guide vane based on bimodulus constitutive relation, including structural design, temperature computation, advanced constitutive equation application, and stress analysis. Based on tensile and compressive tests, the bimodulus properties of two-dimensional CMC material was given, and a novelty bimodulus constitutive model was defined by stress or strain: both of which were developed to achieve a finite element method. Applying the bimodulus constitutive model in a uniaxis compressive test and beam bending model, the different forms of constitutive equations were compared. Then, the bimodulus constitutive equation was used to analyze the stress of the guide vane. Considering the weaving technique of preforms and manufacturing processes, this study discussed the manufacturing feasibility of the CMC guide vane, where the detailed structures were totally duplicated by a metal guide vane. After giving the weaving structure of the guide vane, a new algorithm and plug-in were developed to achieve the assignment of the material’s coordinate system in the guide vane finite element model. By computing the temperature field with a coupling fluid and flow transfer model, the accurate temperature was considered as an important premise for accurate stress analysis. The results indicated that the connection region between the platform and airfoil was vulnerable as a result of stress concentration. Stress analysis by nonlinear constitutive model and guide vane thermal load test got similar conclusion to verify the applicability of bimodulus consititutive equation.