First-order Differential Equations Research Articles

Defining hybrid hierarchical models in πHYFLOW

This paper presents πHyFlow, a modular approach to the process interaction worldview (PI). Traditionally, PI supports a set of interacting processes without enabling modular and hierarchical model definition. πHyFlow basic models define a dynamic set of processes while also enabling a modular interface for supporting model composition. πHyFlow allows an exact representation of continuous signals based on the concepts of dense outputs, and generalized sampling. πHyFlow ability to support accurate models of hybrid systems is presented through a DC-DC converter (DCC) based on a resistor–capacitor (RC) electrical circuit. The DCC output voltage can be modeled by a first-order differential equation that is solved using an exponential time differencing (ETD) integrator. The DCC network model uses a digital on-off controller and an ETD. Simulation results are generated by πHyFlow++, a C++ implementation of the formalism.

An encompassed representation of timescale hierarchies in first-order reaction network

Complex networks are pervasive in various fields such as chemistry, biology, and sociology. In chemistry, first-order reaction networks are represented by a set of first-order differential equations, which can be constructed from the underlying energy landscape. However, as the number of nodes increases, it becomes more challenging to understand complex kinetics across different timescales. Hence, how to construct an interpretable, coarse-graining scheme that preserves the underlying timescales of overall reactions is of crucial importance. Here, we develop a scheme to capture the underlying hierarchical subsets of nodes, and a series of coarse-grained (reduced-dimensional) rate equations between the subsets as a function of time resolution from the original reaction network. Each of the coarse-grained representations guarantees to preserve the underlying slow characteristic timescales in the original network. The crux is the construction of a lumping scheme incorporating a similarity measure in deciphering the underlying timescale hierarchy, which does not rely on the assumption of equilibrium. As an illustrative example, we apply the scheme to four-state Markovian models and Claisen rearrangement of allyl vinyl ether (AVE), and demonstrate that the reduced-dimensional representation accurately reproduces not only the slowest but also the faster timescales of overall reactions although other reduction schemes based on equilibrium assumption well reproduce the slowest timescale but fail to reproduce the second-to-fourth slowest timescales with the same accuracy. Our scheme can be applied not only to the reaction networks but also to networks in other fields, which helps us encompass their hierarchical structures of the complex kinetics over timescales.

A one-dimensional computational model for blood flow in an elastic blood vessel with a rigid catheter.

Strokes are one of the leading causes of death in the United States. Stroke treatment involves removal or dissolution of the obstruction (usually a clot) in the blocked artery by catheter insertion. A computer simulation to systematically plan such patient-specific treatments needs a network of about 105 blood vessels including collaterals. The existing computational fluid dynamic (CFD) solvers are not employed for stroke treatment planning as they are incapable of providing solutions for such big arterial trees in a reasonable amount of time. This work presents a novel one-dimensional mathematical formulation for blood flow modeling in an elastic blood vessel with a centrally placed rigid catheter. The governing equations are first-order hyperbolic partial differential equations, and the hypergeometric function needs to be computed to obtain the characteristic system of these hyperbolic equations. We employed the Discontinuous Galerkin method to solve the hyperbolic system and validated the implementation by comparing it against a well-established 3D CFD solver using idealized vessels and a realistic truncated arterial network. The results showed clinically insignificant differences in steady flow cases, with overall variations between 1D and 3D models remaining below 10%. Additionally, the solver accurately captured wave reflection phenomena at domain discontinuities in unsteady cases. A primary advantage of this model over 3D solvers is its ease in obtaining a discretized geometry of complex vasculatures with multiple arterial branches. Thus, the 1D computational model offers good accuracy and applicability in simulating complex vasculatures, demonstrating promising potential for investigating patient-specific endovascular interventions in strokes.

New oscillation results for first-order nonlinear difference equations with retarded arguments

New oscillation results for first-order nonlinear difference equations with retarded arguments

Bogomol’nyi-like equations in gravity theories

Using the Bogomol’nyi–Prasad–Sommerfield Lagrangian method, we show that gravity theory coupled to matter in various dimensions may possess Bogomol’nyi-like equations, which are first-order differential equations, satisfying the Einstein equations and the Euler–Lagrange equations of classical fields (U(1) gauge and scalar fields). In particular we consider static and spherically symmetric solutions by taking proper ansatzes and then we find an effective Lagrangian density that can reproduce the Einstein equations and the Euler–Lagrange equations of the classical fields. We consider the BPS Lagrangian density to be linear function of first-order derivative of all the fields. From these two Lagrangian desities we are able to obtain the Bogomol’nyi-like equations whose some of solutions are well-known such as Schwarzschild, Reissner–Nordström, Tangherlini black holes, and the recent black holes with scalar hair in three dimensions (Phys. Rev. D 107, 124047). Using these Bogomol’nyi-like equations, we are also able to find new solutions for scalar hair black holes in three and four dimensional spacetime. Furthermore we show that the Bogomol’nyi–Prasad–Sommerfield Lagrangian method can provide a simple alternative proof of black holes uniqueness theorems in any dimension.

Bending of Lloyd’s mirror to eliminate the period chirp in the fabrication of diffraction gratings

We present a new technique to prevent the detrimental period chirp that appears in optical gratings fabricated by laser interference lithography (LIL). The idea is to bend the Lloyd’s mirror in the lithographic setup to eliminate the period chirp already at the step of the grating’s exposure. A new mathematical model was developed to describe the required bending geometry of the mirror. It is shown that this geometry can be described by multiple cross-sections of the mirror, each obtained by the solution of an implicit first-order differential equation. The proposed approach is illustrated on the basis of a concrete example. By slightly bending the Lloyd’s mirror (by ≈ 3.5 mm of maximum deflection over an area of 142 mm × 215 mm) the period chirp of the exposed grating can be eliminated completely.

The non-first-order-factorizable contributions to the three-loop single-mass operator matrix elements [formula omitted] and [formula omitted]

The non-first-order-factorizable contributions1 to the unpolarized and polarized massive operator matrix elements to three-loop order, AQg(3) and ΔAQg(3), are calculated in the single-mass case. For the F12-related master integrals of the problem, we use a semi-analytic method based on series expansions and utilize the first-order differential equations for the master integrals which does not need a special basis of the master integrals. Due to the singularity structure of this basis a part of the integrals has to be computed to O(ε5) in the dimensional parameter. The solutions have to be matched at a series of thresholds and pseudo-thresholds in the region of the Bjorken variable x∈]0,∞[ using highly precise series expansions to obtain the imaginary part of the physical amplitude for x∈]0,1] at a high relative accuracy. We compare the present results both with previous analytic results, the results for fixed Mellin moments, and a prediction in the small-x region. We also derive expansions in the region of small and large values of x. With this paper, all three-loop single-mass unpolarized and polarized operator matrix elements are calculated.

Existence conditions for 2-periodic solutions to a non-homogeneous differential equations with piecewise constant argument

This paper provides a method of finding 2-periodical solutions for the first-order non-homogeneous differential equations with piecewise constant arguments. All existence conditions are described for 2-periodical solutions and obtained explicit formula for these solutions. An example for the problem that has infinitely many solutions is constructed.

General Solution to a Nonlocal Linear Differential Equation of First-Order

General Solution to a Nonlocal Linear Differential Equation of First-Order

Modeling Radioactive Decay Series Phenomenon Involving First-Order Differential Equation and Visualizing Solutions Using Open-Source Software XCOS

Numerous physical and chemical processes can be accurately described by first-order differential equations and can be solved through analytical methods. The solutions to these equations necessitate the specification of initial or boundary conditions. In the realm of physics and chemistry education, the ability to visualize the solutions of differential equations and comprehend their implications for physical/chemical changes holds significant importance. XCOS, an open-source software tool, serves as a valuable resource for both solving differential equations and graphically representing their solutions. Furthermore, XCOS aids in tackling nonlinear differential equations for which analytical solutions are often elusive. This paper delves into radioactive decay and decay series reactions by first-order differential equations. The paper explores the successful resolution and visualization of the phenomenon involving differential equations through XCOS. This approach extends to the manipulation of various parameters within the equations, further fostering inquiry-based learning.

Existence Results for Positive Periodic Solutions to First-Order Neutral Differential Equations

Existence Results for Positive Periodic Solutions to First-Order Neutral Differential Equations

Localized structures in three-field models: Geometrically constrained configurations and the first-order framework

This work deals with models described by three real scalar fields in one spatial dimension. We study the case where two of the three fields engender kinematical modifications, which respond as geometrical constrictions, that can be used to change the center and/or the tails of the kinklike configurations. An important advantage of our procedure is the construction of a method for the obtention of first-order differential equations that solve the equations of motion and give rise to stable localized structures. We illustrate the general procedure investigating several distinct examples, and suggesting some possibilities of applications of practical use, in particular, to the case of domain walls and skyrmions in magnetic material, collisions of kinks and to deal with braneworld scenarios having a warped five-dimensional anti de Sitter geometry, with a single extra dimension of infinite extent.

Compact modeling of hysteresis in organic thin-film transistors

In this work, we propose a model that describes the temporal evolution of the threshold voltage and trapped charge density in Thin-Film Transistors (TFTs) under dynamic conditions, paving the way for the characterization and modeling of memory transistors. The model is expressed as a first-order differential equation for the trapped charge density, which is controlled by a time constant and an independent term proportional to the drain current. The time-dependent threshold voltage is introduced in a previously developed compact model for TFTs with special consideration to the contact effects. The combination of both models and the use of an evolutionary parameter extraction procedure allow for reproducing the experimental dynamic behavior of TFTs. The results of the model and the evolutionary procedure have been validated with published experimental data of pentacene-based transistors. The procedure is able to simultaneously reproduce three kinds of experiments with different initialization routines and constraints in each of them: output and transfer characteristics with hysteresis and current transients characteristics.

Differentiating the Yield of Chemical Reactions Using Parameters in First-Order Kinetic Equations to Identify Elementary Steps That Control the Reactivity from Complicated Reaction Path Networks.

The yield of a chemical reaction is obtained by solving its rate equation. This study introduces an approach for differentiating yields by utilizing the parameters of the rate equation, which is expressed as a first-order linear differential equation. The yield derivative for a specific pair of reactants and products is derived by mathematically expressing the rate constant matrix contraction method, which is a simple kinetic analysis method. The parameters of the rate equation are the Gibbs energies of the intermediates and transition states in the reaction path network used to formulate the rate equation. Thus, our approach for differentiating the yield allows a numerical evaluation of the contribution of energy variation to the yield for each intermediate and transition state in the reaction path network. In other words, a comparison of these values automatically extracts the factors affecting the yield from a complicated reaction path network consisting of numerous reaction paths and intermediates. This study verifies the behavior of the proposed approach through numerical experiments on the reaction path networks of a model system and the Rh-catalyzed hydroformylation reaction. Moreover, the possibility of using this approach for designing ligands in organometallic catalysts is discussed.

A Quantum Formalism for Abstract Dynamical Systems

This paper presents a quantum formulation for classical abstract dynamical systems (ADS), defined by coupled sets of first-order differential equations. They are referred to as “abstract” because their dynamical variables can be of different interrelated natures, not necessarily corresponding to physics, such as populations, socioeconomic variables, behavioral variables, etc. A classical linear Hamiltonian can be derived for ADS by using Dirac’s dynamics for singular Hamiltonian systems. Also, a corresponding first-order Schrödinger equation (which involves the existence of a system Planck constant particular of each system) can be derived from this Hamiltonian. However, Madelung’s reinterpretation of quantum mechanics, followed by the Bohm and Hiley work, produces no further information about the mathematical formulation of ADS. However, a classical quadratic Hamiltonian can also be derived for ADS, as well as a corresponding second-order Schrödinger equation. In this case, the Madelung reinterpretation of quantum mechanics provides a quantum Hamiltonian that does provide the quantum formulation for ADS, which provides new quantum variables interrelated dynamically with the classical variables. An application case is presented: the one-dimensional autonomous system given by the logistic dynamics. The differences between the classical and the quantum trajectories are highlighted in the context of this application case.

Combined Effect of Non-Linear Mixed Convection, and Non-Uniform Heat Source/Sink on Casson Ternary Hybrid Nanofluid Flow Across a Stretched Rotatory Disk

This present study emphasizes the importance of a specific type of fluid called a Casson ternary hybrid nanofluid. Our research explores a novel approach using the combination of several complex phenomena, including magnetohydrodynamic radiative flow, non-linear mixed convection, a non-uniform heat source or sink, a chemical reaction described by the Arrhenius model, and multiple slip effects. The researchers transformed the system of governing equations into a set of first-order ordinary differential equations using appropriate mathematical transformations and then solved them numerically using the bvp4c solver. The study investigated the behavior of velocity profiles, thermal dispersion, concentration dispersion, and heat and mass transfer for different values of the parameters involved. Our results indicated that the rising values of the heat transmission rate escalated by 2.98% for Casson ternary hybrid nanofluid when compared to ternary hybrid nanofluid. Further, ternary hybrid nanofluid had a 7.49% and 6.89% higher heat transmission rate compared to the hybrid nanofluid and conventional nanofluid, respectively. Besides, the heat transmission rate is enhanced by 17.5% and 3.11% respectively under the existence of the mixed convective and non-linear thermal convection parameters. Also, the presence of chemical reaction parameter shows a positive impact on the rate of mass transmission

Diffusive wave model in a finite length channel with a concentrated lateral inflow subject to different types of boundary conditions

The diffusive wave model is one of the simplified forms of Saint-Venant equations, and it is often used instead of the full model. In this paper, we present an analytical solution for the linearized diffusive wave model represented by a simultaneous system of two first-order partial differential equations focused on spatial variation of a lateral inflow in a finite channel. A concentrated lateral inflow from a small-width tributary is considered through the Dirac delta function. We use the Laplace transform method to solve these equations analytically. Two types of upstream boundaries are considered here in the form of a flow-discharge hydrograph and a flow-depth hydrograph, while keeping a flow-depth hydrograph as the downstream boundary. Using unit-step responses of the lateral inflow, the effect of different boundaries on the flow-depth responses and the flow-discharge responses is studied for different values of the Peclet number (Pe). The flow depth is observed to be more sensitive to the downstream boundary and other parameters used in this work. Consideration of the flow depth as the upstream boundary reflects the effect of all the parameters on the unit-step responses presented. These responses are compared with the available semi-infinite channel responses, which are found to be an inappropriate substitute for the finite channel responses for Pe<5 which implies that the downstream boundary cannot be ignored for these cases. However, for the case Pe>5, although the semi-infinite channel responses are found to satisfactorily estimate the discharge along the entire channel, they can approximate the flow depth at the locations closer to the upstream boundary only.

Parametric analysis of pollutant discharge concentration in non-Newtonian nanofluid flow across a permeable Riga sheet with thermal radiation

Proper wastewater disposal is crucial in various manufacturing and ecological systems. This study aims to prevent and regulate pollution in the water supply. It examines how the pollutant discharge concentration affects the flow of non-Newtonian nanofluids (NNNFs) over a porous Riga surface. Two different types of NNNFs, namely, Walter’s B and second-grade fluids, have been examined. The fluid flow is conveyed in the form of a system of partial differential equations (PDEs), which are first reduced to a non-dimensional set of ordinary differential equations (ODEs) and then to first-order differential equations. The numerical approach parametric continuation method is employed to solve these ODEs. It has been noticed that the energy curve declines with increasing numbers of TiO2-nanoparticles (NPs). The effect of the external pollutant source variation factor enriches the concentration of pollutants in both fluid cases. Furthermore, the viscoelastic parameter K1 plays a notable role in determining the behavior of the fluids. Particularly in NNNFs, the variation of K1 enhances the fluid flow, whereas the rise of second-grade fluid factor decreases the velocity of the fluid. Our findings indicate a substantial impact of the parameters under consideration on the concentration of pollutant discharge. Significantly, it was observed that an increase in the amount of NPs and the thermal radiation parameter led to an improvement in the thermal conductivity of the nanofluid, consequently decreasing the concentration of pollutants in the discharge. The nanofluid has greater efficiency in boosting the energy transfer rate of the base fluid. In the case of the second-grade fluid, the energy propagation rate increases up to 6.25%, whereas, in the case of Walter’s fluid B, it increases up to 7.85%.

Modelling Mercury Dynamics in Soils Impacted by Mercury Emission from Artisanal and Small-Scale Mining Activities

ABSTRACT Soil systems are commonly regarded as receptors of anthropogenic mercury contamination and play an essential role in the containment or dispersion of pollution to surface water, groundwater and the atmosphere. Mercury fate and transport are simulated using a local scale model to evaluate mercury dynamics in a soil system impacted by anthropogenic mercury pollution from artisanal and small-scale gold mining. The model assumes first-order differential equations that describe losses principally by evasion and leaching simulated on a yearly time step from thirty years ago to the present and into the future, where two different scenarios are considered. The model predicts that topsoil mercury concentrations may more than double in the next 50 years. Soil mercury is currently a source of mercury to the atmosphere and water but the leaching flux is less than the evasion flux. The soil would accumulate mercury until emission is curtailed and may require more than 500 years for levels to decrease to background values. If current emission rates are maintained, mercury contents of contaminated soil will keep increasing and will reach levels that poses risk to ecological and human health.

THEORETICAL ANALYSIS AND NUMERICAL SOLUTION OF LINEAR AND NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS

In this paper, a collocation method based on the Haar wavelet is presented for the solution of both linear and nonlinear first-order neutral delay differential equations. The Haar functions are used to approximate the first-order derivative, and the approximate solution is obtained by using initial condition and integration. Some examples from the literature are used to test the suggested method efficiency and applicability. A comparison of exact and approximate solutions is given in figures for different numbers of collocation points. The root mean square and maximum absolute errors are calculated for different numbers of collocation points. The rate of convergence is calculated which is approximately equal to 2. The comparison of the present method with the other numerical methods is also given. The results demonstrate that the Haar wavelet collocation method is simple and effective for solving first-order linear and nonlinear neutral delay differential equations.