Simultaneous Differential Equations Research Articles

Analysis of Chemical Kinetics of Multistep Reactions by Mean Reaction Time.

Although methods to derive the rate equation from a kinetic model have been known for over a century, it remains mathematically challenging to derive the rate equation for complex reactions involving multiple steps, as the derivation requires a solution for simultaneous differential equations. Furthermore, the derived kinetic equations are often difficult to intuitively understand. Here, we report a radically different approach to analyze chemical kinetics using the mean reaction time, the average of the time required for the completion of a chemical reaction. This mathematical description of chemical kinetics provides an intuitive understanding of how individual steps in a multistep reaction contributes to the overall kinetics. We demonstrate here the step-by-step approach to derive the formula for the mean reaction time from kinetic models. Surprisingly, one can derive the formula for the mean reaction time without solving simultaneous differential equations or using a steady-state approximation. Being an ensemble-averaged value, the mean reaction time cannot describe the distribution of the reaction time of the individual chemical entities. However, the mean reaction time reveals invaluable insights on how the energy levels of the ground states and the transition states affect the kinetics of the multistep reaction. As proof of principle, we apply the mean reaction time to enzyme kinetics and demonstrate that one can readily derive the expressions of the kinetic parameters (k cat/K M and k cat) even without deriving the Michaelis-Menten equation.

On solutions of single and simultaneous q-shift differential equations in two dimensional complex plane

On solutions of single and simultaneous q-shift differential equations in two dimensional complex plane

A dynamic model of repositioning with a Markov-switching regime

AbstractRepositioning products and services to entice customers is a key management strategy that requires careful planning, timing, and budget allocation. This study is the first to investigate repositioning strategies by considering a Hotelling-type location model where consumer preferences change over time with short- and long-term uncertainty. This requires the firm to adjust its product positioning to continue to appeal to consumers. A key assumption is that the parameters describing the dynamics of consumer preferences are modulated by a Markov-switching regime representing long-term uncertainty. Further, this study investigates the effects of regime shift intensity on repositioning strategies. We numerically solve a simultaneous ordinary differential equation system to derive a firm’s optimal strategy, represented by repositioning thresholds that depend on the regimes. We find that in the two-regime case, one threshold is monotonic with respect to the regime shift intensity, whereas the other can be non-monotonic. This suggests that a firm should simultaneously consider the value of the current regime and option value of waiting for a regime shift, thus effectively demonstrating the significance of the uncertainty associated with regime-switching.

Application of generalized Haar wavelet technique on simultaneous delay differential equations

In this paper, the Haar wavelet technique is applied to a system of simultaneous linear and nonlinear proportional delay differential equations. By implementing this method, the delay differential equations are transformed into a series of linear and nonlinear algebraic equations, respectively. These equations are then solved using numerical code constructed in MATLAB. The obtained approximate solutions are compared with exact solutions, demonstrating the accuracy and efficacy of the Haar wavelet method. Using the fixed point theorem, the existence and uniqueness of the solutions is established. The reliability and efficiency of the proposed technique are illustrated through two numerical examples. Additionally, a comprehensive error analysis is presented and the convergence result is discussed, providing a thorough examination of the robustness of the method. This study not only shows that the Haar wavelet technique is a powerful tool for solving delay differential equations but also sets the groundwork for future research in this area.

Oscillations of the amplitude of the Cooper pair gap induced by interactions between Cooper pairs in the framework of BCS theory

According to BCS theory, superconductivity arises from the weak attraction between electrons. In particular, Cooper pairs are important in determining the properties of superconductivity; therefore, the theoretical description of Cooper pairs is based on the mean-field approximation used in the BCS theory. However, the mean-field approximation cannot estimate the contribution induced by the interactions between Cooper pairs, i.e., many-body terms, accurately. In this article, therefore, using simultaneous differential equations derived from time-dependent Schrödinger equation, we show the detailed description of the Cooper pairs including such interactions, which has not been mentioned in BCS theory so far. We reveal that the time dependence oscillates when interactions between Cooper pairs are accurately considered. Based on theses calculations, the coupling of the Cooper pair gap can become zero, suggesting that the Cooper pairs can be switched on and off in real superconductors in the framework of BCS theory.

CRISPR-Enabled Graphene-Based Bio-Cyber Interface Model for In Vivo Monitoring of Non-Invasive Therapeutic Processes.

In this paper, we present a model of the bio-cyber interface for the Internet of Bio-Nano Things application. The proposed model is inspired by the gains of integrating the Clustered Regularly Interspace Short Palindromic Repeats (CRISPR) technology with the Graphene-Field effect transistor (GFET). The capabilities of the integrated system are harnessed to detect nucleic acids transcribed by another component of the bio-cyber interface, a bioreporter, on being exposed to the signalling molecule of interest. The proposed model offers a label-free real-time signal transduction with multi-symbol signalling capability. We model the entire operation of the interface as a set of simultaneous differential equations representing the process's kinetics. The solution to the model is obtained using a numerical method. Numerical results show that the performance of the interface is influenced by parameters such as the concentrations of the input signalling molecules, the surface receptor on the bioreporter, and the CRISPR complex. The interface's performance also depends considerably on the elimination rate of the signalling molecules from the body. For multi-symbol molecular signalling, the rate of degradation of the transcribed RNAs influences the system's susceptibility to inter-symbol interference.

Leptospirosis Relative Risk Estimates based on Continuous-Time, Discrete-Space Stochastic SIR-L-SI Transmission Model

Leptospirosis is a re-emerging global disease that has become endemic in Malaysia. The transmissions usually occur between animals especially rats to rats and rats to humans. Since, it is an easily contractable disease that can be transmitted directly through contact with infected rat’s urine or indirectly from the environment such as via water and soil, it is very challenging to curb this disease from infecting humans. Poor understanding of the disease and lack of epidemiological data also made leptospirosis is difficult to control. To cope with this problem, a leptospirosis disease transmission model is developed to study the mechanism of leptospirosis disease spread over continuous-time that may help to predict future caused of an outbreak. This study aims to construct a continuous-time and discrete-space stochastic SIR-L-SI (Susceptible, Infected, Recovered Humans-Leptospires in the Environment-Susceptible, Infectious Rats) of leptospirosis disease transmission to estimate the risk involved. A simple method of asymptotic and numerical analyses is applied as an alternative approach for solving simultaneous differential equations in the leptospirosis SIR-L-SI transmission model. The application of the proposed model is demonstrated using leptospirosis data for Malaysia. The results of asymptotic behaviour and numerical analysis provide useful information about susceptible and infective rat and human populations as well as offer relative risk estimates that can be used as one of the control measures in identifying hot-spot areas for this disease.

Proposal of uncertainty analysis methodology for L1PRA using Markov state-transition model

ABSTRACT Following the severe accident at the Fukushima Daiichi Nuclear Power Plant in 2011, the revised nuclear safety regulation in Japan requires continuous safety improvement and states that PRA methods that reflect the latest knowledge should be used in activities related to continuous safety improvement. In this context, the construction of PRA models for the digital RPS (DRPS) has been addressed as an important issue within the Working Group on Risk Assessment (WGRISK) of the OECD/NEA, and several studies have been conducted. And there are challenges in aligning them with the conventional probabilistic risk assessment methodology. In a previous study, the authors developed a simultaneous differential equation describing the relationship between state transitions and state probabilities based on Markov state transition diagrams to calculate them numerically. However, the analytical method for uncertainty analysis commonly used in conventional PRA evaluations is not explicitly presented. The purpose of this study is to provide a methodology for more accurate evaluation of core damage frequency in nuclear power plants equipped with digital RPS, taking into account the uncertainties, and to contribute to the continuous improvement of safety.

Dislocation-density-related constitutive model and its applicability to cyclic deformation and damage behavior of 316LN SS at 823 K

In the present investigation, an in-depth understanding of the relationship between dislocation kinetics and cyclic plasticity, along with the description of yield asymmetry behavior during loading and unloading in nuclear grade 316LN austenitic stainless steel at 823 K, was examined by employing a constitutive framework using the evolution of physically-based internal-state variables. The formulation encompassed a set of first-order simultaneous differential equations that governed the evolution of dislocation density, mean free path of dislocations, and back stress. These variables were interdependent with the power law which described the kinetics of plastic strain rate. The evolution equations were numerically integrated, and the simulated data were fitted to the experimental data up to stabilized cycle at a temperature of 823 K and total strain amplitudes of ±0.004, ±0.006, ±0.008 and ± 0.010. The predicted cyclic stress response as a function of the cyclic number displayed the continuous cyclic hardening from the first cycle to the initiation of stabilized cycle. Further, good agreement between the predicted and experimental hysteresis loops was found at different cycle numbers. The predicted dislocation density and tensile peak values of back/effective stress increased with cyclic hardening and approached the value of saturation at the stabilized cycle. On the other hand, a rapid decrease in mean free path with cycle number was noticed. An increase in total strain amplitude resulted in a systematic increase in dislocation density, peak stress, back stress and effective stress. At the total strain amplitudes above ±0.006, the predicted higher rates of dislocation accumulation and annihilation, and the faster evolution rate of back stress towards saturation led to the development of stable dislocation substructure configurations with a clear distinction between dislocation high/low dense regions. Furthermore, the applicability of the model was extended by including an empirical relationship between the damage variable and cycle number within the power law, thereby enabling predictions of the cyclic stress-strain response beyond the stabilized cycle.

Applications of Fractional-Laplace Transformation in the Field of Electrical Engineering

This study examines the various ways that fractional Laplace transform can be used to solve three different kinds of mathematical equations: the equation of analysis of electric circuits, simultaneous differential equations, and the heat conduction equation. This article how to use the fractional Laplace transform to calculate heat flow in semi-infinite solids in the context of heat conduction. The answers that are developed offer important information about how temperatures vary across time and space. The essay also examines how to analyse electrical circuits using the Fractional Laplace transform. This method allows researchers to measure significant electrical parameters including charge and current, which improves their comprehension of circuit dynamics. Practical examples are included throughout the essay to show how useful the Fractional Laplace transform is in various fields. As a result of the answers found using this methodology, researchers and engineers working in the fields of heat conduction, system dynamics, and circuit analysis can gain important new knowledge. In conclusion, this study explains the applicability and effectiveness of the fractional Laplace transform in resolving a variety of mathematical equations. It is a vital tool for researchers because it may be used in a wide range of scientific and engineering areas.

THERMALLY-OPTICALLY-THERMALLY STIMULATED LUMINESCENCE

A three-stage energy band model was studied. The model consists of electrons thermally stimulated from the ground state to the first excited state, after which they were optically stimulated into the second excited state and they were finally stimulated thermally into the conduction band. A set of simultaneous differential equations was generated from the models and three assumed conditions were applied to this model, which they were solved analytically and analytical expressions were obtained. The same set of simultaneous equations were solved numerically using ode 15s MATLAB solver. When considering first-order peaks, the kinetic parameters obtained were found to be in good agreement with the analytical expressions. But when considering non first-order peaks, the kinetic parameters obtained numerically were not in good agreement with the analytical expressions and explanations had been given. Second-order peaks could not be obtained despite careful selection of the kinetic parameters because the traps were quickly saturated and the quasi- equilibrium conditions assumed could no longer be satisfied. The stability of the excited TA-OSL signals produced by the model was also studied. The real stability of the excited TA-OSL signals produced by this model was found to be about 46 million years.

Outlining the impact of structural properties of a multilayered porous channel with oscillating velocities and fields

In this study, we analyze the multilayered viscous potential flow, in which the effects of primary and combined resonances on the interfacial stability due to oscillating velocities and a periodic field are carried out. Fluids are assumed to be immiscible and incompressible with different electrical and dynamic properties and move through a porous nature. Viscosity occurs at the surface through the surface dynamic condition, where it is assumed to be weak in the fluid bulk and can be ignored. Given the boundary and surface conditions, the linear solutions of the motion and field equations led to simultaneous differential equations with complex periodic coefficients, which are a generalization of Mathieu’s equations. The special case of uniform velocity and the constant field is discussed where a quadruple algebraic equation has complex coefficients is obtained. The method of multiple time scales is applied to obtain analytical approximate solutions and perform the stability criteria for both the non-resonant and resonant states. The impacts of flow velocities and associated electric field properties, as well as the permeability of porous media and fluid sheet geometry, were taken into consideration. Through numerical discretization by varying the values of the quantities of the model, essential conclusions can be made that support an understanding of the stabilization process. For the primary resonance, it is found that there is an important role in increasing the thickness of the layers to the stability of the fluid sheet, while the porosity of the layers has a dual role phenomenon depending on the thickness of the upper layer. In the case of combined resonance, it was mentioned that increasing the dielectric constant between the lower layer and middle one tends to assist the system to instability, but it is worth noting that it supports it slowly, on the other hand, the electric horizontal field helps the system to be more stable.

Bayesian kinetic modeling for tracer-based metabolomic data

BackgroundStable Isotope Resolved Metabolomics (SIRM) is a new biological approach that uses stable isotope tracers such as uniformly 13C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{13}C$$\\end{document}-enriched glucose (13C6\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$^{13}C_6$$\\end{document}-Glc) to trace metabolic pathways or networks at the atomic level in complex biological systems. Non-steady-state kinetic modeling based on SIRM data uses sets of simultaneous ordinary differential equations (ODEs) to quantitatively characterize the dynamic behavior of metabolic networks. It has been increasingly used to understand the regulation of normal metabolism and dysregulation in the development of diseases. However, fitting a kinetic model is challenging because there are usually multiple sets of parameter values that fit the data equally well, especially for large-scale kinetic models. In addition, there is a lack of statistically rigorous methods to compare kinetic model parameters between different experimental groups.ResultsWe propose a new Bayesian statistical framework to enhance parameter estimation and hypothesis testing for non-steady-state kinetic modeling of SIRM data. For estimating kinetic model parameters, we leverage the prior distribution not only to allow incorporation of experts’ knowledge but also to provide robust parameter estimation. We also introduce a shrinkage approach for borrowing information across the ensemble of metabolites to stably estimate the variance of an individual isotopomer. In addition, we use a component-wise adaptive Metropolis algorithm with delayed rejection to perform efficient Monte Carlo sampling of the posterior distribution over high-dimensional parameter space. For comparing kinetic model parameters between experimental groups, we propose a new reparameterization method that converts the complex hypothesis testing problem into a more tractable parameter estimation problem. We also propose an inference procedure based on credible interval and credible value. Our method is freely available for academic use at https://github.com/xuzhang0131/MCMCFlux.ConclusionsOur new Bayesian framework provides robust estimation of kinetic model parameters and enables rigorous comparison of model parameters between experimental groups. Simulation studies and application to a lung cancer study demonstrate that our framework performs well for non-steady-state kinetic modeling of SIRM data.

Elastic deformation effect on carboxymethyl cellulose water-based (TiO2–Ti6Al4V) hybrid nanoliquid over a stretching sheet with an induced magnetic field

An analysis is carried out to study the two-dimensional stagnant point flow of titanium dioxide -titanium alloy /carboxymethyl cellulose (CMC) water-based hybrid nanoliquid over an expanding surface characterized by the induced magnetic field (IMF). Further, the consideration of thermal buoyancy and the additional nonuniform heat source is involved in the modeling. The intent of this work has significant novelty due to the elastic deformation behavior over a stretching surface. The governing mathematical equations of the problem are reduced to a set of simultaneous ordinary differential equations and are elucidated numerically by employing the Runge-Kutta-Fehlbergs fourth-and-fifth (RKF-45) order process combined with a shooting approach. This article examines the properties of change in parameters on the functions including temperature profiles, induced magnetic field, and velocity profiles of hybrid nanoliquid, shown through graphs. The impact of the Elastic deformation coefficient reduces the temperature where as magnetic field enhance the velocity profile. Magnetic parameter boosts skin friction coefficient with an increase in the buoyancy effect. Momentum profile and induced magnetic field are affected most by the Magnetic Prandtl number

MHD NANOFLUID FLOW PAST AN INCLINED PLATE WITH SORET AND DUFOUR EFFECTS

A steady incompressible nanofluid flow past an inclined permeable plate is numerically studied, including radiation, viscous dissipation, and Soret and Dufour effects. A coupled non-linear simultaneous differential similarity equation is created by non-dimensionalizing the ruling partial differential equations. Then, nonlinear coupled equations with transformed boundary conditions are resolved by the shooting method based on the Runge-Kutta fourth-order method. The Prandtl number, Grashof number, Schmidt number, magnetic parameter, and Soret and Dufour effects are just a few of the controlling flow parameters for which computations are done. Graphs are used to illustrate how various flow factors affect momentum, energy, and concentration equations. The impacts of these variables on skin friction coefficients, Nusselt number, and Sherwood number are given in tabular form. The concentration profile increases with the Soret number and the reverse trend is observed with the Dufour number. It is further noted that heat-mass transfer rate reduces in the boundary layer due to an increase in the values of Soret or a decrease in the values of Dufour.

RESPONSE OF A FIRST-ORDER SERIES SYSTEM TO THE TRANSFER FUNCTION, THE LIQUID LEVEL OF TWO INTERACTING TANKS

This article presents the experimental and theoretical results of the discharge and filling of water between two torispherical tanks connected in series at different heights, the model that was used was developed with equations reported in the literature for geometries of torispherical tanks and it was necessary to develop two more equations in the laboratory to calculate the discharge and fill volumetric flows for the tanks used. The unloading and filling model was obtained with transitory mass balances resulting in two simultaneous linear differential equations where the unloading and filling heights are related as a function of the operation time. The model was solved numerically, and the experimental results were compared against those of the proposed model, finally the transfer function for this system of interacting tanks was found. KEYWORDS: Transient mass balances in two atmospheric interacting tanks at different heights and equal torispherical geometries. The discharge and fill model of two interacting tanks as a function of time. The volumes of partially filled torispherical tanks. The transfer function of a pulse in the discharge pipe that affects the volumetric flow within the discharge tank.

Harvesting electricity from random vibrations via a nonlinear energy sink

Integration of a vibration absorber and an energy harvester is a promising approach to achieve simultaneously vibration reductions and vibratory energy harvesting. The present work investigates random responses of a nonlinear energy sink (NES) combined with a giant magnetostrictive energy harvester (GMEH). When a structure is modeled as a one-degree-of-freedom oscillator subjected to a random excitation, the structure with the integrated NES-GMEH is governed by two simultaneous nonlinear stochastic differential equations coupled with a nonlinear algebraic equation. The Ferrari formula is applied to decouple the algebraic equations from the stochastic differential-algebraic equations. The resulting stochastic algebraic differential equations are transformed into the steady Fokker-Planck-Kolmogorov equations. A fourth finite difference scheme is used to solve numerically the FPK equations. The FPK equations based solutions are qualitatively verified via Monte Carlo simulations. The parametric effects on vibration suppression and electricity production are examined, where the electricity is indicated by the square expectations of voltage and power. The responses of the structure, the structure with the NES only and the structure with both NES-GMEH are compared under the same Gaussian white noise and system parameters. It is found that the integrated NES-GMEH system achieves the best vibration suppression. It is demonstrated that a properly designed NES-GMEH device can simultaneously suppress vibration and produce electricity under Gaussian white noise excitations.

Vibrations of Phase-Lags on Electro-Magneto Nonlocal Elastic Solid with Voids in Generalized Thermoelastic Cylinder/Disk

The stress-strain-temperature relations, strain-displacement relations and governing equations have been addressed for electro-magneto transversely isotropic nonlocal elastic hollow cylinder with voids in the reference of 3-phase-lag effect of heat conduction. The strength of the magnetic field proceeds in the direction of the z-axis of the hollow cylinder/disk. The simultaneous differential equations have been eliminated by applying elimination technique to obtain unknown field functions such as dilatation, equilibrated voids volume fraction, temperature, displacement and stresses. Free vibration analysis has been explored by applying stress free and thermally insulated/isothermal boundaries. Analytical results are verified by employing numerically analyzed results for unknown field functions and presented graphically for the vibrations of stress free field functions such as thermoelastric damping, frequencies and frequency-shift. The results have been authenticated by analyzing analytical and numerical results with existing literature with earlier published work. The study of present paper based on 3-phase-lag model of generalized thermoelasticity may receive better approach to allow voids and relaxation time parameters, which have many applications in the field of science, technology and engineering. The study may also be useful in the area of seismology for mining and drilling in the earth’s crust.
 HIGHLIGHTS
 
 The analysis for electro-magneto transversely isotropic nonlocal thermoelastic hollow cylinder with voids in the reference of 3-phase-lag effect of heat conduction
 The strength of the magnetic field proceeds in the direction of the z-axis of the nonlocal thermoelastic hollow cylinder/disk
 Free vibration analysis has been explored by applying stress free and thermally insulated/isothermal boundaries
 Analytical results are verified by employing numerically analyzed results for unknown field functions and presented graphically for the vibrations of stress free field functions such as damping, frequencies and frequency-shift
 
 GRAPHICAL ABSTRACT

Monetary and fiscal policy in a nonlinear model of public debt

In this paper we study the dynamic relationship between the public debt ratio and the real interest rate. Specifically, by means of a macroeconomic model of simultaneous difference equations – one for the debt ratio and the other for the real interest rate – we focus on the role of monetary policy, fiscal policy and risk premium in affecting the stability of the debt ratio and the existence of steady states, if any. We show that, in a dynamic framework, fiscal rules may not be enough to control the pattern of the debt ratio, and the adoption of a monetary policy, in the form of an interest rate rule, is necessary to control the pattern of the debt ratio for assuring its sustainability over time. Notably, the creation or disappearance of steady states, or periodic (stable) cycles, can generate scenarios of multistability. While we obtain clear evidence that an active monetary policy has a stabilizing effect on both the real interest rate and the debt ratio, we also find that, in some scenarios, fiscal policy is not sufficient to avoid explosive patterns of the debt ratio.

Development of a multi-scale model to simulate mesenchymal stem cell osteogenic differentiation within hydrogels in a rotating wall bioreactor

Demand for mesenchymal stem cell (MSC)-based therapies for various clinical applications has been steadily increasing. Suitable implants can be obtained by in-vitro osteogenic differentiation of stem cells to meet this demand, at least in part. This paper proposes a multiscale population balance model for the osteogenic differentiation of mesenchymal stem cells encapsulated within alginate-gelatin hydrogels in a rotating wall bioreactor. The mathematical model incorporates key genes and metabolic pathways linked to proliferation and osteogenesis and captures spatial as well as cell cycle heterogeneity within the hydrogels inside the bioreactor. The discretized formulation of the model consists of 12,563 simultaneous ordinary differential equations. Concentrations of key metabolites (e.g., glucose, pyruvate, glutamine, lactate) were compared with experimental data from static well plate cultures, rendering the model predictions as adequate. Gene activation (Runx2 and osteonectin) in alginate-gelatin-bead-encapsulated cells was slightly delayed compared with well-plate cultures. Global sensitivity analysis revealed that parameters related to gene expression (decay rates and gene transcription activation constants) carry the most significance and should be the focus of future parameter estimation. Hydrogel bead size does not meaningfully impact simulation outcomes for bead diameters in the 2–3 mm range. The mathematical model proposed in this work can function as a foundation for model-based bioreactor and bioprocess optimization when coupled with a high-performance computing system.