Number Of Differential Equations Research Articles

Reduced Desensitization Formulation for Optimal Control Problems

Solutions to nonlinear optimal control problems (OCPs) exhibit dispersions under model uncertainties and it is desirable to generate optimal solutions that exhibit less sensitivity to model uncertainties. We propose a novel solution desensitization method dubbed “Reduced Desensitization Formulation (RDF)” by leveraging non-uniqueness of the solution of the costate differential equations when a hybrid indirect-direct optimization method is used. A key property of the RDF method is a significant reduction in the number of differential equations needed for generating desensitized solutions. This feature facilitates the generation of open-loop desensitized trajectories and makes the methodology applicable to OCPs with a larger number of uncertain parameters. To demonstrate the utility of the RDF method, three important classes of trajectory optimization problems are considered with uncertainty in the thrust magnitude of the propulsion system: (1) minimum-fuel low-thrust interplanetary rendezvous maneuvers, (2) low-thrust orbit-raising maneuvers, and (3) minimum-fuel high-thrust rocket-landing problems. For the considered problems with bang-bang control profiles, an analysis is presented on the change in the number of control switches between sensitive and desensitized optimal solutions. Numerical results demonstrate desensitization of the considered performance indices with respect to the thrust magnitude of the propulsion system.

Barometers Behaving Badly II: a Critical Evaluation of Cpx-Only and Cpx-Liq Thermobarometry in Variably-Hydrous Arc Magmas

ABSTRACT The chemistry of erupted clinopyroxene crystals (±equilibrium liquids) have been widely used to deduce the pressures and temperatures of magma storage in volcanic arcs. However, the large number of different equations parameterizing the relationship between mineral and melt compositions and intensive variables such as pressure and temperature yield vastly different results, with implications for our interpretation of magma storage conditions. We use a new test dataset composed of the average Clinopyroxene-Liquid (Cpx-Liq) compositions from N = 543 variably hydrous experiments at crustal conditions (1 bar to 17 kbar) to assess the performance of different thermobarometers and identify the most accurate and precise expressions for application to subduction zone magmas. First, we assess different equilibrium tests, finding that comparing the measured and predicted Enstatite-Ferrosillite and KD (using Fet in both phases) are the most useful tests in arc magmas, whereas CaTs, CaTi and Jd tests have limited utility. We then apply further quality filters based on cation sums (3.95–4.05), number of analyses (N > 5) and the presence of reported H2O data in the quenched experimental glass (hereafter ‘liquid’) to obtain a filtered dataset (N = 214). We use this filtered dataset to compare calculated versus experimental pressures and temperatures for different combinations of thermobarometers. A number of Cpx-Liq thermometers perform very well when liquid H2O contents are known, although the Cpx composition contributes little to the calculated temperature relative to the liquid composition. Most Cpx-only thermometers perform very badly, greatly overestimating temperatures for hydrous experiments. These two findings demonstrate that the Cpx chemistry alone holds very little temperature information in hydrous systems. Most Cpx-Liq and Cpx-only barometers show similar performance to one another (mostly yielding root mean square errors [RMSEs] of 2–3.5 kbar), although the best Cpx-only barometers currently outperform the best Cpx-Liq barometers. We also assess the sensitivity of different equations to melt H2O contents, which are poorly constrained in many natural systems. Overall, this work demonstrates that Cpx-based barometry on individual Cpx only provides sufficient resolution to distinguish broad storage regions in continental arcs (e.g. upper, mid, lower crust). Significant averaging of Cpx compositions from experiments reported at similar pressures can reduce RMSEs to ~1.3–1.9 kbar. We hope our findings motivate the substantial amount of experimental and analytical work that is required to obtain precise and accurate estimates of magma storage depths from Cpx ± Liq equilibrium in volcanic arcs.

Isotropic stellar model in mimetic theory

We investigate how to derive an isotropic stellar model in the frame of mimetic gravitational theory. Recently, this theory has gained big interest due to its difference from Einstein’s general relativity (GR), especially in the domain non-vacuum solutions. In this regard, we apply the field equation of mimetic gravitational theory to a spherically symmetric ansatz and obtain an over determined system of non-linear differential equations in which the number of differential equations is less than the unknown functions. To overcome the over determined system we suppose a specific form of the temporal component of the metric potential, g_{tt}, and assume the vanishing of the anisotropic condition to derive the form of the spatial component of the metric, g_{rr}. In this regard, we discuss the possibility to derive a stellar isotropic model that is in agreement with observed pulsars. To examine the stability of the isotropic model we use the Tolman–Oppenheimer–Volkoff equation and the adiabatic index. Moreover, we test the model using a wide range of observed values of masses and radii of pulsars and show that they fit in a good way.

Parallel scalable simulations of biological neural networks using TensorFlow: A beginner’s guide

Biological neural networks are often modeled as systems of coupled, nonlinear, ordinary or partial differential equations. The number of differential equations used to model a network increases with the size of the network and the level of detail used to model individual neurons and synapses. As one scales up the size of the simulation, it becomes essential to utilize powerful computing platforms. While many tools exist that solve these equations numerically, they are often platform-specific. Further, there is a high barrier of entry to developing flexible platform-independent general-purpose code that supports hardware acceleration on modern computing architectures such as GPUs/TPUs and Distributed Platforms. TensorFlow is a Python-based open-source package designed for machine learning algorithms. However, it is also a scalable environment for a variety of computations, including solving differential equations using iterative algorithms such as Runge-Kutta methods. In this article and the accompanying tutorials, we present a simple exposition of numerical methods to solve ordinary differential equations using Python and TensorFlow. The tutorials consist of a series of Python notebooks that, over the course of five sessions, will lead novice programmers from writing programs to integrate simple one-dimensional ordinary differential equations using Python to solving a large system (1000’s of differential equations) of coupled conductance-based neurons using a highly parallelized and scalable framework. Embedded with the tutorial is a physiologically realistic implementation of a network in the insect olfactory system. This system, consisting of multiple neuron and synapse types, can serve as a template to simulate other networks.

Dynamical modeling for non-Gaussian data with high-dimensional sparse ordinary differential equations

Ordinary differential equations (ODE) have been widely used for modeling dynamical complex systems. For high-dimensional ODE models where the number of differential equations is large, it remains challenging to estimate the ODE parameters and to identify the sparse structure of the ODE models. Most existing methods exploit the least-square based approach and are only applicable to Gaussian observations. However, as discrete data are ubiquitous in applications, it is of practical importance to develop dynamic modeling for non-Gaussian observations. New methods and algorithms are developed for both parameter estimation and sparse structure identification in high-dimensional linear ODE systems. First, the high-dimensional generalized profiling method is proposed as a likelihood-based approach with ODE fidelity and sparsity-inducing regularization, along with efficient computation based on parameter cascading. Second, two versions of the two-step collocation methods are extended to the non-Gaussian set-up by incorporating the iteratively reweighted least squares technique. Simulations show that the profiling procedure has excellent performance in latent process and derivative fitting and ODE parameter estimation, while the two-step collocation approach excels in identifying the sparse structure of the ODE system. The usefulness of the proposed methods is also demonstrated by analyzing three real datasets from Google trends, stock market sectors, and yeast cell cycle studies.

Routh reducibility and controllability of unstable mechanical systems

Routh reduction presents the minimum number of differential equations that uniquely describe the state of nonlinear mechanical systems where the state variables can be separated into essential ones and cyclic ones. This work extends Routh reducibility for a relevant set of controlled mechanical systems. A chain of theorems is presented for identifying the conditions when reduced order rank conditions can be applied for determining the Kalman controllability of Routh reducible mechanical systems where actuation takes place along the cyclic coordinates only, while some of the essential coordinates and their derivatives are observed. Four mechanical examples represent the advantages of using reduced rank conditions to check and/or to exclude linear controllability in such systems.

On a class of difference equations with interlacing indices

Some results on the long-term behavior of solutions to a class of difference equations, which includes numerous nonlinear difference equations of various orders that attracted some attention in the last 15 years, are presented. We also present a natural connection among these difference equations, compare some results on the equations with some other ones in the literature, and give a list of a considerable number of difference equations which can be treated in a similar way.

Resting Energy Expenditure of Master Athletes: Accuracy of Predictive Equations and Primary Determinants.

Resting energy expenditure (REE) is determined mainly by fat-free mass (FFM). FFM depends also on daily physical activity. REE normally decreases with increased age due to decreases in FFM and physical activity. Measuring REE is essential for estimating total energy expenditure. As such, there are a number of different equations in use to predict REE. In recent years, an increasing number of older adults continue to participate in competitive sports creating the surge of master athletes. It is currently unclear if these equations developed primarily for the general population are also valid for highly active, older master athletes. Therefore, we tested the validity of six commonly-used equations for predicting REE in master athletes. In conjunction with the World Masters Athletic Championship in Malaga, Spain, we measured REE in 113 master athletes by indirect calorimetry. The most commonly used equations to predict REE [Harris & Benedict (H&B), World Health Organization (WHO), Müller (MÜL), Müller-FFM (MÜL-FFM), Cunningham (CUN), and De Lorenzo (LOR)] were tested for their accuracies. The influences of age, sex, height, body weight, FFM, training hours per week, phase angle, ambient temperature, and athletic specialization on REE were determined. All estimated REEs for the general population differed significantly from the measured ones (H&B, WHO, MÜL, MÜL-FFM, CUN, all p < 0.005). The equation put forward by De Lorenzo provided the most accurate prediction of REE for master athletes, closely followed by FFM-based Cunningham’s equation. The accuracy of the remaining commonly-used prediction equations to estimate REE in master athletes are less accurate. Body weight (p < 0.001), FFM (p < 0.001), FM (p = 0.007), sex (p = 0.045) and interestingly temperature (p = 0.004) are the significant predictors of REE. We conclude that REE in master athletes is primarily determined by body composition and ambient temperature. Our study provides a first estimate of energy requirements for master athletes in order to cover adequately athletes’ energy and nutrient requirements to maintain their health status and physical performance.

Об упрощении сложных уравнений применением блочных элементов

There are several approaches aimed at simplifying complex partial differential equations or their systems involved in the formulation of boundary value problems by introducing simpler, but in a larger number of differential equations. Their solutions allow us to describe solutions to complex boundary value problems. However, to implement this approach, it is necessary to construct solutions of simplified boundary value problems for arbitrary boundary conditions in solvability spaces boundary value problem. In some cases, this can be done using the block element method. The block element method, which has a topological basis, reveals both global and local properties of solutions to boundary value problems for partial differential equations. At the same time, it can be used to study and solve more complex boundary value problems by applying relations that describe certain equations of the continuum by means of relatively simple equations, for example, Helmholtz. To do this, we need to construct solutions of the Helmholtz equations that satisfy boundary conditions that contain completely arbitrary values, rather than partial values, set at the boundary of functions. In relation to the Helmholtz equations, this is achieved using the block element method. Examples of constructing solutions to boundary value problems for Helmholtz equation for Dirichlet and Neumann problems and a comparative analysis of solutions are given in this article.

Maximum entropy approach to reliability analysis based epidemic disease model

This study analyzes the dynamic model of epidemic disease system based on the maximum entropy approach to reliability. The epidemic disease system is assumed to be composed of several classes, namely susceptible class, infective class, resistance class, exposed class and so on. The dynamics of the populations in each class are depicted by a number of differential equations, where the populations transmitting between the classes are the averages of transmitting populations in microscopic stochastic transmission process. In microscopic transmission process, several basic variables are assumed to be random, such as incubation period, hospitalization duration and so on. To infer the probability distributions of these variables is a main task of this study. Inspired by the recent study of reliability, the maximum entropy based approach is applied to determine the parameters in the dynamic model. The basic idea of this study is presented below. In this study, degrade (hazard) function, which is a fundamental quantity in the disciplines of risk and reliability analysis, is associated with death rate, incidence rate and healing rate. Specifically, death rate and healing rate of the infective people are associated with hazard rate and repair rate of the repaired system in reliability theory, respectively; the incidence rate is associated to the hazard rate during incubation period. By means of the maximum entropy principle, the moments of the period from onset to death, the period from onset to recovery and the incubation period are fused to infer the most probable death rate, incidence rate and healing rate. Applying the maximum-entropy based statistical inference, the information of macroscopic transmission process is fused, such as average values, fluctuations and median values. To the best of our knowledge, the traditional fitting approaches to determine parameters usually rely on the information of macroscopic phenomenon. It is different from these fitting approaches that the maximum-entropy based approach applied in this study relies on the information of microscopic process. Thus this approach is adapted to practice scenario where the limitations of information access and number of samples both exist. And the parameter determination is independent on the choice of the macroscopic epidemic disease dynamic model. After applying the maximum entropy principle based inference and the SEIR (susceptible-infective-exposed-removed) model, several discussions associated with coronavirus disease 2019 (COVID-19) are made. With the help of recent information of microscopic transmission process, the parameters in SEIR model are determined directly by the maximum-entropy principle. Then with numerical calculation, dynamics of the populations in infective class, resistance class and exposed class are obtained. Besides, some typical phenomena are revealed by the analytical and numerical results. For example, the calculation shows that the peak of the infectious ratio in transmission process is unique; the infectious ratio of steady state tends to zero when the immunity duration is much larger than the healing period. Additionally, limited immunity duration model is also considered. The relationship between the infectious ratio of steady state and the proportion of immunization duration to healing period is presented.

Ordinary differential equation approximation of gamma distributed delay model.

In many models of pharmacodynamic systems with delays, a delay of an input is introduced by means of the convolution with the gamma distribution. An approximation of the convolution integral of bound functions based on a system of ordinary differential equations that utilizes properties of the binomial series has been introduced. The approximation converges uniformly on every compact time interval and an estimate of the approximation error has been found [Formula: see text] where [Formula: see text] is the number of differential equations and [Formula: see text] is the shape parameter of the gamma distribution. The accuracy of approximation has been tested on a set of input functions for which the convolution is known explicitly. For tested functions, [Formula: see text] has resulted in anaccurate approximation, if [Formula: see text]. However, if [Formula: see text] the error of approximation decreases slowly with increasing [Formula: see text], and [Formula: see text] might be necessary to achieve acceptable accuracy. Finally, the approximation was applied to estimate parameters for the distributed delay model of chemotherapy-induced myelosuppression from previously published WBC count data in rats treated with 5-fluorouracil.

Critical collapse of ultrarelativistic fluids: Damping or growth of aspherical deformations

We perform fully nonlinear numerical simulations to study aspherical deformations of the critical self-similar solution in the gravitational collapse of ultra-relativistic fluids. Adopting a perturbative calculation, Gundlach predicted that these perturbations behave like damped or growing oscillations, with the frequency and damping (or growth) rates depending on the equation of state. We consider a number of different equations of state and degrees of asphericity and find very good agreement with the findings of Gundlach for polar $\ell = 2$ modes. For sufficiently soft equations of state, the modes are damped, meaning that, in the limit of perfect fine-tuning, the spherically symmetric critical solution is recovered. We find that the degree of asphericity has at most a small effect on the frequency and damping parameter, or on the critical exponents in the power-law scalings. Our findings also confirm, for the first time, Gundlach's prediction that the $\ell = 2$ modes become unstable for sufficiently stiff equations of state. In this regime the spherically symmetric self-similar solution can no longer be recovered by fine-tuning to the black-hole threshold, and one can no longer expect power-law scaling to hold to arbitrarily small scales.

A Promising Method for Calculating True Steady-State Metabolite Concentrations in Large-Scale Metabolic Reaction Network Models.

The calculation of steady-state metabolite concentrations in metabolic reaction network models is the first step in the sensitivity analysis of a metabolic reaction system described by differential equations. However, this calculation becomes very difficult when the number of differential equations is more than 100. In the present study, therefore, we investigated a calculation procedure for obtaining true steady-state metabolite concentrations both efficiently and accurately even in large-scale network models. For convenience, a linear pathway model composed of a simple Michaelis-Menten rate law and two TCA cycle models were used as case studies. The calculation procedure is as follows: first solve the differential equations by a numerical method for solving initial-value problems until the upper several digits of the calculated values stabilize, and then use these values as initial guesses for a root-finding technique. An intensive investigation indicates that the S-system technique, finding roots in logarithmic space and providing a broader convergence region, is superior to the Newton-Raphson technique, and the algorithm using the S-system technique successfully provides true steady-state values with machine accuracy even with 1,500 differential equations. The complex-step method is also shown to contribute to shortening the calculation time and enhancing the accuracy. The program code has been deposited to https://github.com/BioprocessdesignLab/Steadystateconc.

Lumping of degree-based mean-field and pair-approximation equations for multistate contact processes.

Contact processes form a large and highly interesting class of dynamic processes on networks, including epidemic and information-spreading networks. While devising stochastic models of such processes is relatively easy, analyzing them is very challenging from a computational point of view, particularly for large networks appearing in real applications. One strategy to reduce the complexity of their analysis is to rely on approximations, often in terms of a set of differential equations capturing the evolution of a random node, distinguishing nodes with different topological contexts (i.e., different degrees of different neighborhoods), such as degree-based mean-field (DBMF), approximate-master-equation (AME), or pair-approximation (PA) approaches. The number of differential equations so obtained is typically proportional to the maximum degree k_{max} of the network, which is much smaller than the size of the master equation of the underlying stochastic model, yet numerically solving these equations can still be problematic for large k_{max}. In this paper, we consider AME and PA, extended to cope with multiple local states, and we provide an aggregation procedure that clusters together nodes having similar degrees, treating those in the same cluster as indistinguishable, thus reducing the number of equations while preserving an accurate description of global observables of interest. We also provide an automatic way to build such equations and to identify a small number of degree clusters that give accurate results. The method is tested on several case studies, where it shows a high level of compression and a reduction of computational time of several orders of magnitude for large networks, with minimal loss in accuracy.

Dynamic model of a self-excited induction generator with fundamental stray load and iron losses

This paper considers a dynamic model of a self-excited induction generator that takes into account the fundamental stray load and iron losses. The model is de?scribed with the same number of differential equations as the conventional induc?tion machine model. Determination of the stray load and iron losses resistances does not involve any tests other than those imposed by the international standards nor does it require any details about induction machine materials or geometry. The dynamic analysis has been carried out for the case of a wind turbine-driven self-excited induction generator. The steady-state analysis, on the other hand, has been carried out for the case of a load-independent prime mover. The considered advanced model, aside from being compared with the conventional model, has been experimentally validated for two different-efficiency induction machines, both rated 1.5 kW.

Nonlinear System Identification of a Refrigeration System

Applications of advanced control algorithms are important in the refrigeration field to achieve low-energy costs and accurate set-point tracking. However, the designing and tuning of control systems depend on dynamic mathematical models. Approaches like analytical modeling can be time-consuming because they usually lead to a large number of differential equations with unknown parameters. In this work, the application of system identification with the fast recursive orthogonal least square (FROLS) algorithm is proposed as an alternative to analytical modeling to develop a process dynamic model. The evaporating temperature (EVT), condensing temperature (CDT) and useful superheat (USH) are the outputs of interest for this system; covariance analysis of the candidate inputs shows that the model should be single-input–single-output (SISO). Good simulation results are obtained with two different validation data, with average output errors of 0.0343 (EVT model), 0.0079 (CDT model) and 0.1578 (USH model) for one of the datasets, showing that this algorithm is a valid alternative for modeling refrigeration systems.

Degree of Disequilibrium analysis for automatic selection of kinetic constraints in the Rate-Controlled Constrained-Equilibrium method

The Rate-Controlled Constrained-Equilibrium (RCCE) model reduction scheme for chemical kinetics provides acceptable accuracies with a number of differential equations much lower than the number of species in the underlying Detailed Kinetic Model (DKM). To yield good approximations, however, the method requires accurate identification of the rate controlling constraints. So far, a drawback of the RCCE scheme has been the absence of a fully automatable and systematic procedure that is capable of identifying the best constraints for a given range of thermodynamic conditions and a required level of approximation. In this paper, we propose a new methodology for such identification based on a simple algebraic analysis of the results of a preliminary simulation of the underlying DKM, which is focused on the behaviour of the degrees of disequilibrium (DoD) of the individual chemical reactions. The new methodology is based on computing an Approximate Reduced Row Echelon Form of the Actual Degrees of Disequilibrium (ARREFADD) with respect to a preset tolerance level. An alternative variant is to select an Approximate Singular Value Decomposition of the Actual Degrees of Disequilibrium (ASVDADD). Either procedure identifies a low dimensional subspace in the DoD space, from which the actual DoD traces do not depart beyond a fixed distance related to the preset tolerance (ARREFADD methodology) or to the first neglected singular value of the matrix of DoD traces (ASVDADD methodology). The effectiveness and robustness of the method is demonstrated for the case of a very rapid supersonic nozzle expansion of the products of hydrogen and methane oxycombustion and for the case of methane/oxygen ignition. The results are in excellent agreement with DKM predictions. For both variants of the method, we provide a simple Matlab code implementing the proposed constraint selection algorithm.

Complete hierarchies of SIR models on arbitrary networks with exact and approximate moment closure

We first generalise ideas discussed by Kiss et al. (2015) to prove a theorem for generating exact closures (here expressing joint probabilities in terms of their constituent marginal probabilities) for susceptible-infectious-removed (SIR) dynamics on arbitrary graphs (networks). For Poisson transmission and removal processes, this enables us to obtain a systematic reduction in the number of differential equations needed for an exact ‘moment closure’ representation of the underlying stochastic model. We define ‘transmission blocks’ as a possible extension of the block concept in graph theory and show that the order at which the exact moment closure representation is curtailed is the size of the largest transmission block. More generally, approximate closures of the hierarchy of moment equations for these dynamics are typically defined for the first and second order yielding mean-field and pairwise models respectively. It is frequently implied that, in principle, closed models can be written down at arbitrary order if only we had the time and patience to do this. However, for epidemic dynamics on networks, these higher-order models have not been defined explicitly. Here we unambiguously define hierarchies of approximate closed models that can utilise subsystem states of any order, and show how well-known models are special cases of these hierarchies.

A method of inverse differential operators using ortogonal polynomials and special functions for solving some types of differential equations and physical problems

A general operational method, which is based on the developed technique of the inverse derivative operator, for solving a wide range of problems described by some classes of differential equations is represented. The inverse derivative operators for solving a number of differential equations are constructed and used. The operational identities are derived with the use of the inverse derivative operator, integral transformations, and generalized forms of orthogonal polynomials and special functions. Examples of solving various partial differential equations, such as equations of heat conduction and diffusion, as well as the Fokker-Planck equation, etc. are given. The application of the operational approach to solving a number of physical problems, among them problems related to the motion of charged particles in external field, is demonstrated.

On a system of nonlinear differential equations of high dimension

Under consideration is the Cauchy problem for a system of nonlinear differential equations of high dimension. It is proved that when a number of differential equations is sufficiently large, the last component of a solution to the Cauchy problem is an approximate solution to the initial value problem for a delay differential equation.