Some properties of solutions of a functional-differential equation of second order with delay.
Ontology highlight
ABSTRACT:
SUBMITTER: Ilea VA
PROVIDER: S-EPMC3934606 | biostudies-other | 2014
REPOSITORIES: biostudies-other
Similar Datasets
Project description:The stochastic simulation algorithm (SSA) describes the time evolution of a discrete nonlinear Markov process. This stochastic process has a probability density function that is the solution of a differential equation, commonly known as the chemical master equation (CME) or forward-Kolmogorov equation. In the same way that the CME gives rise to the SSA, and trajectories of the latter are exact with respect to the former, trajectories obtained from a delay SSA are exact representations of the underlying delay CME (DCME). However, in contrast to the CME, no closed-form solutions have so far been derived for any kind of DCME. In this paper, we describe for the first time direct and closed solutions of the DCME for simple reaction schemes, such as a single-delayed unimolecular reaction as well as chemical reactions for transcription and translation with delayed mRNA maturation. We also discuss the conditions that have to be met such that such solutions can be derived.
| S-EPMC4528653 | biostudies-literature
Project description:We study the nonlinear integrable equation, u t + 2((u x u xx )/u) = ?u xxx , which is invariant under scaling of dependent variable and was called the SIdV equation (see Sen et al. 2012 Commun. Nonlinear Sci. Numer. Simul. 17, 4115-4124 (doi:10.1016/j.cnsns.2012.03.001)). The order-n kink solution u [n] of the SIdV equation, which is associated with the n-soliton solution of the Korteweg-de Vries equation, is constructed by using the n-fold Darboux transformation (DT) from zero 'seed' solution. The kink-type solutions generated by the onefold, twofold and threefold DT are obtained analytically. The key features of these kink-type solutions are studied, namely their trajectories, phase shifts after collision and decomposition into separate single kink solitons.
| S-EPMC6731723 | biostudies-literature
Project description:Fractional derivative or fractional calculus plays a significant role in theoretical modeling of scientific and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.
| S-EPMC3869418 | biostudies-literature
Project description:We use the polynomial least squares method (PLSM), which allows us to compute analytical approximate polynomial solutions for a very general class of strongly nonlinear delay differential equations. The method is tested by computing approximate solutions for several applications including the pantograph equations and a nonlinear time-delay model from biology. The accuracy of the method is illustrated by a comparison with approximate solutions previously computed using other methods.
| S-EPMC4134833 | biostudies-other
Project description:Second order linear differential equations can be used as models for regulation since under a range of parameter values they can account for return to equilibrium as well as potential oscillations in regulated variables. One method that can estimate parameters of these equations from intensive time series data is the method of Latent Differential Equations (LDE). However, the LDE method can exhibit bias in its parameters if the dimension of the time delay embedding and thus the width of the convolution kernel is not chosen wisely. This article presents a simulation study showing that a constrained fourth order Latent Differential Equation (FOLDE) model for the second order system almost completely eliminates bias as long as the width of the convolution kernel is less than two thirds the period of oscillations in the data. The FOLDE model adds two degrees of freedom over the standard LDE model but significantly improves model fit.
| S-EPMC7518521 | biostudies-literature
Project description:Dengue is considered non-endemic to mainland China. However, travellers frequently import the virus from overseas and local mosquito species can then spread the disease in the population. As a consequence, mainland China still experiences large dengue outbreaks. Temperature plays a key role in these outbreaks: it affects the development and survival of the vector and the replication rate of the virus. To better understand its implication in the transmission risk of dengue, we developed a delay differential equation model that explicitly simulates temperature-dependent development periods and tested it with collected field data for the Asian tiger mosquito, Aedes albopictus. The model predicts mosquito occurrence locations with a high accuracy (Cohen's κ of 0.78) and realistically replicates mosquito population dynamics. Analysing the infection dynamics during the 2014 dengue outbreak that occurred in Guangzhou showed that the outbreak could have lasted for another four weeks if mosquito control interventions had not been undertaken. Finally, we analyse the dengue transmission risk in mainland China. We find that southern China, including Guangzhou, can have more than seven months of dengue transmission per year while even Beijing, in the temperate north, can have dengue transmission during hot summer months. The results demonstrate the importance of using detailed vector and infection ecology, especially when vector-borne disease transmission risk is modelled over a broad range of climatic zones.
| S-EPMC7996998 | biostudies-literature
Project description:Chiral organic compounds are excellent second-order nonlinear optical (NLO) materials due to their inherent non-symmetric electronic structures combined with the advantages of organic compounds. At present, density functional theory (DFT) has become a powerful tool for predicting the properties of novel materials. In this paper, based on chiral lemniscular [16]cycloparaphenylene, three novel compounds are designed by introduction of donor/acceptor units and their combinations. The geometrical/electronic structure, electronic absorption, and the second-order NLO properties of these compounds have been systematically investigated by DFT/TDDFT theory. The simulated UV-Vis/CD spectra of compound 1 are in good agreement with the experimental ones, enabling us to assign their electronic transition characteristics and absolute configuration with high confidence. The investigations show that energy gaps, absorption wavelength and second-order NLO response may be effectively tuned by the introduction of the donor or acceptor units or their combinations. For instance, the second-order NLO value of compound 4 is about 207 times as large as the average second-order polarizability of the organic molecule urea. Thus, the studied compounds are expected to be potential large second-order NLO materials.
| S-EPMC9051605 | biostudies-literature
Project description:In the "footsteps illusion", light and dark squares travel at constant speed across black and white stripes. The squares appear to move faster and slower as their contrast against the stripes varies. We now demonstrate some second-order footsteps illusions, in which all edges are defined by colors or textures-even though luminance-based neural motion detectors are blind to such edges.
| S-EPMC4975121 | biostudies-literature
Project description:We develop a second-order correction to commonly used density functional approximations (DFAs) to eliminate the systematic delocalization error. The method, based on the previously developed global scaling correction (GSC), is an exact quadratic correction to the DFA for the fractional charge behavior and uses the analytical second derivatives of the total energy with respect to fractional occupation numbers of the canonical molecular orbitals. For small and medium-size molecules, this correction leads to ground-state orbital energies that are a highly accurate approximation to the corresponding quasiparticle energies. It provides excellent predictions of ionization potentials, electron affinities, photoemission spectrum, and photoexcitation energies beyond previous approximate second-order approaches, thus showing potential for broad applications in computational spectroscopy.
| S-EPMC9367128 | biostudies-literature
Project description:Constrained fourth-order latent differential equation (FOLDE) models have been proposed (e.g., Boker et al. 2020) as alternative to second-order latent differential equation (SOLDE) models to estimate second-order linear differential equation systems such as the damped linear oscillator model. When, however, only a relatively small number of measurement occasions T are available (i.e., [Formula: see text]), the recommendation of which model to use is not clear (Boker et al. 2020). Based on a data set, which consists of [Formula: see text] observations of daily stress for [Formula: see text] individuals, we illustrate that FOLDE can help to choose an embedding dimension, even in the case of a small T. This is of great importance, as parameter estimates depend on the embedding dimension as well as on the latent differential equations model. Consequently, the wavelength as quantity of potential substantive interest may vary considerably. We extend the modeling approaches used in past research by including multiple subjects, by accounting for individual differences in equilibrium, and by including multiple instead of one single observed indicator.
| S-EPMC7826318 | biostudies-literature