215 research outputs found
Half-linear differential equations : regular variation, principal solutions, and asymptotic classes
We are interested in the structure of the solution space of second-order half-linear differential equations taking into account various classifications regarding asymptotics of solutions. We focus on an exhaustive analysis of the relations among several types of classes which include the classes constructed with respect to the values of the limits of solutions and their quasiderivatives, the classes of regularly varying solutions, the classes of principal and nonprincipal solutions, and the classes of the solutions that obey certain asymptotic formulae. Many of our observations are new even in the case of linear differential equations, and we provide also the revision of existing results
Half-linear differential equations: Regular variation, principal solutions, and asymptotic classes
We are interested in the structure of the solution space of second-order half-linear differential equations taking into account various classifications regarding asymptotics of solutions. We focus on an exhaustive analysis of the relations among several types of classes which include the classes constructed with respect to the values of the limits of solutions and their quasiderivatives, the classes of regularly varying solutions, the classes of principal and nonprincipal solutions, and the classes of the so-lutions that obey certain asymptotic formulae. Many of our observations are new even in the case of linear differential equations, and we provide also the revision of existing results
Research in the general area of non-linear dynamical systems Final report, 8 Jun. 1965 - 8 Jun. 1967
Nonlinear dynamical systems research on systems stability, invariance principles, Liapunov functions, and Volterra and functional integral equation
New developments in Functional and Fractional Differential Equations and in Lie Symmetry
Delay, difference, functional, fractional, and partial differential equations have many applications in science and engineering. In this Special Issue, 29 experts co-authored 10 papers dealing with these subjects. A summary of the main points of these papers follows:Several oscillation conditions for a first-order linear differential equation with non-monotone delay are established in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, whereas a sharp oscillation criterion using the notion of slowly varying functions is established in A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. The approximation of a linear autonomous differential equation with a small delay is considered in Approximation of a Linear Autonomous Differential Equation with Small Delay; the model of infection diseases by Marchuk is studied in Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Exact solutions to fractional-order FokkerâPlanck equations are presented in New Exact Solutions and Conservation Laws to the Fractional-Order FokkerâPlanck Equations, and a spectral collocation approach to solving a class of time-fractional stochastic heat equations driven by Brownian motion is constructed in A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise. A finite difference approximation method for a space fractional convection-diffusion model with variable coefficients is proposed in Finite Difference Approximation Method for a Space Fractional ConvectionâDiffusion Equation with Variable Coefficients; existence results for a nonlinear fractional difference equation with delay and impulses are established in On Nonlinear Fractional Difference Equation with Delay and Impulses. A complete Noether symmetry analysis of a generalized coupled LaneâEmdenâKleinâGordonâFock system with central symmetry is provided in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, and new soliton solutions of a fractional Jaulent soliton Miodek system via symmetry analysis are presented in New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis
Intrinsic and extrinsic thermodynamics for stochastic population processes with multi-level large-deviation structure
A set of core features is set forth as the essence of a thermodynamic
description, which derive from large-deviation properties in systems with
hierarchies of timescales, but which are \emph{not} dependent upon conservation
laws or microscopic reversibility in the substrate hosting the process. The
most fundamental elements are the concept of a macrostate in relation to the
large-deviation entropy, and the decomposition of contributions to
irreversibility among interacting subsystems, which is the origin of the
dependence on a concept of heat in both classical and stochastic
thermodynamics. A natural decomposition is shown to exist, into a relative
entropy and a housekeeping entropy rate, which define respectively the
\textit{intensive} thermodynamics of a system and an \textit{extensive}
thermodynamic vector embedding the system in its context. Both intensive and
extensive components are functions of Hartley information of the momentary
system stationary state, which is information \emph{about} the joint effect of
system processes on its contribution to irreversibility. Results are derived
for stochastic Chemical Reaction Networks, including a Legendre duality for the
housekeeping entropy rate to thermodynamically characterize fully-irreversible
processes on an equal footing with those at the opposite limit of
detailed-balance. The work is meant to encourage development of inherent
thermodynamic descriptions for rule-based systems and the living state, which
are not conceived as reductive explanations to heat flows
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