Related papers: A gentle introduction to the differential equation…

In chemical reaction network theory, ordinary differential equations are used to model the temporal change of chemical species concentration. As the functional form of these ordinary differential equations systems is derived from an…

This article proposes a dynamical system modeling approach for the analysis of longitudinal data of self-regulated systems experiencing multiple excitations. The aim of such an approach is to focus on the evolution of a signal (e.g., heart…

We give an introduction to discrete functional analysis techniques for stationary and transient diffusion equations. We show how these techniques are used to establish the convergence of various numerical schemes without assuming…

In this paper we present a method of discrete modeling and analysis of multi-level dynamics of complex large-scale hierarchical dynamic systems subject to external dynamic control mechanism. In a model each state describes parallel dynamics…

Partial differential equations with discrete (concentrated) state-dependent delays in the space of continuous functions are investigated. In general, the corresponding initial value problem is not well posed, so we find an additional…

The energy method can be used to identify well-posed initial boundary value problems for quasi-linear, symmetric hyperbolic partial differential equations with maximally dissipative boundary conditions. A similar analysis of the discrete…

We propose new numerical approach to non-conservative dynamical systems. Our method being of low order, enhances qualitative performance of standard discrete gradient algorithm, thank to new concept of a reservoir. Paper is of explanatory…

The numerical solution methods for partial differential equation (PDE) solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods…

The martingale method is used to establish concentration inequalities for a class of dependent random sequences on a countable state space, with the constants in the inequalities expressed in terms of certain mixing coefficients. Along the…

Here we introduce some new classes of discrete stable random variables, which are useful for understanding of a new general notion of stability of random variables called us as casual stability. There are given some examples of casual and…

In solving diffusion problems, it is common to consider the finite difference equation to be an approximation to the differential equation. Nevertheless, history shows that the finite difference equation is primitive and that the…

This article introduces the splitting method to systems responding to rough paths as external stimuli. The focus is on nonlinear partial differential equations with rough noise but we also cover rough differential equations. Applications to…

Ordinary differential equations (ODEs) are the primary means to modelling dynamical systems in many natural and engineering sciences. The number of equations required to describe a system with high heterogeneity limits our capability of…

This short, self-contained article seeks to introduce and survey continuous-time deep learning approaches that are based on neural ordinary differential equations (neural ODEs). It primarily targets readers familiar with ordinary and…

Differential complexes such as the de Rham complex have recently come to play an important role in the design and analysis of numerical methods for partial differential equations. The design of stable discretizations of systems of partial…

In this article, I introduce the differential equation model and review their frequentist and Bayesian computation methods. A numerical example of the FitzHugh-Nagumo model is given.

A computational tool for coarse-graining nonlinear systems of ordinary differential equations in time is discussed. Three illustrative model examples are worked out that demonstrate the range of capability of the method. This includes the…

We present a microscopic approach to quantum dissipation and sketch the derivation of the kinetic equation describing the evolution of a simple quantum system in interaction with a complex quantum system. A typical quantum complex system is…

We begin by introducing the main ideas of the paper under discussion, and we give a brief description of the method proposed. Next, we discuss an alternative approach based on B-spline expansion, and lastly we make some comments on the…

An exact discretization method is being developed for solving linear systems of ordinary fractional-derivative differential equations with constant matrix coefficients (LSOFDDECMC). It is shown that the obtained linear discrete system in…