Related papers: A gentle introduction to the differential equation…
A nonlinear partial differential equation is a nonlinear relationship between an unknown function and how it changes due to two or more input variables. A numerical method reduces such an equation to arithmetic for quick visualization, but…
In this article we review classical and recent results in anomalous diffusion and provide mechanisms useful for the study of the fundamentals of certain processes, mainly in condensed matter physics, chemistry and biology. Emphasis will be…
This paper proposes a dynamical Variable-separation method for solving parameter-dependent dynamical systems. To achieve this, we establish a dynamical low-rank approximation for the solutions of these dynamical systems by successively…
We consider a hidden Markov model, where the signal process, given by a diffusion, is only indirectly observed through some noisy measurements. The article develops a variational method for approximating the hidden states of the signal…
We present an ``equation-free'' multiscale approach to the simulation of unsteady diffusion in a random medium. The diffusivity of the medium is modeled as a random field with short correlation length, and the governing equations are cast…
There has recently been considerable interest in using a nonstandard piecewise approximation to formulate fractional order differential equations as difference equations that describe the same dynamical behaviour and are more amenable to a…
To study discrete dynamical systems of different types --- deterministic, statistical and quantum --- we develop various approaches. We introduce the concept of a system of discrete relations on an abstract simplicial complex and develop…
In this article we develop a new methodology to prove weak approximation results for general stochastic differential equations. Instead of using a partial differential equation approach as is usually done for diffusions, the approach…
We propose a method to reduce the computational effort to solve a partial differential equation on a given domain. The main idea is to split the domain of interest in two subdomains, and to use different approximation methods in each of the…
Stochastic differential equations describe well many physical, biological and sociological systems, despite the simplification often made in their derivation. Here the usage of simple stochastic differential equations to characterize and…
A multi-linear variable separation approach is developed to solve a differential-difference Toda equation. The semi-discrete form of the continuous universal formula is found for a suitable potential of the differential-difference Toda…
Random invariant manifolds often provide geometric structures for understanding stochastic dynamics. In this paper, a dynamical approximation estimate is derived for a class of stochastic partial differential equations, by showing that the…
We obtain large and moderate deviation estimates, as well as concentration inequalities, for a class of nonuniformly expanding maps with stretched exponential decay of correlations. In the large deviation regime, we also exhibit examples…
We propose certain approach of solving two-dimensional non-stationary and stationary advection-diffusion-reaction boundary value problems through their reduction to the set of corresponding one-dimensional problems. This method leverages…
A version of the Dynamical Systems Gradient Method for solving ill-posed nonlinear monotone operator equations is studied in this paper. A discrepancy principle is proposed and justified. A numerical experiment was carried out with the new…
Using the damped pendulum system we introduce the averaging method to study the periodic solutions of a dynamical system with small perturbation. We provide sufficient conditions for the existence of periodic solutions with small amplitude…
The primary emphasis of this work on kinetics is to illustrate the a posteriori approach to applications, where focus on data leads to novel outcomes, rather than the a priori tendencies of applied analysis which imposes constructs on the…
In this paper, we introduce and study a class of resolvent dynamical systems to investigate some inertial proximal methods for solving mixed variational inequalities. These proposed methods along with their discretizations and derived rates…
In this work, we address the numerical identification of entanglement in dynamical scenarios. To this end, we consider different programs based on the restriction of the evolution to the set of separable (i.e., non-entangled) states,…
The discrete-dipole approximation (DDA) is a flexible technique for computing scattering and absorption by targets of arbitrary geometry. In this paper we perform systematic study of various non-stationary iterative (conjugate gradient)…