Related papers: General Matrix-Valued Inhomogeneous Linear Stochas…
A general method for solving linear differential equations of arbitrary order, is used to arrive at new representations for the solutions of the known differential equations, both without and with a source term. A new quasi-solvable…
One obtains a probabilistic representation for the entropic generalized solutions to a nonlinear Fokker-Planck equation in $\mathbb R^d$ with multivalued nonlinear diffusion term as density probabilities of solutions to a nonlinear…
The present paper is concerned with a space-time homogenization problem for nonlinear diffusion equations with periodically oscillating (in space and time) coefficients. Main results consist of a homogenization theorem (i.e., convergence of…
We derive an efficient stochastic algorithm for inverse problems that present an unknown linear forcing term and a set of nonlinear parameters to be recovered. It is assumed that the data is noisy and that the linear part of the problem is…
This paper is a survey of methods for solving smooth (strongly) monotone stochastic variational inequalities. To begin with, we give the deterministic foundation from which the stochastic methods eventually evolved. Then we review methods…
Asymptotic solutions are derived for inhomogeneous differential equations having a large real or complex parameter and a simple turning point. They involve Scorer functions and three slowly varying analytic coefficient functions. The…
We present a method for linear stability analysis of systems with parametric uncertainty formulated in the stochastic Galerkin framework. Specifically, we assume that for a model partial differential equation, the parameter is given in the…
We study approximation of non-autonomous linear differential equations with variable delay over infinite intervals. We use piecewise constant argument to obtain a corresponding discrete difference equation. The study of numerical…
Direct solution of simultaneous linear equations is regarded to be slow for large systems of equations and requires special treatment to avoid numerical instability. A new method is proposed that addresses the numerical instability without…
We exhibit an alternative method for solving inhomogeneous second--order linear ordinary dynamic equations on time scales, based on reduction of order rather than variation of parameters. Our form extends recent (and long-standing) analysis…
A new method for solving numerically stochastic partial differential equations (SPDEs) with multiple scales is presented. The method combines a spectral method with the heterogeneous multiscale method (HMM) presented in [W. E, D. Liu, and…
We develop the rough path counterpart of It\^o stochastic integration and - differential equations driven by general semimartingales. This significantly enlarges the classes of (It\^o / forward) stochastic differential equations treatable…
The existence and uniqueness of measure-valued solutions to stochastic nonlinear, non-local Fokker-Planck equations is proven. This type of stochastic PDE is shown to arise in the mean field limit of weakly interacting diffusions with…
This article is devoted to the study of solutions of non-homogenous linear differential equations having entire coefficients. We get all non-trivial solutions of infinite order of equation $f^{(n)}+a_{n-1}(z)f^{(n-1)}+\ldots…
The detailed construction of the general solution of a second order non-homogenous linear operatordifference equation is presented. The wide applicability of such an equation as well as the usefulness of its resolutive formula is shown by…
We consider linear and quadratic integrals of motion for general variable quadratic Hamiltonians. Fundamental relations between the eigenvalue problem for linear dynamical invariants and solutions of the corresponding Cauchy initial value…
The regularity and characterization of solutions to degenerate, quasilinear SPDE is studied. Our results are two-fold: First, we prove regularity results for solutions to certain degenerate, quasilinear SPDE driven by Lipschitz continuous…
Matrix determinants play an important role in data analysis, in particular when Gaussian processes are involved. Due to currently exploding data volumes, linear operations - matrices - acting on the data are often not accessible directly…
In this paper we consider multivariate time series obtained as solution to multidimensional nonlinear stochastic difference equations whose coefficients are allowed to be locally degenerate and to present discontinuities. We provide simple…
As a continuation of our previous work \cite{KV2} the aim of the recent paper is to investigate the solutions of special inhomogeneous linear functional equations by using spectral synthesis in translation invariant closed linear subspaces…