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We consider several examples of nonautonomous systems of difference equations coming from semi-classical orthogonal polynomials via recurrence coefficients and ladder operators, with respect to various generalisations of Laguerre and…

We consider a wide class of fully nonlinear integro-differential equations that degenerate when the gradient of the solution vanishes. By using compactness and perturbation arguments, we give a complete characterization of the regularity of…

In this paper, we consider a class of fully nonlinear equations on closed smooth Riemannian manifolds, which can be viewed as an extension of $\sigma_k$ Yamabe equation. Moreover, we prove local gradient and second derivative estimates for…

We investigate the existence and uniqueness of solutions for second-order semi-linear partial differential equations defined on a Riemannian manifold $M$. By combining differential geometry and analysis techniques, we establish the…

We propose a natural gradient term for a class of second-order partial differential equations of the form \begin{equation}\nonumber M(x,Du,D^2u)+g(u)N(x,Du, D^2u)+f(x,u)=0 \;\;\mbox{in}\;\; \Omega, \end{equation} where…

Let $k$ be a differential field of characteristic zero with an algebraically closed field of constants. In this article, we provide a classification of first order differential equations over $k$ and study the algebraic dependence of…

We study existence, uniqueness, norm estimates and asymptotic time behaviour (in some cases can be claimed to be sharp) for the solution of a general evolutionary integral (differential) equation of scalar type on a locally compact…

This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It presents a new framework using these equations as a central tool for computation and…

We define a new grading, that we call the "level grading", on the algebra of polynomials generated by the derivatives $u_{k+i}=\partial^{k+i}u/\partial x^{k+i}$ over the ring $K^{(k)}$ of $C^{\infty}$ functions of $u,u_1,...,u_k$. This…

We prove that a general class of nonlinear, non-autonomous ODEs in Fr\'echet spaces are close to ODEs in a specific normal form, where closeness means that solutions of the normal form ODE satisfy the original ODE up to a residual that…

The classification, both up to isomorphism or up to equivalence, of the gradings on a finite dimensional nonassociative algebra A over an algebraically closed field F, such that its group scheme of automorphisms is smooth, is shown to be…

It seems that in the current age, computers, computation, and data have an increasingly important role to play in scientific research and discovery. This is reflected in part by the rise of machine learning and artificial intelligence,…

In this paper we characterise the global stability, global boundedness and recurrence of solutions of a scalar nonlinear stochastic differential equation. The differential equation is a perturbed version of a globally stable autonomous…

A linear system of difference equations and a nonlinear perturbation are considered, we obtain sufficient conditions to ensure the topological equivalence between them, namely, the linear part satisfies a property of dichotomy on the…

We find that to the dynamics of a given dissipative system a $p=1$ differential form can be associated with a general decomposition into a potential term and a non-potential residual part. If the residual part is absent the form is closed…

In this paper, autonomous differential equations type delta of order one on integral domains are defined. For this we will use the autonomous ring defined on the Hurwitz expansion ring of exponential generating functions with coefficients…

In this paper, we propose a class of stochastic exponential discrete gradient schemes for SDEs with linear and gradient components in the coefficients. The root mean-square errors of the schemes are analyzed, and the structure-preserving…

Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…

We study global regularity of nonlinear systems of partial differential equations depending on the symmetric part of the gradient with Dirichlet boundary conditions. These systems arise from variational problems in plasticity with power…

A class of fourth--order neutral type difference equations with quasidifferences and deviating arguments is considered. Our approach is based on studying the considered equation as a system of a four--dimensional difference system. The…