Related papers: Differential-Algebraic Equations and Beyond: From …
The paper introduces a new finite element numerical method for the solution of partial differential equations on evolving domains. The approach uses a completely Eulerian description of the domain motion. The physical domain is embedded in…
In this paper we review the simulation method of the non-smooth contact dynamics. This technique was designed to solve the unilateral and frictional contact problem for a large number of rigid bodies and has proved to be especially valuable…
This paper is devoted to the study of some connections between coadjoint orbits in infinite dimensional Lie algebras, isospectral deformations and linearization of dynamical systems. We explain how results from deformation theory,…
In this paper, we propose a novel algebraic and geometric description for the dissipative dynamics. Our formulation bears some similarity to the Poisson structure for non-dissipative systems. We develop a canonical description for…
We derive an exact solution for a simple non-autonomous delay differential equation (DDE) over the entire real-time axis, representing it as a sum of Gaussian-shaped dynamics with distinct peak positions. This marks the first explicit…
The aim of these notes is to present an accessible overview of some topics in classical algebraic geometry which have applications to aspects of discrete integrable systems. Precisely, we focus on surface theory on the algebraic geometry…
Input-affine dynamical systems often arise in control and modeling scenarios, such as the data-driven case when state-derivative observations are recorded under bounded noise. Common tasks in system analysis and control include optimal…
In this note, we address the following question: Why certain nonassociative algebra structures emerge in the regularity theory of elliptic type PDEs and also in constructing nonclassical and singular solutions? The aim of the paper is…
Symbolic dynamics is a coarse-grained description of dynamics. By taking into account the ``geometry'' of the dynamics, it can be cast into a powerful tool for practitioners in nonlinear science. Detailed symbolic dynamics can be developed…
Motivated by the modeling of three-dimensional fluid turbulence, we define and study a class of stochastic partial differential equations (SPDEs) that are randomly stirred by a spatially smooth and uncorrelated in time forcing term. To…
This article proposes modifications of the Parareal algorithm for its application to higher index differential algebraic equations (DAEs). It is based on the idea of applying the algorithm to only the differential components of the equation…
We give an abstract formulation of the formal theory partial differential equations (PDEs) in synthetic differential geometry, one that would seamlessly generalize the traditional theory to a range of enhanced contexts, such as…
Fractional differential equations (FDEs) are an extension of the theory of fractional calculus. However, due to the difficulty in finding analytical solutions, there have not been extensive applications of FDEs until recent decades. With…
The aim of this article is to highlight the interest to apply Differential Geometry and Mechanics concepts to chaotic dynamical systems study. Thus, the local metric properties of curvature and torsion will directly provide the analytical…
This paper presents a geometric variational discretization of compressible fluid dynamics. The numerical scheme is obtained by discretizing, in a structure preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups…
End-to-end learning of dynamical systems with black-box models, such as neural ordinary differential equations (ODEs), provides a flexible framework for learning dynamics from data without prescribing a mathematical model for the dynamics.…
In this work, we present a result on the local existence and uniqueness of solutions to nonlinear Partial Differential-Algebraic Equations (PDAEs). By applying established theoretical results, we identify the conditions that guarantee the…
The quest for analytical solutions to differential equations has traditionally been constrained by the need for extensive mathematical expertise. Machine learning methods like genetic algorithms have shown promise in this domain, but are…
This paper studies formulations of second-order elliptic partial differential equations in nondivergence form on convex domains as equivalent variational problems. The first formulation is that of Smears \& S\"uli [SIAM J.\ Numer.\ Anal.\…
Reduced-order models have long been used to understand the behavior of nonlinear partial differential equations (PDEs). Naturally, reduced-order modeling techniques come at the price of computational accuracy for a decrease in computation…