Related papers: Discrete integrable systems and condensation algor…
We study the discretization of (almost-)Dirac structures using the notion of retraction and discretization maps on manifolds. Additionally, we apply the proposed discretization techniques to obtain numerical integrators for port-Hamiltonian…
We study discretization of Darboux integrable systems. The discretization is done by using $x$- or $y$-integrals of the considered systems. New examples of semi-discrete Darboux integrable systems are obtained.
This paper deals with the design of discrete-time algorithms for the robust filtering differentiator. Two discrete-time realizations of the filtering differentiator are introduced. The first one, which is based on an exact discretization of…
Coarse-grained models that preserve hydrodynamics provide a natural approach to study collective properties of soft-matter systems. Here, we demonstrate that commonly used integration schemes in dissipative particle dynamics give rise to…
This work studies limits of Pfaffian systems, a class of first-order PDEs appearing in the Feynman integral calculus. Such limits appear naturally in the context of scattering amplitudes when there is a separation of scale in a given set of…
Recently, continuous-time dynamical systems have proved useful in providing conceptual and quantitative insights into gradient-based optimization, widely used in modern machine learning and statistics. An important question that arises in…
We introduce a new $C^1$ algorithm for the rigorous integration of dissipative partial differential equations. The algorithm is designed for computer-assisted proofs that require rigorous control of both solutions and their derivatives with…
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
Based on the continuous time random walk, we derive the Fokker-Planck equations with Caputo-Fabrizio fractional derivative, which can effectively model a variety of physical phenomena, especially, the material heterogeneities and structures…
We study the discretization of Darboux integrable systems. The discretization is done using $x$-, $y$-integrals of the considered continuous systems. New examples of semi-discrete Darboux integrable systems are obtained.
This paper discusses the properties and the numerical discretizations of the fractional substantial integral $$I_s^\nu f(x)=\frac{1}{\Gamma(\nu)} \int_{a}^x{\left(x-\tau\right)^{\nu-1}}e^{-\sigma(x-\tau)}{f(\tau)}d\tau,\nu>0, $$ and the…
In this work we study arbitrary-order hybrid discretizations of Friedrichs systems. Friedrichs systems provide a framework that goes beyond the standard classification of partial differential equations into hyperbolic or elliptic, and are…
Discrete variational methods show excellent performance in numerical simulations of mechanical systems. In this paper, we adapt discrete variational integrators for the case of mechanical systems with double-bracket dissipation. In…
We represent an integration algorithm combining the characteristics method and Hopf-Cole transformation. This algorithm allows one to partially integrate a large class of multidimensional systems of nonlinear Partial Differential Equations…
In analogy to the definition of the lambda-determinant, we define a one-parameter deformation of the Dodgson condensation formula for Pfaffians. We prove that the resulting rational function is a polynomial with weights given by the…
We present a finite element scheme for fractional diffusion problems with varying diffusivity and fractional order. We consider a symmetric integral form of these nonlocal equations defined on general geometries and in arbitrary bounded…
Partition functions for dimers on closed oriented surfaces are known to be alternating sums of Pfaffians of Kasteleyn matrices. In this paper, we obtain the formula for the coefficients in terms of discrete spin structures.
An interesting family of geometric integrators for Lagrangian systems can be defined using discretizations of the Hamilton's principle of critical action. This family of geometric integrators is called variational integrators. In this…
The main result of this paper is a Pfaffian formula for the partition function of the dimer model on a graph G embedded in a closed, possibly non-orientable surface S. This formula is suitable for computational purposes, and it is obtained…
In this paper we design discrete port-Hamiltonian systems systematically in two different ways, by applying discrete gradient methods and splitting methods respectively. The discrete port-Hamiltonian systems we get satisfy a discrete notion…