Related papers: Generalized invariant manifolds for integrable equ…

We formulate probabilistic numerical approximations to solutions of ordinary differential equations (ODEs) as problems in Gaussian process (GP) regression with non-linear measurement functions. This is achieved by defining the measurement…

We study some classes of semi-linear differential equations including both well-posed and ill-posed cases that can generate cocycles (or cocycle correspondences with generating cocycles). Under exponential dichotomy condition with other…

In a recent paper [TMP, 200:1 (2019), 966--984] by the authors, a series of integrable discrete autonomous equations on a square lattice with a non-standard structure of generalized symmetries is constructed. We build modified series by…

A new class of integrable mappings and chains is introduced. Corresponding $(1+2)$ integrable systems invariant with respect to such discrete transformations are presented in an explicit form. Their soliton-type solutions are constructed in…

Mixed-dimensional partial differential equations arise in several physical applications, wherein parts of the domain have extreme aspect ratios. In this case, it is often appealing to model these features as lower-dimensional manifolds…

Geometric integration theory can be employed when numerically solving ODEs or PDEs with constraints. In this paper, we present several one-step algorithms of various orders for ODEs on a collection of spheres. To demonstrate the versatility…

For a system of partial differential equations admitting point, contact, or higher symmetries, the framework of invariant reduction systematically computes how invariant geometric structures, such as conservation laws, presymplectic…

The generalized symmetry method is applied to a class of completely discrete equations including the Adler-Bobenko-Suris list. Assuming the existence of a generalized symmetry, we derive a few integrability conditions suitable for testing…

Stochastic diffusion equations are crucial for modeling a range of physical phenomena influenced by uncertainties. We introduce the generalized finite difference method for solving these equations. Then, we examine its consistency,…

Despite the promise of scientific machine learning (SciML) in combining data-driven techniques with mechanistic modeling, existing approaches for incorporating hard constraints in neural differential equations (NDEs) face significant…

Partial differential equation (PDE) models are widely used in engineering and natural sciences to describe spatio-temporal processes. The parameters of the considered processes are often unknown and have to be estimated from experimental…

We present a new solver for coupled nonlinear elliptic partial differential equations (PDEs). The solver is based on pseudo-spectral collocation with domain decomposition and can handle one- to three-dimensional problems. It has three…

The primary purpose of this paper concerns the relation of (compact) generalized manifolds to finite Poincar\'{e} duality complexes (PD complexes). The problem is that an arbitrary generalized manifold $X$ is always an ENR space, but it is…

For a system of partial differential equations (PDEs) $F = 0$ admitting a local (point, contact, or higher) symmetry $X$ with the characteristic $\varphi$, invariant solutions satisfy the reduced system $F = \varphi = 0$. We propose a…

We present an exposition of a method of discretizing ordinary differential equations while preserving their Lie point symmetries. This method is very general and can be applied to any ODE with a nontrivial symmetry group. The method is…

Among the methods for solving ODE-IVPs, the class of General Linear Methods (GLMs) is able to encompass most of them, ranging from Linear Multistep Formulae (LMF) to RK formulae. Moreover, it is possible to obtain methods able to overcome…

We consider generalized gradients in the general context of $G$-structures. They are natural first order differential operators acting on sections of vector bundles associated to irreducible $G$-representations. We study their geometric…

We carry out the generalized symmetry classification of polylinear autonomous discrete equations defined on the square, which belong to a twelve-parametric class. The direct result of this classification is a list of equations containing no…

For curved projective manifolds we introduce a notion of a normal tractor frame field, based around any point. This leads to canonical systems of (redundant) coordinates that generalise the usual homogeneous coordinates on projective space.…

In this article we present some integrability conditions for partial difference equations obtained using the formal symmetries approach. We apply them to find integrable partial difference equations contained in a class of equations…