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This article is focused on two related topics within the study of partial differential equations (PDEs) that illustrate a beautiful connection between dynamics, topology, and analysis: stability and spatial dynamics. The first is a property…
Travelling and rotating waves are ubiquitous phenomena observed in time dependent PDEs modelling the combined effect of dissipation and non-linear interaction. From an abstract viewpoint they appear as relative equilibria of an equivariant…
Damping of structures and systems is often dominated by frictional dissipation in connections, the prediction of which remains a longstanding scientific challenge. Previous studies have shown that the actual topography of contact interfaces…
Direct methods for the simulation of optimal control problems apply a specific discretization to the dynamics of the problem, and the discrete adjoint method is suitable to calculate corresponding conditions to approximate an optimal…
We examine the evolution of a bistable reaction in a one-dimensional stretching flow, as a model for chaotic advection. We derive two reduced systems of ordinary differential equations (ODE's) for the dynamics of the governing…
Fourth-order accurate compact schemes for variable coefficient convection diffusion equations are considered. A sufficient condition for the stability of the fully discrete problem is derived using a difference equation based approach. The…
In this article, we study the free vibration and the mechanical buckling of plates using a three dimensional consistent approach based on the scaled boundary finite element method. The in-plane dimensions of the plate are modeled by…
A new field of numerical astrophysics is introduced which addresses the solution of large, multidimensional structural or slowly-evolving problems (rotating stars, interacting binaries, thick advective accretion disks, four dimensional…
Finite dimensional solutions to a class of stochastic partial differential equations are obtained extending the differential constraints method for deterministic PDE to the stochastic framework. A geometrical reformulation of the stochastic…
In this paper we present a continuation method which transforms spatially distributed ODE systems into continuous PDE. We show that this continuation can be performed both for linear and nonlinear systems, including multidimensional, space-…
It has recently been shown that the evolution of a linear Partial Differential Equation (PDE) can be more conveniently represented in terms of the evolution of a higher spatial derivative of the state. This higher spatial derivative (termed…
It was recently shown by the authors that a semilinear elliptic equation can be represented as an infinite-dimensional dynamical system in terms of boundary data on a shrinking one-parameter family of domains. The resulting system is…
A thin and narrow rectangular plate having the two short edges hinged and the two long edges free is considered. A nonlinear nonlocal evolution equation describing the deformation of the plate is introduced: well-posedness and existence of…
This paper investigates model-order reduction methods for geometrically nonlinear structures. The parametrisation method of invariant manifolds is used and adapted to the case of mechanical systems expressed in the physical basis, so that…
Partial Integral Equations (PIEs) have been used to represent both systems with delay and systems of Partial Differential Equations (PDEs) in one or two spatial dimensions. In this paper, we show that these results can be combined to obtain…
The infinite time-evolving block decimation (iTEBD) algorithm [Phys. Rev. Lett. 98, 070201 (2007)] allows to simulate unitary evolution and to compute the ground state of one-dimensional quantum lattice systems in the thermodynamic limit.…
Reconstructing PDE solutions from sparse observations is a core challenge in scientific computing. We present FM4PDE, a flow-matching generative framework that learns the joint distribution of PDE coefficients (or initial states) and…
This article investigates the non-stationary reaction-diffusion-advection equation, emphasizing solutions with internal layers and the associated inverse problems. We examine a nonlinear singularly perturbed partial differential equation…
In this paper we investigate the asymptotic stability of a fourth-order PDE with a fading memory forcing term and boundary conditions arising from a flexible robotics model. We carry on our study by using an abstract formulation of the…
An interface description and numerical simulations of model A kinetics are used for the first time to investigate the intra-surface kinetics of phase ordering on corrugated surfaces. Geometrical dynamical equations are derived for the…