Related papers: Non-Autonomous Inertial Manifold Reduction
We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show…
In the midst of the neural network's success in solving partial differential equations, tackling eigenvalue problems using neural networks remains a challenging task. However, the Physics Constrained-General Inverse Power Method Neural…
The optimization problems with simple bounds are an important class of problems. To facilitate the computation of such problems, an unconstrained-like dynamic method, motivated by the Lyapunov control principle, is proposed. This method…
This paper develops validated computational methods for studying infinite dimensional stable manifolds at equilibrium solutions of parabolic PDEs, synthesizing disparate errors resulting from numerical approximation. To construct our…
A nonlocal contact problem for two-dimensional linear elliptic equations is stated and investigated. The method of separation of variables is used to find the solution of a stated problem in case of Poisson's equation. Then the more general…
Delayed neural field models can be viewed as a dynamical system in an appropriate functional analytic setting. On two dimensional rectangular space domains, and for a special class of connectivity and delay functions, we describe the…
In this paper, we consider the inverse boundary problems of recovering the time-dependent nonlinearity and damping term for a semilinear wave equation on a Riemannian manifold. The Carleman estimate and the construction of Gaussian beams…
We propose a method for computing the Lyapunov exponents of renewal equations (delay equations of Volterra type) and of coupled systems of renewal and delay differential equations. The method consists in the reformulation of the delay…
The renormalization method based on the Taylor expansion for asymptotic analysis of differential equations is generalized to difference equations. The proposed renormalization method is based on the Newton-Maclaurin expansion. Several basic…
Dripping water from a faucet is a typical example exhibiting rich nonlinear phenomena. For such a system, the time stamps at which water drops separate from the faucet can be directly observed in real experiments, and the time series of…
We establish the exponential decay of the solutions of the damped wave equations in one-dimensional space where the damping coefficient is a nowhere-vanishing function of space. The considered PDE is associated with several dynamic boundary…
A promising step from linear towards nonlinear data-driven control is via the design of controllers for linear parameter-varying (LPV) systems, which are linear systems whose parameters are varying along a measurable scheduling signal.…
Nonlinear dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. However, many popular methods can fail dramatically, even on simple two-dimensional manifolds, due to problems such as…
In this paper we propose a Local Orthogonal Decomposition method (LOD) for elliptic partial differential equations with inhomogeneous Dirichlet- and Neumann boundary conditions. For this purpose, we present new boundary correctors which…
While modern representation learning relies heavily on global error signals, decentralized algorithms driven by local interactions offer a fundamental distributed alternative. However, the macroscopic convergence properties of these…
Linear nonautonomous/random parabolic partial differential equations are considered under the Dirichlet, Neumann or Robin boundary conditions, where both the zero order coefficients in the equation and the coefficients in the boundary…
Invariant manifolds are fundamental tools for describing and understanding nonlinear dynamics. In this paper, we present a theory of stable and unstable manifolds for infinite dimensional random dynamical systems generated by a class of…
Applications of variational methods are typically restricted to conservative systems. Some extensions to dissipative systems have been reported too but require ad hoc techniques such as the artificial doubling of the dynamical variables.…
Coefficient inverse problems related to identifying the right-hand side of an equation with use of additional information is of interest among inverse problems for partial differential equations. When considering non-stationary problems,…
We consider the inverse problem of reconstructing the interior boundary curve of a doubly connected domain from the knowledge of the temperature and the thermal flux on the exterior boundary curve. The use of the Laguerre transform in time…