Related papers: The discrete energy method in numerical relativity…
We consider finite element approximations of unique continuation problems subject to elliptic equations in the case where the normal derivative of the exact solution is known to reside in some finite dimensional space. To give quantitative…
In this paper we present numerical methods - finite differences and finite elements - for solution of partial differential equation of fractional order in time for one-dimensional space. This equation describes anomalous diffusion which is…
We consider a parabolic partial differential equation with Dirichlet boundary conditions and measure or $L^1$ data. The key difficulty consists in a presence of a monotone operator~$A$ subjected to a non-standard growth condition,…
We address the approximation of entropy solutions to initial-boundary value problems for nonlinear strictly hyperbolic conservation laws using neural networks. A general and systematic framework is introduced for the design of efficient and…
Observations and magnetohydrodynamic simulations of solar and stellar atmospheres reveal an intermittent behavior or steep gradients in physical parameters, such as magnetic field, temperature, and bulk velocities. The numerical solution of…
Nonlinear two-point boundary value problems arise in numerous areas of application. The existence and number of solutions for various cases has been studied from a theoretical standpoint. These results generally rely upon growth conditions…
We develop two local energy methods for distributed parameter port-Hamiltonian (pH) systems on one-dimensional spatial domains. The methods are applied to derive a characterization of exponential stability directly in terms of the energy…
In order to understand the impact of random influences at physical boundary on the evolution of multiscale systems, a stochastic partial differential equation model under a fast random dynamical boundary condition is investigated. The…
We consider both discrete and continuous "uncertain horizon" deterministic control processes, for which the termination time is a random variable. We examine the dynamic programming equations for the value function of such processes,…
In differential equation discovery algorithms, numerical differentiation is usually a fixed preliminary step. Current methods improve robustness with data subsampling and sparsity but often ignore the variability from the differentiation…
A new method for numerical solving of boundary problem for ordinary differential equations with slowly varying coefficients which is aimed at better representation of solutions in the regions of their rapid oscillations or exponential…
This paper is concerned with fully discrete finite element methods for approximating variational solutions of nonlinear stochastic elastic wave equations with multiplicative noise. A detailed analysis of the properties of the weak solution…
Nitsche's method is a popular approach to implement Dirichlet-type boundary conditions in situations where a strong imposition is either inconvenient or simply not feasible. The method is widely applied in the context of unfitted finite…
Alternative approach for description of the non-equilibrium phenomena arising in solids at a severe external loading is analyzed. The approach is based on the new form of kinetic equations in terms of the internal and modified free energy.…
Following our earlier work, we construct statistical discrete geometry by applying statistical mechanics to discrete (Regge) gravity. We propose a coarse-graining method for discrete geometry under the assumptions of atomism and background…
We present a fully discrete finite element method for the interior null controllability problem subject to the wave equation. For the numerical scheme, piece-wise affine continuous elements in space and finite differences in time are…
A nonlinear partial differential equation is a nonlinear relationship between an unknown function and how it changes due to two or more input variables. A numerical method reduces such an equation to arithmetic for quick visualization, but…
We derive new boundary conditions and implementation procedures for nonlinear initial boundary value problems that lead to energy and entropy bounded solutions. A step-by-step procedure for general nonlinear hyperbolic problems on…
This paper proposes a decoupled numerical scheme of the time-dependent Ginzburg--Landau equations under the temporal gauge. For the magnetic potential and the order parameter, the discrete scheme adopts the second type Ned${\rm…
The numerical methods for differential equation solution allow obtaining a discrete field that converges towards the solution if the method is applied to the correct problem. Nevertheless, the numerical methods have the restricted class of…