Related papers: Approximate Solutions to Second-Order Parabolic Eq…
We consider a small interacting sample coupled to several non-interacting leads. Initially, the system is at thermal equilibrium. At some instant $t_0$ the system is set into the so called partition-free transport scenario by turning on a…
A method is given to obtain the Green's function for the Poisson equation in any arbitrary integer dimension under periodic boundary conditions. We obtain recursion relations which relate the solution in d-dimensional space to that in…
This paper investigates the existence and uniqueness of mild solutions, as well as the approximate controllability, of a class of fractional evolution equations with nonlocal conditions in Hilbert spaces. Sufficient conditions for…
We introduce a discretization/approximation scheme for reflected stochastic partial differential equations driven by space-time white noise through systems of reflecting stochastic differential equations. To establish the convergence of the…
In this study a spatio-temporal approach for the solution of the time-dependent Boltzmann (transport) equation is derived. Finding the exact solution using the Boltzmann equation for the general case is generally an open problem and…
In this paper, we present a discrete-type approximation scheme to solve continuous-time optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration. The approximations satisfy…
In this article, we present a comprehensive framework for constructing smooth, localized solutions in systems of semi-linear partial differential equations, with a particular emphasis to the Gray-Scott model. Specifically, we construct a…
We propose local space-time approximation spaces for parabolic problems that are optimal in the sense of Kolmogorov and may be employed in multiscale and domain decomposition methods. The diffusion coefficient can be arbitrarily rough in…
We give a proof of dynamical localization in the form of exponential decay of spatial correlations in the time evolution for the one-dimensional continuum Anderson model via the fractional moments method. This follows via exponential decay…
We show that the evolution equation of the effective potential in the auxiliary mass method corresponds to a leading approximation of a certain series. This series is derived from an evolution equation of an effective action using a…
This paper is concerned with the analysis of a new stable space-time finite element method (FEM) for the numerical solution of parabolic evolution problems in moving spatial computational domains. The discrete bilinear form is elliptic on…
We propose a time-space discretization scheme for quasi-linear parabolic PDEs. The algorithm relies on the theory of fully coupled forward--backward SDEs, which provides an efficient probabilistic representation of this type of equation.…
We consider the numerical approximation of a general second order semi--linear parabolic partial differential equation. Equations of this type arise in many contexts, such as transport in porous media which is fundamental in many…
In this article, we derive the scalar parametrized Klein-Gordon equation from the formal information theory framework. The least biased probability distribution is obtained, and the scalar equation is recast in terms of a Fokker-Planck…
In this article, we develop a new method to approximate numerically the fractional Laplacian of functions defined on $\mathbb R$, as well as some more general singular integrals. After mapping $\mathbb R$ into a finite interval, we…
In this paper we develop and analyse domain decomposition methods for linear systems of equations arising from conforming finite element discretisations of positive Maxwell-type equations. Convergence of domain decomposition methods rely…
We analyze fully implicit and linearly implicit backward difference formula (BDF) methods for quasilinear parabolic equations, without making any assumptions on the growth or decay of the coefficient functions. We combine maximal parabolic…
Deep neural networks are powerful tools for approximating functions, and they are applied to successfully solve various problems in many fields. In this paper, we propose a neural network-based numerical method to solve partial differential…
A fully discrete approximation of the semi-linear stochastic wave equation driven by multiplicative noise is presented. A standard linear finite element approximation is used in space and a stochastic trigonometric method for the temporal…
In this paper, we use an implicit two-derivative deferred correction time discretization approach and combine it with a spatial discretization of the discontinuous Galerkin spectral element method to solve (non-)linear PDEs. The resulting…