Related papers: Solution decomposition for the nonlinear Poisson-B…
Starting from the 3D Gross-Pitaevskii equation we revisit the dimensional reduction to an effective one-dimensional wave-equation that describes the longitudinal dynamics of a Bose condensate in an axially-symmetric external potential.…
Monte Carlo PDE solvers have become increasingly popular for solving heat-related partial differential equations in geometry processing and computer graphics due to their robustness in handling complex geometries. While existing methods can…
In this paper within a field-theoretical approach taking into account explicitly a co-solvent with a nonzero dipole and a polarizability tensor, we derive a modified Poisson-Boltzmann equation. Applying the modified Poisson-Boltzmann…
An integral equation method is presented for the 1D steady-state Poisson-Nernst-Planck equations modeling ion transport through membrane channels. The differential equations are recast as integral equations using Green's 3rd identity…
We consider a non-linear parabolic partial differential equation (PDE) on $\mathbb R^d$ with a distributional coefficient in the non-linear term. The distribution is an element of a Besov space with negative regularity and the non-linearity…
The Poisson-Boltzmann (PB) theory is widely used to depict ionic systems. As a mean-field theory, the PB theory neglects the correlation effect in the ionic atmosphere and leads to deviations from experimental results as the concentration…
The elliptic 2-Hessian equation is a fully nonlinear partial differential equation (PDE) that is related to intrinsic curvature for three dimensional manifolds. We introduce two numerical methods for this PDE: the first is provably…
We introduce the stochastic Boltzmann equation (SBE) as an approach for exploring the spin dynamics of magnetic molecules coupled to a stochastic environment. The SBE is a time-evolution equation for the probability density of the spin…
The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators suggests that formal uncertainty quantification can also be performed in this context. Competing statistical…
Implicit-solvent models are widely used to study the electrostatics in dissolved biomolecules, which are parameterized using force fields. Standard force fields treat the charge distribution with point charges, however, other force fields…
In this paper we present a novel fast method to solve Poisson equation in an arbitrary two dimensional region with Neumann boundary condition. The basic idea is to solve the original Poisson problem by a two-step procedure: the first one…
With a small parameter $\epsilon$, Poisson-Nernst-Planck (PNP) systems over a finite one-dimensional (1D) spatial domain have steady state solutions, called 1D boundary layer solutions, which profiles form boundary layers near boundary…
Solving partial differential equations (PDEs) is the canonical approach for understanding the behavior of physical systems. However, large scale solutions of PDEs using state of the art discretization techniques remains an expensive…
Recently, a nonlinear Poisson equation has been introduced to model nonlinear and nonlocal hyperpolarization effects in electrostatic solute-solvent interaction for biomolecular solvation analysis. Due to a strong nonlinearity associated…
The non-equilibrium steady states of a semi-infinite quasi-one-dimensional univalent binary electrolyte solution, characterised by non-vanishing electric currents, are investigated by means of Poisson-Nernst-Planck (PNP) theory. Exact…
Numerical analysis is conducted for a generalized particle method for a Poisson equation. Unique solvability is derived for the discretized Poisson equation by introducing a connectivity condition for particle distributions. Moreover, by…
The Poisson-Nernst-Planck (PNP) equations are fundamental for modeling ion transport in electrochemical systems, capturing the intricate interplay of concentration gradients, electric fields, and ion fluxes essential for applications such…
A nonlinear Poisson--Boltzmann equation with transmission boundary conditions at the interface between two materials is investigated. The model describes the electrostatic potential generated by a vector of ion concentrations in a periodic…
A macroscopic model to describe the dynamics of ion transport in ion channels is the Poisson-Nernst-Planck(PNP) equations. In this paper, we develop a finite-difference method for solving PNP equations, which is second-order accurate in…
The aim of this article is to analyze numerical schemes using two-layer neural networks with infinite width for the resolution of the high-dimensional Poisson-Neumann partial differential equations (PDEs) with Neumann boundary conditions.…