Nonlocal Poisson Equation Research Articles

Investigation of the solvability of boundary value problems for the nonlocal Poisson equation with periodic conditions in circular domains

In this paper, boundary value problems with transformed arguments are studied in the unit ball. The transformation of the arguments is specified using the involution type mapping. These mappings participate both in the equation and in the boundary conditions. The equation under consideration is a nonlocal analog of the Poisson equation. Boundary conditions are specified as a relationship between the value of the desired function in the upper hemisphere and the value of the lower hemisphere. These conditions generalize the known periodic conditions for spherical regions. When studying boundary value problems, the properties of involutive mappings are used. The problems under consideration are solved by reducing them to analogues of boundary value problems with periodic conditions for the classical Poisson equation. Using well-known statements for periodic problems for the problems under consideration, theorems on the existence and uniqueness of solutions are proved. Exact conditions for the solvability of the problems under study are found. Spectral questions related to periodic problems are also studied. Eigenfunctions and eigenvalues of these problems are found.

Weak averaging principle for multiscale stochastic dynamical systems driven by stable processes

We study the averaging principle for a family of multiscale stochastic dynamical systems. The fast and slow components of the systems are driven by two independent stable Lévy noises, whose stable indexes may be different. The homogenizing index r0 of slow components has a relation with the stable index α1 of the noise of fast components given by 0<r0<2−2/α1. By first studying a nonlocal Poisson equation and then constructing suitable correctors, we obtain that the slow components weakly converge to a Lévy process as the scale parameter goes to zero.

On the Solvability of Some Boundary Value Problems for the Nonlocal Poisson Equation with Boundary Operators of Fractional Order

In this paper, in the class of smooth functions, integration and differentiation operators connected with fractional conformable derivatives are introduced. The mutual reversibility of these operators is proved, and the properties of these operators in the class of smooth functions are studied. Using transformations generalizing involutive transformations, a nonlocal analogue of the Laplace operator is introduced. For the corresponding nonlocal analogue of the Poisson equation, the solvability of some boundary value problems with fractional conformable derivatives is studied. For the problems under consideration, theorems on the existence and uniqueness of solutions are proved. Necessary and sufficient conditions for solvability of the studied problems are obtained, and integral representations of solutions are given.

On solvability of the Dirichlet and Neumann boundary value problems for the Poisson equation with multiple involution

Transformations of the involution type are considered in the space $R^l$, $l\geq 2$. The matrix properties of these transformations are investigated. The structure of the matrix under consideration is determined and it is proved that the matrix of these transformations is determined by the elements of the first row. Also, the symmetry of the matrix under study is proved. In addition, the eigenvectors and eigenvalues of the matrix under consideration are found explicitly. The inverse matrix is also found and it is proved that the inverse matrix has the same structure as the main matrix. The properties of the nonlocal analogue of the Laplace operator are introduced and studied as applications of the transformations under consideration. For the corresponding nonlocal Poisson equation in the unit ball, the solvability of the Dirichlet and Neumann boundary value problems is investigated. A theorem on the unique solvability of the Dirichlet problem is proved, an explicit form of the Green's function and an integral representation of the solution are constructed, and the order of smoothness of the solution of the problem in the Hölder class is found. Necessary and sufficient conditions for the solvability of the Neumann problem, an explicit form of the Green's function, and the integral representation are also found.

Online drift estimation for jump-diffusion processes

We show the convergence of an online stochastic gradient descent estimator to obtain the drift parameter of a continuous-time jump-diffusion process. The stochastic gradient descent follows a stochastic path in the gradient direction of a function to find a minimum, which in our case determines the estimate of the unknown drift parameter. We decompose the deviation of the stochastic descent direction from the deterministic descent direction into four terms: the weak solution of the non-local Poisson equation, a Riemann integral, a stochastic integral, and a covariation term. This decomposition is employed to prove the convergence of the online estimator and we use simulations to illustrate the performance of the online estimator.

Online Drift Estimation for Jump-Diffusion Processes

We show the convergence of an online stochastic gradient descent estimator to obtain the drift parameter of a continuous-time jump-diffusion process. The stochastic gradient descent follows a stochastic path in the gradient direction of a function to find a minimum, which in our case determines the estimate of the unknown drift parameter. We decompose the deviation of the stochastic descent direction from the deterministic descent direction into four terms: the weak solution of the non-local Poisson equation, a Riemann integral, a stochastic integral, and a covariation term. This decomposition is employed to prove the convergence of the online estimator and we use simulations to illustrate the performance of the online estimator.

Vlasov--Poisson System Tackled by Particle Simulation Utilizing Boundary Element Methods

This paper presents a grid-free simulation algorithm for the fully three-dimensional Vlasov--Poisson system for collisionless electron plasmas. We employ a standard particle method for the numerical approximation of the distribution function. Whereas the advection of the particles is grid-free by its very nature, the computation of the acceleration involves the solution of the nonlocal Poisson equation. To circumvent a volume mesh, we utilize the fast boundary element method, which reduces the three-dimensional Poisson equation to a system of linear equations on its two-dimensional boundary. This gives rise to fully populated matrices which are approximated by the $\mathcal H^2$-technique, reducing the computational time from quadratic to linear complexity. The approximation scheme based on interpolation has been shown to be robust and flexible, allowing a straightforward generalization to vector-valued functions. In particular, the Coulomb forces acting on the particles are computed in linear complexity. In first numerical tests, we validate our approach with the help of classical nonlinear plasma phenomena. Furthermore, we show that our method is able to simulate electron plasmas in complex three-dimensional domains with mixed boundary conditions in linear complexity.

On the solvability of the main boundary value problems for a nonlocal Poisson equation

Solvability of the main boundary value problems for the nonlocal Poisson equation is studied. Existence and uniqueness theorems for the considered problems are obtained. The necessary and sufficient solvability conditions for all problems are given and integral representations for the solutions are constructed.

Exact solutions of the newton-schrödinger equation, infinite derivative gravity and schwarzschild atoms

Exact solutions to the stationary spherically symmetric Newton-Schrodinger equation are proposed in terms of integrals involving generalized Gaussians. The energy eigenvalues are also obtained in terms of these integrals which agree with the numerical results in the literature. A discussion of infinite derivative-gravity follows which allows generalizing the Newton-Schrodinger equation by replacing the ordinary Poisson equation with a modified non-local Poisson equation associated with infinite-derivative gravity. We proceed to replace the nonlinear Newton-Schrodinger equation for a non-linear quantum-like BohmPoisson equation involving Bohm’s quantum potential, and where the fundamental quantity is no longer the wave-function Ψ but the real-valued probability density ρ . Finally, we discuss how the latter equations reflect a nonlinear feeding loop mechanism between matter and geometry which allows us to envisage a “Schwarzschild atom” as a spherically symmetric probability cloud of matter which curves the geometry, and in turn, the geometry back-reacts on this matter cloud perturbing its initial distribution over the space, which in turn will affect the geometry, and so forth until static equilibrium is reached.

The Quasi-Neutral Limit in Optimal Semiconductor Design

We study the quasi-neutral limit in an optimal semiconductor design problem constrained by a nonlinear, nonlocal Poisson equation modelling the drift diffusion equations in thermal equilibrium. While a broad knowledge on the asymptotic links between the different models in the semiconductor model hierarchy exists, there are so far no results on the corresponding optimization problems available. Using a variational approach we end up with a bi-level optimization problem, which is thoroughly analysed. Further, we exploit the concept of Gamma-convergence to perform the quasi-neutral limit for the minima and minimizers. This justifies the construction of fast optimization algorithms based on the zero space charge approximation of the drift-diffusion model. The analytical results are underlined by numerical experiments confirming the feasibility of our approach.

Electromechanical behavior, end enhancements and radial elasticity of single-walled CNTs: A physically-consistent model based on nonlocal charges

We simulate the electromechanical behavior of freestanding and clamped carbon nanotubes (CNTs) subjected to an electric potential and displaying charge end enhancements. In this case, CNT deformation is a consequence of the electrostatic pressure associated with the charge density. Whereas experiments investigate several micron long CNTs, computational models consider nanometer-sized CNTs. Hence, experimental and computational results, often, cannot be compared. Analytical solutions appear thus useful as reference solutions for finite element analysis, alternative to atomistic simulations, and complementary to experimental tests. In this work, the electromechanical behavior of CNTs is computed by using the nonlocal Poisson equation obtained extending the charge transport equations for spatially variable currents. Based on a certain equivalence between gradient and integral formulations, we solve the problem governing the electromechanical problem of CNTs regarded as cylinders subjected to electrostatic forces. The present procedure automatically captures charge end enhancements, thus overcoming the necessity of introducing suitable charge end correction factors. The analytical expressions of the radial displacement of freestanding CNTs, and the deflection of bent CNTs are obtained. For armchair single wall CNTs, the results obtained are consistent with experiments, and molecular mechanics and atomistic simulations. Size dependence of electric and mechanical properties is assessed.

Numerical study of Hermite-Gaussian beams in nonlocal thermal media

Based on the nonlocal nonlinear Schrdinger equation and Poisson equation of thermal diffusion, using the slip-step Fourier algorithm and multi-grid method, we numerically investigated the propagation properties of Hermite-Guassian beams in the nonlocal thermal media. The results show that low-order Hermite-Gaussian beams can propagate stably, in contrast with the unstable propagation of high-order Hermite-Gaussian beams. The worse the stability is, the higher the order is. The effect of the boundary of the sample with different cross sections on the propagation properties of Hermite-Guassian beam is also discussed in detail. We found that propagation properties in square geometry are in agreement with those in Snyder-Mitchell model. However, in rectangular sample, the evolution of intensity distribution of Hermite-Gaussian beams differs seriously from that in the square sample.

Electro‐reaction‐diffusion systems with nonlocal constraints

AbstractThe paper deals with equations modelling the redistribution of charged particles by reactions, drift and diffusion processes. The corresponding model equations contain parabolic PDEs for the densities of mobile species, ODEs for the densities of immobile species, a possibly nonlinear, nonlocal Poisson equation and some nonlocal constraints. Based on applications to semiconductor technology these equations have to be investigated for non‐smooth data and kinetic coefficients which depend on the state variables.In two space dimensions we discuss the steady states of the system, we prove energy estimates, global a priori estimates and give a global existence result. (© 2004 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Global estimates and asymptotics for electro reaction diffusion systems in heterostructures

We treat a wide class of electro reaction diffusion systems with nonsmooth data in two dimensional domains. Forced by applications in semiconductor technology a nonlinear and nonlocal Poisson equation is involved. We state global existence, uniqueness and asymptotic properties of solutions to the evolution problem. Essential tools in our investigations are energetic estimates, Moser iteration, regularization techniques and results for electro diffusion systems with weakly nonlinear volume and boundary source terms. Especially, we discuss the connection between the existence of global lower bounds for the chemical potentials and the property that the energy functional decays exponentially to its equilibrium value as time tends to infinity

An advanced continuum medium model for treating solvation effects: Nonlocal electrostatics with a cavity

The Born–Kirkwood–Onsager (BKO) model of solvation, where a solute molecule is positioned inside a cavity cut into a solvent, which is considered as a dielectric continuum, is studied within the bounds of nonlocal electrostatics. The nonlocal cavity model is explicitly formulated and the corresponding nonlocal Poisson equation is reduced to an integral equation describing the behavior of the charge density induced in the medium. It is found that the presence of a cavity does not create singularities in the total electrostatic potential and its normal derivatives. Such singularities appear only in the local limit and are completely dissipated by nonlocal effects. The Born case of a spherical cavity with a point charge at its centre is investigated in detail. The corresponding one-dimensional integral Poisson equation is solved numerically and values for the solvation energy are determined. Several tests of this approach are presented: (a) We show that our integral equation reduces in the local limit to the chief equation of the local BKO theory. (b) We provide certain approximations which enable us to obtain the solution corresponding to the preceding nonlocal treatment of Dogonadze and Kornyshev (DK). (c) We make a comparison with the results of molecular solvation theory (mean spherical approximation), as applied to the calculation of solvation energies of spherical ions.