CARLEMAN'S FORMULA OF A SOLUTIONS OF THE POISSON EQUATION IN BOUNDED DOMAIN

We suggest an explicit continuation formula for a solution to the Cauchy problem for the Poisson equation in a domain from its values and values of its normal derivative on a part of the boundary. We construct the continuation formula of this problem based on the Carleman--Yarmuhamedov function method.

Numerical simulation of thermal behavior in a naturally ventilated greenhouse

Inside a greenhouse, during the day, the temperature rises very quickly, while the plants have to face temperatures that rise to more than 35[Formula: see text]C. The plant closes its pores to limit sweating and stops growing. As soon as it gets hot, it is therefore necessary to ventilate the greenhouse. In this context, this research aims to investigate the behavior of the natural ventilation on the internal climate of the tunnel greenhouse, which contains two openings in the roof. The effect of the position of the openings on heat transfer is considered, thus promoting photosynthesis and plant growth. The vorticity transport equation, the Poisson equation and the energy equation are discretized by using the finite volume method. Two-dimensional simulations that described laminar flows in a steady state were carried out. Flows are studied for a range of parameters: the Rayleigh number, Ra, [Formula: see text], and three positions of opening ventilation. The results reveal that the ventilation through the top opening position allows the best creation of heat exchanges between the air inside the greenhouse and its atmosphere, which serves to conserve the plant under a favorable climate that allows its growth.

The Prelimit Generator Comparison Approach of Stein’s Method

This paper uses the generator comparison approach of Stein’s method to analyze the gap between steady-state distributions of Markov chains and diffusion processes. The “standard” generator comparison approach starts with the Poisson equation for the diffusion, and the main technical difficulty is to obtain bounds on the derivatives of the solution to the Poisson equation, also known as Stein factor bounds. In this paper we propose starting with the Poisson equation of the Markov chain; we term this the prelimit approach. Although one still needs Stein factor bounds, they now correspond to finite differences of the Markov chain Poisson equation solution rather than the derivatives of the solution to the diffusion Poisson equation. In certain cases, the former are easier to obtain. We use the [Formula: see text] model as a simple working example to illustrate our approach.

PENYELESAIAN PERSAMAAN POISSON MENGGUNAKAN METODE HOMOTOPI PERTUBASI

In this paper, we discuss the solution of the Poisson equation with some initial condition. We use the homotopy pertubation method to get the solution.. The homotopy pertubation method is a combination of the homotopy method and the pertubation method. The solution of the equation is assumed to be in the form of a power series. The result is by using the homotopy pertubation method for the diffution equation, the solution is the same with the exact solution.

Charged particles density distribution in the cathode fall region of the glow discharge in helium

In the present work, the mechanism of formation and propagation of the group of high energy electrons in the cathode regions of a glow discharge in helium is discussed. Using the method of the Monte Carlo collisions simulation, the beam electron energy distribution function in the cathode fall region of a glow discharge has been determined in the gas pressure range of 30−70 Pa. It is shown that the electron distribution function at the end of the cathode fall region contains a lot of electrons which have no any collisions and have energies close to the cathode fall potential. On the basis of the obtained results the distribution of the ion density was simulated using the Poisson equation. It is shown that the ion density distribution stays almost constant in the cathode fall region. The beam and ion density increased with the pressure growth.

Non-relativistic ten-dimensional minimal supergravity

We construct a non-relativistic limit of ten-dimensional $$ \mathcal{N} $$ N = 1 supergravity from the point of view of the symmetries, the action, and the equations of motion. This limit can only be realized in a supersymmetric way provided we impose by hand a set of geometric constraints, invariant under all the symmetries of the non-relativistic theory, that define a so-called ‘self-dual’ Dilatation-invariant String Newton-Cartan geometry. The non-relativistic action exhibits three emerging symmetries: one local scale symmetry and two local conformal supersymmetries. Due to these emerging symmetries the Poisson equation for the Newton potential and two partner fermionic equations do not follow from a variation of the non-relativistic action but, instead, are obtained by a supersymmetry variation of the other equations of motion that do follow from a variation of the non-relativistic action. We shortly discuss the inclusion of the Yang-Mills sector that would lead to a non-relativistic heterotic supergravity action.

Infinite Schrödinger networks

Finite-difference models of partial differential equations such as Laplace or Poisson equations lead to a finite network. A discretized equation on an unbounded plane or space results in an infinite network. In an infinite network, Schrödinger operator (perturbed Laplace operator, $q$-Laplace) is defined to develop a discrete potential theory which has a model in the Schrödinger equation in the Euclidean spaces. The relation between Laplace operator $\Delta$-theory and the $\Delta_q$-theory is investigated. In the $\Delta_q$-theory the Poisson equation is solved if the network is a tree and a canonical representation for non-negative $q$-superharmonic functions is obtained in general case.

Stability features of steady-state solutions for a diode with electron and ion counter-streams

Stability features of steady-state solutions for a diode with counter-streaming electron and ion flows are studied. For this purpose, the time-dependent problem for an exponential potential perturbation with complex frequency is considered. By linearization of the Poisson equation and electron and ion densities integrodifferential equation for the potential perturbation amplitude is derived. In the case of uniform unperturbed potential distribution an explicit solution of this equation is obtained. Eigen modes of the perturbation are studied. The limiting value of the diode length above which steady state solutions in question are unstable is found. The obtained analytical Eigen modes coincide with the result of numerical simulation of the potential perturbation evolution.