scholarly journals Solving Poisson Equation by Distributional HK-Integral: Prospects and Limitations

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Amila J. Maldeniya ◽  
Naleen C. Ganegoda ◽  
Kaushika De Silva ◽  
Sanath K. Boralugoda
Keyword(s):  
Boundary Condition ◽  
Half Space ◽  
Poisson Equation ◽  
Dirichlet Boundary Condition ◽  
Linear Functional ◽  
Dirichlet Boundary ◽  
Two Dimensional ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Test Function

In this paper, we present some properties of integrable distributions which are continuous linear functional on the space of test function D ℝ 2 . Here, it uses two-dimensional Henstock–Kurzweil integral. We discuss integrable distributional solution for Poisson’s equation in the upper half space ℝ + 3 with Dirichlet boundary condition.

COMPLETE STABILITY FOR STANDARD CELLULAR NEURAL NETWORKS

1999 ◽  
Vol 09 (05) ◽  
pp. 909-918 ◽  
Author(s):  
SONG-SUN LIN ◽  
CHIH-WEN SHIH
Keyword(s):  
Neural Networks ◽  
Boundary Condition ◽  
Symmetric Space ◽  
Dirichlet Boundary Condition ◽  
Periodic Boundary ◽  
Dirichlet Boundary ◽  
Periodic Boundary Conditions ◽  
Cellular Neural Networks ◽  
Two Dimensional ◽  
Space Variant

We consider cellular neural networks with symmetric space-variant feedback template. The complete stability is proved via detailed analysis on the energy function. The proof is presented for the two-dimensional case with Dirichlet boundary condition. It can be extended to other dimensions with minor adjustments. Modifications to the cases of Neumann and periodic boundary conditions are also mentioned.

REINFORCEMENT OF THE POISSON EQUATION BY A THIN LAYER

2011 ◽  
Vol 21 (05) ◽  
pp. 1153-1192 ◽  
Author(s):  
JINGYU LI ◽  
KAIJUN ZHANG
Keyword(s):  
Boundary Condition ◽  
Shear Modulus ◽  
Poisson Equation ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Neumann Boundary ◽  
Robin Boundary Condition ◽  
Oscillation Period ◽  
The Poisson Equation ◽  
Robin Boundary

We consider the problem of reinforcing an elastic medium by a strong, rough, thin external layer. This model is governed by the Poisson equation with homogeneous Dirichlet boundary condition. We characterize the asymptotic behavior of the solution as the shear modulus of the layer goes to infinity. We find that there are four types of behaviors: the limiting solution satisfies Poisson equation with Dirichlet boundary condition, Robin boundary condition or Neumann boundary condition, or the limiting solution does not exist. The specific type depends on the integral of the load on the medium, the curvature of the interface and the scaling relations among the shear modulus, the thickness and the oscillation period of the layer.

Singularities of two-dimensional exterior solutions of the Helmholtz equation

1971 ◽  
Vol 69 (1) ◽  
pp. 175-188 ◽  
Author(s):  
R. F. Millar
Keyword(s):  
Boundary Condition ◽  
Convex Hull ◽  
Helmholtz Equation ◽  
Dirichlet Boundary Condition ◽  
Harmonic Functions ◽  
Dirichlet Boundary ◽  
Two Dimensional ◽  
Helmholtz Equations

AbstractA technique for locating possible singularities of two-dimensional ex-terior harmonic functions was discussed in a previous paper. In the present work, the method is generalized to exterior solutions of the Helmholtz equation. Although the procedure deviates in some of its details from the earlier exposition, the conclusions are similar. In particular, it is verified that solutions of the Laplace and Helmholtz equations that satisfy the same Dirichlet boundary condition on the same boundary, possess the same convex hull of singularities. The possibility of extending the method to more general equations is raised.

The product of the distributions and

1972 ◽  
Vol 71 (1) ◽  
pp. 123-130 ◽  
Author(s):  
B. Fisher
Keyword(s):  
Compact Support ◽  
Linear Functional ◽  
Null Sequence ◽  
Generalized Function ◽  
Continuous Linear ◽  
Test Functions ◽  
Continuous Linear Functional ◽  
Test Function ◽  
Fixed Domain

In the following a distribution or a generalized function f is denned, as by Gelfand and Shilov(2) or by Temple (3), as a continuous linear functional on the space K of infinitely differentiable test functions ø having compact support. The value of f at a test function ø will be denoted by (f, ø).A sequence of test functions {øn} is said to be a null sequence if(1) the support of each øn is contained in some fixed domain D independent of n,(2) the sequence converges uniformly to zero in D, as n tends to infinity, for all m.

Image Segmentation Based on Poisson Equation

2013 ◽  
Vol 284-287 ◽  
pp. 3131-3134
Author(s):  
Zhi Heng Zhou ◽  
Hui Qiang Zhong
Keyword(s):  
Boundary Condition ◽  
Image Segmentation ◽  
Poisson Equation ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Segmentation Method ◽  
Snake Model ◽  
The Poisson Equation ◽  
Binary Segmentation ◽  
Edge Based

Image segmentation is an important part of the image processing. Currently, image segmentation methods are mainly the threshold-based segmentation method, the region-based segmentation method, the edge-based segmentation method and the Snake model based on energy function etc. This paper presents a novel image segmentation method based on the Poisson equation. The goal of the segmentation method is to divide the image into two homogeneous parts, the boundary portion and the non-boundary portion, which have similar gray values in homogeneous part. The key of the method is to build a Poisson equation with Dirichlet boundary condition. It sets a gradient threshold as the Dirichlet boundary condition of the Poisson equation, and gets a binary image by retaining the image boundary and smoothing the non-image boundary. Then simple binary segmentation will be able to get the image boundary. The experimental results show that this segmentation method can get accurate image boundaries for non-noise images and the weak noise images.

scholarly journals Two-dimensional dilaton gravity theory and lattice Schwarzian theory

2019 ◽  
Vol 34 (29) ◽  
pp. 1950176
Author(s):  
Su-Kuan Chu ◽  
Chen-Te Ma ◽  
Chih-Hung Wu
Keyword(s):  
Boundary Condition ◽  
Cosmological Constant ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Gravity Theory ◽  
Boundary Theory ◽  
Two Dimensional ◽  
Dilaton Field ◽  
Dilaton Gravity ◽  
Cosmological Constants

We report a holographic study of a two-dimensional dilaton gravity theory with the Dirichlet boundary condition for the cases of nonvanishing and vanishing cosmological constants. Our result shows that the boundary theory of the two-dimensional dilaton gravity theory with the Dirichlet boundary condition for the case of nonvanishing cosmological constants is the Schwarzian term coupled to a dilaton field, while for the case of vanishing cosmological constant, a theory does not have a kinetic term. We also include the higher derivative term [Formula: see text], where [Formula: see text] is the scalar curvature that is coupled to a dilaton field. We find that the form of the boundary theory is not modified perturbatively. Finally, we show that a lattice holographic picture is realized up to the second-order perturbation of boundary cutoff [Formula: see text] under a constant boundary dilaton field and the nonvanishing cosmological constant by identifying the lattice spacing [Formula: see text] of a lattice Schwarzian theory with the boundary cutoff [Formula: see text] of the two-dimensional dilaton gravity theory.

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