Local Solvability of a Linear System with a Fractional Derivative in Time in a Boundary Condition

2015 ◽  
Vol 18 (4) ◽  
Author(s):  
Nataliya Vasylyeva
Keyword(s):  
Boundary Condition ◽  
Linear System ◽  
Fractional Derivative ◽  
Classical Solution ◽  
Poisson Equation ◽  
Existence And Uniqueness ◽  
Local Solvability ◽  
The Poisson Equation ◽  
The Right ◽  
Coercive Estimates

AbstractIn this paper we analyze a linear system for the Poisson equation with a boundary condition comprising the fractional derivative in time and the right-hand sides depended on time. First, we prove existence and uniqueness of the classical solution to this problem, and provide the coercive estimates of the solution. Second, based on the obtained results we establish one-to-one solvability to a linear system of a general form in the H¨older spaces.

Grad-Shafranov reconstruction of the in-plane magnetic field potential in the X-point vicinity: boundary-layer approximation

2020 ◽  
Author(s):  
Daniil Korovinskiy ◽  
Andrey Divin ◽  
Vladimir Semenov ◽  
Nikolai Erkaev ◽  
Stefan Kiehas
Keyword(s):  
Boundary Layer ◽  
Poisson Equation ◽  
Field Potential ◽  
Boundary Layer Approximation ◽  
The Poisson Equation ◽  
Right Hand ◽  
Reconnection Region ◽  
Out Of Plane ◽  
Shafranov Equation ◽  
The Right

<p>The problem of steady symmetrical two-dimensional magnetic reconnection is addressed in terms of the EMHD approximation. In the immediate vicinity of the X-point, this approach has been proven to be an appropriate frame for the reconstruction problem, expressed, particularly, by the Poisson equation for the magnetic potential <em>A</em>, where the right-hand side contains the out-of-plane electron current density with reversed sign. With boundary conditions fixed at some curve (the satellite trajectory), and assuming the right-hand side to be a function of <em>A</em>, one arrives at an ill-posed problem for the Grad-Shafranov equation. The further simplification of the problem may be achieved by using the boundary layer approximation, since magnetic configuration in reconnection region is highly stretched. The benchmark reconstruction of PIC-simulation data, using four numerical techniques, has shown that the main contribution for inaccuracy arises from replacing the Poisson equation by the Grad-Shafranov one. A boundary layer approximation, in turn, does not affect the accuracy significantly; in some cases this approach can appear even the most appropriate. </p>

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.

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.

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