scholarly journals Exact solution for the Poisson field in a semi-infinite strip

2017 ◽  
Vol 473 (2200) ◽  
pp. 20160908
Author(s):  
Yossi Cohen ◽  
Daniel H. Rothman
Keyword(s):  
Poisson Equation ◽  
Analytic Solutions ◽  
Length Scale ◽  
Infinite Strip ◽  
Physical Processes ◽  
The Poisson Equation ◽  
Exact Analytic Solutions ◽  
Square Root Singularity ◽  
Poisson Networks ◽  
Poisson Field

The Poisson equation is associated with many physical processes. Yet exact analytic solutions for the two-dimensional Poisson field are scarce. Here we derive an analytic solution for the Poisson equation with constant forcing in a semi-infinite strip. We provide a method that can be used to solve the field in other intricate geometries. We show that the Poisson flux reveals an inverse square-root singularity at a tip of a slit, and identify a characteristic length scale in which a small perturbation, in a form of a new slit, is screened by the field. We suggest that this length scale expresses itself as a characteristic spacing between tips in real Poisson networks that grow in response to fluxes at tips.

scholarly journals A Numerical Solution Using an Adaptively Preconditioned Lanczos Method for a Class of Linear Systems Related with the Fractional Poisson Equation

2008 ◽  
Vol 2008 ◽  
pp. 1-26 ◽  
Author(s):  
M. Ilić ◽  
I. W. Turner ◽  
V. Anh
Keyword(s):  
Linear System ◽  
Linear Systems ◽  
Poisson Equation ◽  
Matrix Function ◽  
Fractional Power ◽  
Analytic Solutions ◽  
Linear System Of Equations ◽  
Exact Analytic Solutions ◽  
The Matrix ◽  
Symmetric Positive Definite Matrices

This study considers the solution of a class of linear systems related with the fractional Poisson equation (FPE) (−∇2)α/2φ=g(x,y) with nonhomogeneous boundary conditions on a bounded domain. A numerical approximation to FPE is derived using a matrix representation of the Laplacian to generate a linear system of equations with its matrix A raised to the fractional power α/2. The solution of the linear system then requires the action of the matrix function f(A)=A−α/2 on a vector b. For large, sparse, and symmetric positive definite matrices, the Lanczos approximation generates f(A)b≈β0Vmf(Tm)e1. This method works well when both the analytic grade of A with respect to b and the residual for the linear system are sufficiently small. Memory constraints often require restarting the Lanczos decomposition; however this is not straightforward in the context of matrix function approximation. In this paper, we use the idea of thick-restart and adaptive preconditioning for solving linear systems to improve convergence of the Lanczos approximation. We give an error bound for the new method and illustrate its role in solving FPE. Numerical results are provided to gauge the performance of the proposed method relative to exact analytic solutions.

CRYSTALLINE GRAVITY

2009 ◽  
Vol 24 (18n19) ◽  
pp. 3243-3255 ◽  
Author(s):  
GERARD 't HOOFT
Keyword(s):  
Finite Number ◽  
Lorentz Group ◽  
Degrees Of Freedom ◽  
Holonomy Group ◽  
Discrete Subgroup ◽  
Analytic Solutions ◽  
Space Time ◽  
Exact Analytic Solutions ◽  
Finite Domains ◽  
Dimensional Crystal

Matter interacting classically with gravity in 3+1 dimensions usually gives rise to a continuum of degrees of freedom, so that, in any attempt to quantize the theory, ultraviolet divergences are nearly inevitable. Here, we investigate a theory that only displays a finite number of degrees of freedom in compact sections of space-time. In finite domains, one has only exact, analytic solutions. This is achieved by limiting ourselves to straight pieces of string, surrounded by locally flat sections of space-time. Next, we suggest replacing in the string holonomy group, the Lorentz group by a discrete subgroup, which turns space-time into a 4-dimensional crystal with defects.

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