Related papers: Discrete Green's functions for products of regular…
The discrete Green's functions are the pseudoinverse (or the inverse) of the Laplacian (or its variations) of a graph. In this paper, we will give combinatorial interpretations of Green's functions in terms of enumerating trees and forests…
The discrete Green's function (without boundary) $\mathbb{G}$ is a pseudo-inverse of the combinatorial Laplace operator of a graph $G=(V,E)$. We reveal the intimate connection between Green's function and the theory of exact stopping rules…
We study a discrete model of the Laplacian in $\mathbb{R}^2$ that preserves the geometric structure of the original continual object. This means that, speaking of a discrete model, we do not mean just the direct replacement of differential…
The Green's function of the discrete Sch\"odinger operator on a finite graph is considered. This setting reproduces Laplacian and signless Laplacian by adjusting appropriate potentials. We show two ways of the expression for the Green's…
The basic mathematical properties of Green's functions used in statistical mechanics as well as the equations defining these functions and the techniques of solving these equations are reviewed. An approach is presented called the…
This article is devoted to deduce the expression of the Green's function related to a general constant coefficients fractional difference equation coupled to Dirichlet conditions. In this case, due to the points where some of the fractional…
We prove estimates for the Green's function of the discrete bilaplacian in squares and cubes in two and three dimensions which are optimal except possibly near the corners of the square and the edges and corners of the cube. The main idea…
We construct Green's functions for divergence form, second order parabolic systems in non-smooth time-varying domains whose boundaries are locally represented as graph of functions that are Lipschitz continuous in the spatial variables and…
Starting with the Green's functions found for normal diffusion, we construct exact time-dependent Green's functions for subdiffusive equation (with fractional time derivatives), with the boundary conditions involving a linear combination of…
We consider a two-dimensional diffusion process in a two-layered plane, governed by distinct covariance matrices in the upper and lower half-planes and by two drift vectors pointed away from the $x$-axis. We first analyze the case where the…
The paper presents an analysis of the dynamic behaviour of discrete flexural systems composed of Euler--Bernoulli beams. The canonical object of study is the discrete Green's function, from which information regarding the dynamic response…
A method to calculate exact Green's functions on lattices in various dimensions is presented. Expressions in terms of generalized hypergeometric functions in one or more variables are obtained for various examples by relating the resolvent…
In this short article, we non-perturbatively derive a recursive formula for the Green's function associated with finitely many point Dirac delta potentials in one dimension. We also extend this formula to the case for the Dirac delta…
Combining the study of the simple random walk on graphs, generating functions (especially Green functions), complex dynamics and general complex analysis we introduce a new method of spectral analysis on self-similar graphs. We give an…
A linear singularly perturbed convection-diffusion problem with characteristic layers is considered in three dimensions. Sharp bounds for the associated Green's function and its derivatives are established in the $L_1$ norm. The dependence…
The structure of diagonal singularities of Green functions of partial differential operators of even order acting on smooth sections of a vector bundle over a Riemannian man ifold is studied. A special class of operators formed by the…
Motivated by the question of defining a $p$-adic string worldsheet action in genus one, we define a Laplacian operator on the Tate curve, and study its Green's function. We show that the Green's function exists. We provide an explicit…
The Green's function method has applications in several fields in Physics, from classical differential equations to quantum many-body problems. In the quantum context, Green's functions are correlation functions, from which it is possible…
We introduce a technique to obtain pointwise upper and lower bounds for the Green's function of elliptic operators whose principal part is the Laplacian and that include a drift term diverging near the boundary like a power of the inverse…
This paper presents an extended version of the article [Franz, S., Kopteva, N.: J. Differential Equations, 252 (2012)]. The main improvement compared to the latter is in that here we additionally estimate the mixed second-order derivative…