The Green Function for Linear Second Order Dynamic Equations

2021 ◽  
pp. 213-282
Author(s):  
Svetlin G. Georgiev ◽  
Khaled Zennir
Keyword(s):  
Green Function ◽  
Second Order ◽  
Dynamic Equations ◽  
The Green Function

scholarly journals Second-Order Regularity Estimates for Singular Schrödinger Equations on Convex Domains

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Xiangxing Tao
Keyword(s):  
Green Function ◽  
Singular Potential ◽  
Coefficient Matrix ◽  
Second Order ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Convex Domains ◽  
Regularity Estimates ◽  
The Green Function ◽  
Atomic Hardy Space

LetΩ⊂ℝnbe a nonsmooth convex domain and letfbe a distribution in the atomic Hardy spaceHatp(Ω); we study the Schrödinger equations-div⁡(A∇u)+Vu=finΩwith the singular potentialVand the nonsmooth coefficient matrixA. We will show the existence of the Green function and establish theLpintegrability of the second-order derivative of the solution to the Schrödinger equation onΩwith the Dirichlet boundary condition forn/(n+1)<p≤2. Some fundamental pointwise estimates for the Green function are also given.

A global random walk on grid algorithm for second order elliptic equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Karl K. Sabelfeld ◽  
Dmitrii Smirnov
Keyword(s):  
Differential Equation ◽  
Random Walk ◽  
Green Function ◽  
Elliptic Equations ◽  
Symmetry Property ◽  
Variable Coefficients ◽  
Second Order ◽  
Grid Algorithm ◽  
The Green Function ◽  
Second Order Elliptic Equations

Abstract We suggest in this paper a global random walk on grid (GRWG) method for solving second order elliptic equations. The equation may have constant or variable coefficients. The GRWS method calculates the solution in any desired family of m prescribed points of the gird in contrast to the classical stochastic differential equation based Feynman–Kac formula, and the conventional random walk on spheres (RWS) algorithm as well. The method uses only N trajectories instead of mN trajectories in the RWS algorithm and the Feynman–Kac formula. The idea is based on the symmetry property of the Green function and a double randomization approach.

CONFORMAL INVARIANCE AND OPEN STRING IN QED

1986 ◽  
Vol 01 (08) ◽  
pp. 475-483 ◽  
Author(s):  
I.V. KOLOKOLOV ◽  
M. YA. PALCHIK
Keyword(s):  
Green Function ◽  
Conformal Invariance ◽  
Open String ◽  
Dynamic Equations ◽  
Conformal Theory ◽  
The Green Function

The properties of gauge strings [Formula: see text] in the conformal Euclidean QED are analyzed. The Green function 〈Aμ(x3)ψ(x1x2)〉 is found. The nontrivial criterion of conformal theory self-consistence is derived and verified. The average 〈ψ(x1x2)〉 is shown to have a form of free massless QED propagator by means of dynamic equations.

Exponentially Convergent Approximation to the Elliptic Solution Operator

2006 ◽  
Vol 6 (4) ◽  
pp. 386-404 ◽  
Author(s):  
Ivan. P. Gavrilyuk ◽  
V.L. Makarov ◽  
V.B. Vasylyk
Keyword(s):  
Green Function ◽  
Boundary Value Problem ◽  
Elliptic Boundary ◽  
Boundary Value ◽  
Solution Operator ◽  
Hyperbolic Operator ◽  
Elliptic Solution ◽  
Elliptic Boundary Value ◽  
Sine Family ◽  
The Green Function

AbstractWe develop an accurate approximation of the normalized hyperbolic operator sine family generated by a strongly positive operator A in a Banach space X which represents the solution operator for the elliptic boundary value problem. The solution of the corresponding inhomogeneous boundary value problem is found through the solution operator and the Green function. Starting with the Dunford — Cauchy representation for the normalized hyperbolic operator sine family and for the Green function, we then discretize the integrals involved by the exponentially convergent Sinc quadratures involving a short sum of resolvents of A. Our algorithm inherits a two-level parallelism with respect to both the computation of resolvents and the treatment of different values of the spatial variable x ∈ [0, 1].

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