GREEN'S FUNCTIONS ON FRACTALS

Fractals ◽  
2000 ◽  
Vol 08 (04) ◽  
pp. 385-402 ◽  
Author(s):  
JUN KIGAMI ◽  
DANIEL R. SHELDON ◽  
ROBERT S. STRICHARTZ
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Initial Value Problem ◽  
Computer Experiments ◽  
Recursive Formula ◽  
Absolute Maximum ◽  
Initial Value ◽  
Sharp Estimates ◽  
Self Similar ◽  
The Given

For a regular harmonic structure on a post-critically finite (p.c.f.) self-similar fractal, the Dirichlet problem for the Laplacian can be solved by integrating against an explicitly given Green's function. We give a recursive formula for computing the values of the Green's function near the diagonal, and use it to give sharp estimates for the decay of the Green's function near the boundary. We present data from computer experiments searching for the absolute maximum of the Green's function for two different examples, and we formulate two radically different conjectures for where the maximum occurs. We also investigate a local Green's function that can be used to solve an initial value problem for the Laplacian, giving an explicit formula for the case of the Sierpinski gasket. The local Green's function turns out to be unbounded, and in fact not even integrable, but because of cancelation, it is still possible to form a singular integral to solve the initial value problem if the given function satisfies a Hölder condition.

scholarly journals Solution of Ordinary Differential Equation Using Green's Function & Sturm - Liouville Problem

2019 ◽  
pp. 241-244
Author(s):  
S. Angel Auxzaline Mary ◽  
T. Ramesh
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Green’S Function ◽  
Green's Function ◽  
Initial Value Problem ◽  
Series Representation ◽  
Liouville Problem ◽  
Initial Value ◽  
Sturm Liouville

In this paper, we describe Green's function to determine the importance of this function, i.e. Boundary & Initial Value problem, Sturm-Liouville Problem. Along with the series representation of Green's Function.

scholarly journals Quasilinearization of the Initial Value Problem for Difference Equations with “Maxima”

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
S. Hristova ◽  
A. Golev ◽  
K. Stefanova
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Initial Value Problem ◽  
Linear Difference Equation ◽  
Successive Approximations ◽  
Time Interval ◽  
Initial Value ◽  
Past Time ◽  
The Given ◽  
Unknown Solution

The object of investigation of the paper is a special type of difference equations containing the maximum value of the unknown function over a past time interval. These equations are adequate models of real processes which present state depends significantly on their maximal value over a past time interval. An algorithm based on the quasilinearization method is suggested to solve approximately the initial value problem for the given difference equation. Every successive approximation of the unknown solution is the unique solution of an appropriately constructed initial value problem for a linear difference equation with “maxima,” and a formula for its explicit form is given. Also, each approximation is a lower/upper solution of the given mixed problem. It is proved the quadratic convergence of the successive approximations. The suggested algorithm is realized as a computer program, and it is applied to an example, illustrating the advantages of the suggested scheme.

Estimates of the Green’s Function and its Regular Part on Heisenberg Group Domains

2011 ◽  
Vol 11 (3) ◽  
Author(s):  
Najoua Gamara ◽  
Habiba Guemri
Keyword(s):  
Heisenberg Group ◽  
Green’S Function ◽  
Green's Function ◽  
Forthcoming Paper ◽  
Regular Part ◽  
Kohn Laplacian ◽  
Preliminary Work ◽  
Characteristic Points ◽  
The Given ◽  
Euclidean Domains

AbstractThis paper is a preliminary work on Heisenberg group domains, devoted to the study of the Green’s function for the Kohn Laplacian on domains far away from the set of characteristic points. We give some estimates of the Green’s function, its regular part and their derivatives analogous to those proved by A. Bahri, Y.Y. Li, O. Rey in [1], and O. Rey in [16] for Euclidean domains. While the study of such functions on the set of characteristic points of the given domain will be discussed in a forthcoming paper.

GORKOV'S ANSATZ IN SUPERCONDUCTIVITY THEORY

1965 ◽  
Vol 43 (12) ◽  
pp. 2142-2149 ◽  
Author(s):  
A. J. Coleman ◽  
S. Pruski
Keyword(s):  
Density Matrix ◽  
Green’S Function ◽  
Green's Function ◽  
Order Parameter ◽  
Green’S Functions ◽  
Second Order ◽  
Bcs Theory ◽  
Initial Value ◽  
Ginzburg Landau ◽  
Superconductivity Theory

By means of Green's functions methods, Gorkov derived the BCS theory on the basis of the Ansatz that the correlation part of the second-order Green's function could be factored in the form χχ* where χ is a two-particle function closely related to the Ginzburg–Landau order-parameter. Since the density matrix is an initial value of a Green's function, Gorkov's Ansatz is equivalent to an assumption about the 2-matrix. The present paper considers circumstances in which the Gorkov Ansatz is exactly satisfied by a system of a definite number of fermions.

scholarly journals ρ-PROJECTIVE AND σ-INVOLUNTARY VARIATIONAL INEQUALITIES AND IMPROVED PROJECTION METHOD FOR PROJECTED DYNAMICAL SYSTEM

2021 ◽  
Vol 9 (2) ◽  
pp. 18-24
Author(s):  
Siddharth Mitra ◽  
Prasanta Kumar Das
Keyword(s):  
Dynamical System ◽  
Variational Inequality ◽  
Variational Inequalities ◽  
Gradient Method ◽  
Initial Value Problem ◽  
Equivalence Theorem ◽  
Variational Inequality Problems ◽  
Initial Value ◽  
Type Method ◽  
The Given

Purpose of study: To introduce the concept of projective and involuntary variational inequality problems of order  and  respectively. To study the equivalence theorem between these problems. To study the projected dynamical system using self involutory variational inequality problems. Methodology: Improved extra gradient method is used. Main Finding: Using a self-solvable improved extra gradient method we solve the variational inequalities. The algorithm of the projected dynamical system is provided using the RK-4 method whose equilibrium point solves the involutory variational inequality problems. Application of this study: Runge-Kutta type method of order 2 and 4 is used for the initial value problem with the given projected dynamical system with the help of self involutory variational inequality problems. The originality of this study:  The concept of self involutory variational inequality problems, projective and involuntary variational inequality problems of order  and  respectively are newly defined.

scholarly journals Monotone-Iterative Method for the Initial Value Problem with Initial Time Difference for Differential Equations with “Maxima”

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
S. Hristova ◽  
A. Golev
Keyword(s):  
Differential Equations ◽  
Initial Value Problem ◽  
Upper And Lower Solutions ◽  
Successive Approximations ◽  
Initial Time ◽  
Time Interval ◽  
Monotone Iterative Method ◽  
Initial Value ◽  
Monotone Iterative ◽  
The Given

The object of investigation of the paper is a special type of functional differential equations containing the maximum value of the unknown function over a past time interval. An improved algorithm of the monotone-iterative technique is suggested to nonlinear differential equations with “maxima.” The case when upper and lower solutions of the given problem are known at different initial time is studied. Additionally, all initial value problems for successive approximations have both initial time and initial functions different. It allows us to construct sequences of successive approximations as well as sequences of initial functions, which are convergent to the solution and to the initial function of the given initial value problem, respectively. The suggested algorithm is realized as a computer program, and it is applied to several examples, illustrating the advantages of the suggested scheme.

Evolution of near-inertial waves

1995 ◽  
Vol 301 ◽  
pp. 269-294 ◽  
Author(s):  
R. C. Kloosterziel ◽  
P. Müller
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Initial Value Problems ◽  
Initial Conditions ◽  
Fluid Layer ◽  
Normal Modes ◽  
Buoyancy Frequency ◽  
Infinite Domain ◽  
Initial Value ◽  
Flux Boundary Condition

The three-dimensional evolution of near-inertial internal gravity waves is investigated for the case of a laterally unbounded fluid layer of constant finite depth. A general Green's function formulation is derived which can be used to solve initial value problems or study the effect of forcing. The Green's function is expanded in vertical normal modes, and is very singular. Convolutions with finite-sized initial conditions lead however to well-behaved solutions. Expansions in similarity solutions of the diffusion equation are shown to be an alternative for finding exact solutions to initial value problems, with respect to one normal mode. For the case of constant buoyancy frequency normal modes expansions are shown to be equivalent to expansions in an alternative series of which the first term is the response on the infinite domain, all the others being corrections to account for the no-flux boundary condition on the upper and lower boundaries.

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