Integral Solution of Diffusion Equation: Part 2—Boundary Conditions of Second and Third Kinds

1987 ◽  
Vol 109 (3) ◽  
pp. 557-562 ◽  
Author(s):  
A. Haji-Sheikh ◽  
R. Lakshminarayanan
Keyword(s):  
Diffusion Equation ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Green’S Function ◽  
Green's Function ◽  
Buried Pipe ◽  
Square Enclosure ◽  
Complex Boundary ◽  
The Galerkin Method ◽  
Numerical Complexity

An analytical solution of the diffusion equation using the Galerkin method to calculate the eigenvalues is currently available for boundary conditions of the first kind. This paper includes algebraic techniques to accommodate boundary conditions of the second and third kinds. Several case studies are presented to illustrate the utility and accuracy of the procedure. Selected examples either have no exact solutions or their exact solutions have not been cited because of the mathematical or numerical complexity. The illustrations include transient conduction in hemielliptical solids with either external convective surfaces or convective bases, and buried pipe in a square enclosure. Whenever possible, symbolic programming is used to carry out the differentiations and integrations. In some cases, however, the integrations must be strictly numerical. It is also demonstrated that a Green’s function can be defined to accommodate many geometries with nonorthogonal boundaries subject to more complex boundary conditions for which an exact Green’s function does not exist.

A Liouville Type Theorem for Poly-harmonic System with Dirichlet Boundary Conditions in a Half Space

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Zhao Liu ◽  
Wei Dai
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Half Space ◽  
Green's Function ◽  
Dirichlet Boundary ◽  
Type Theorem ◽  
Dirichlet Boundary Conditions ◽  
Liouville Type ◽  
Liouville Type Theorem ◽  
Harmonic System

AbstractIn this paper, we consider the following poly-harmonic system with Dirichlet boundary conditions in a half space ℝwherewhereis the Green’s function in ℝ

scholarly journals Green's Function for A Piecewise Continous Potential via Integral Equations Method

2018 ◽  
Vol 24 (2) ◽  
pp. 20-35
Author(s):  
Benali Brahim ◽  
Mohammed Tayeb Meftah ◽  
Rai Vandana
Keyword(s):  
Boundary Conditions ◽  
Green’S Function ◽  
Integral Equations ◽  
Green's Function ◽  
Hamiltonian Operator ◽  
Potential Part ◽  
Piecewise Continuous ◽  
Discrete Spectra

The aim of this work is to provide Green's function for the Schrodingerequation. The potential part in the Hamiltonian is piecewise continuous operator.It is a zero operator on a disk of radius "a" and a constant V0 outside this disk (intwo dimensions). We have used, to construct the Green's function, the technique ofthe integral equations. We have respected the boundary conditions of the problem.The discrete spectra of the Hamiltonian operator have been also derived.

Sign Properties of Green's Functions For Two Classes of Boundary Value Problems

1987 ◽  
Vol 30 (1) ◽  
pp. 28-35 ◽  
Author(s):  
P. W. Eloe
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Green’S Function ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Difference Equations ◽  
Green's Functions ◽  
Green’S Functions ◽  
Boundary Value ◽  
Partial Derivatives

AbstractLet G(x,s) be the Green's function for the boundary value problem y(n) = 0, Ty = 0, where Ty = 0 represents boundary conditions at two points. The signs of G(x,s) and certain of its partial derivatives with respect to x are determined for two classes of boundary value problems. The results are also carried over to analogous classes of boundary value problems for difference equations.

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