Efficient representations of Green’s functions for some elliptic equations with piecewise-constant coefficients

2010 ◽  
Vol 8 (1) ◽  
pp. 53-72 ◽  
Author(s):  
Yuri A. Melnikov
Keyword(s):  
Elliptic Equations ◽  
Green's Functions ◽  
Green’S Functions ◽  
Piecewise Constant ◽  
Constant Coefficients

Indefinite Green's Functions and Elementary Solutions

1963 ◽  
Vol 6 (1) ◽  
pp. 71-103 ◽  
Author(s):  
G. F. D. Duff ◽  
R. A. Ross
Keyword(s):  
Green's Functions ◽  
Green’S Functions ◽  
Applied Mathematics ◽  
Integral Transforms ◽  
Source Distribution ◽  
Asymptotic Estimates ◽  
Elementary Functions ◽  
Constant Coefficients ◽  
Linear Differential ◽  
Superposition Of Solutions

Linear differential equations both ordinary and partial are often studied by means of Green's functions. One reason for this is that linearity permits superposition of solutions. A Green's function describes the "effect" of a point source, and the description of line, surface, or volume sources is achieved by superposing, that is to say, integrating, this function over the source distribution.For equations with constant coefficients the use of integral transforms permits the calculation of such source functions in the form of integrals. Only in the simplest cases is explicit evaluation by elementary functions possible, and this has perforce led to the use of asymptotic estimates, which so thoroughly pervade the domain of applied mathematics.

scholarly journals Green's functions and Riemann's method

1965 ◽  
Vol 14 (4) ◽  
pp. 293-302 ◽  
Author(s):  
A. G. Mackie
Keyword(s):  
Fundamental Solution ◽  
Green’S Function ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Elliptic Equations ◽  
Green's Functions ◽  
Green’S Functions ◽  
Hyperbolic Equations ◽  
Boundary Value ◽  
Auxiliary Function

Methods for solving boundary value problems in linear, second order, partial differential equations in two variables tend to be somewhat rigidly partitioned in some of the standard text-books. Problems for elliptic equations are sometimes solved by finding the fundamental solution which is defined as a solution with a given singularity at a certain point. Another approach is by way of Green's functions which are usually defined as solutions of the original homogeneous equations now made inhomogeneous by the introduction of adelta function on the right hand side. The Green's function coincides with the fundamental solution for elliptic equations but exhibits a totally different type of singularity for parabolic or hyperbolic equations. Boundary value problems for hyperbolic equations can often by solved by Riemann's method which depends on the existence of an auxiliary function called the Riemann or sometimes the Riemann-Green function. The main object of this paper is to show the close relationship between Riemann's method and the method of Green's functions. This not only serves to unify different methods of solution of boundary value problems but also provides an additional method of determining Riemann functions for given hyperbolic equations. Before establishing these relationships we shall survey the general approach to boundary value problems through the use of the Green's function.

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