J. Basic. Appl. Sci. Res., 2(9)8865-8876, 2012
© 2012, TextRoad Publication
ISSN 2090-4304
Journal of Basic and Applied
Scientific Research
www.textroad.com
*Corresponding Author: Mehdi Delkhosh, Islamic Azad University, Bardaskan Branch, Department of
Mathematics, Bardaskan, Iran, Email:mehdidelkhosh@yahoo.com
Green’s Function and its Applications
Mehdi Delkhosh, Mohammad Delkhosh
2
, Mohsen Jamali
3
1)
Islamic Azad University, Bardaskan Branch, Department of Mathematics, Bardaskan, Iran
2)
Islamic Azad University, Bardaskan Branch, Department of Computer, Bardaskan, Iran
3)
Islamic Azad University, Bardaskan Branch, Department of Computer, Bardaskan, Iran
ABSTRACT
Green's function, a mathematical function that was introduced by George Green in 1793 to 1841.
Green’s functions used for solving Ordinary and Partial Differential Equations in different dimensions
and for time-dependent and time-independent problem, and also in physics and mechanics, specifically in
quantum field theory, electrodynamics and statistical field theory, to refer to various types of correlation
functions.
In this paper, we describe some of the applications of Green's function in sciences, to determine the
importance of this function. i.e. Boundary and Initial Value problem, Wave Equation, Kirchhoff
Diffusion Equation, Diffraction Theory, Helmholtz Equation and etc.
KEYWORDS: Green’s Function, Boundary Value problem, Wave Equation, Kirchhoff Diffusion
Equation, Diffraction Theory, Helmholtz Equation, Diffusion Equation, Laplace Equations, Poisson
Equations, Bessel Equations, Sturm-Liouville Differential Equation.
AMS 2000: 34K17, 34K10, 34A30, 34B05, 46L60, 47B25
1. INTERODUCTION
George Green (14 July 1793 – 31 May 1841) was largely self-taught British mathematical physicist
who wrote "An Essay on the Application of Mathematical Analysis to the Theories of Electricity and
Magnetism (Green, 1828)". The essay introduced several important concepts, among them a theorem
similar to the modern Green's theorem, the idea of potential functions as currently used in physics, and
the concept of what are now called Green's functions. George Green was the first person to create a
mathematical theory of electricity and magnetism and his theory formed the foundation for the work of
other scientists such as James Clerk Maxwell, William Thomson, and others. His work ran parallel to that
of the great mathematician Gauss (potential theory).
1.1. Definition
We assume the following ordinary differential equation in the interval b a, , is given [3, 4, 5, 6, 9, 25,
and 26]:
b x a x r y x P y x P y Ly ) ( ) ( ) (
2 1
(1)
where x r x P x P ), ( ), (
2 1
are continue and differentiable on b a, (analytic on b a, ).
If two independent solution ) (
1
x y and ) (
2
x y are available for the homogeneous equation 0 Ly , then
2 2 1 1
) (
y C y C y
h
a general solution of 0 Ly on b a, .
Now, we suing the method of variation of parameters to get particular solution of ) ( x r Ly on b a, .
If we substitute
2 2 1 1
y v y v y into the equation ) ( x r Ly and we assume 0
2 2 1 1
y v y v , we obtain
dx
y y W
x r y
v dx
y y W
x r y
v
) , (
) ( .
,
) , (
) ( .
2 1
1
2
2 1
2
1
Or
dx
y y W
x r y
y dx
y y W
x r y
y y
p
) , (
) ( .
) , (
) ( .
2 1
1
2
2 1
2
1
) (
(2)
Or, let
0
x be any point on the interval b a, and let the indefinite integrals be replaced by definite
integrals with respect to a dummy variable
from
0
x to x :
x
x
p
d r
y y y y
x y y x y y
y
0
) (
) ( ). ( ) ( ). (
) ( ). ( ) ( ). (
1 2 2 1
1 2 2 1 ) (
(3)
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