<< Chapter < Page Chapter >> Page >
v t = f - p ρ + ν 2 v , velocity perpendicular to velocity gradient v t = ν 2 v , if f and p vanish

Green's function, convolution, and superposition

A property of linear PDEs is that if two functions are each a solution to a PDE, then the sum of the two functions is also a solution of the PDE. This property of superposition can be used to derive solutions for general boundary, initial conditions, or distribution of sources by the process of convolution with a Green's function. The student is encouraged to read P. M. Morse and H. Feshbach, Methods of Theoretical Physics , 1953 for a discussion of Green's functions.

The Green's function is used to find the solution of an inhomogeneous differential equation and/or boundary conditions from the solution of the differential equation that is homogeneous everywhere except at one point in the space of the independent variables. (The initial condition is considered as a subset of boundary conditions here.) When the point is on the boundary, the Green's function may be used to satisfy inhomogeneous boundary conditions; when it is out in space, it may be used to satisfy the inhomogeneous PDE.

The concept of Green's solution is most easily illustrated for the solution to the Poisson equation for a distributed source ρ ( x , y , z ) throughout the volume. The Green's function is a solution to the homogeneous equation or the Laplace equation except at ( x o , y o , z o ) where it is equal to the Dirac delta function. The Dirac delta function is zero everywhere except in the neighborhood of zero. It has the following property.

- f ( ξ ) δ ( ξ - x ) d ξ = f ( x )

The Green's function for the Poisson equation in three dimensions is the solution of the following differential equation

2 G = - δ ( x - x o ) = - δ ( x - x o ) δ ( y - y o ) δ ( z - z o ) G ( x x o ) = 1 4 π x - x o

It is a solution of the Laplace equation except at x = x o where it has a singularity, i.e., it has a point source. The solution of the Poisson equation is determined by convolution.

u ( x ) = G ( x x o ) ρ ( x o ) d x o d y o d z o

Suppose now that one has an elliptic problem in only two dimensions. One can either solve for the Green's function in two dimensions or just recognize that the Dirac delta function in two dimensions is just the convolution of the three-dimensional Dirac delta function with unity.

δ ( x - x o ) δ ( y - y o ) = - δ ( x - x o ) δ ( y - y o ) δ ( z - z o ) d z o

Thus the two-dimensional Green's function can be found by convolution of the three dimensional Green's function with unity.

G ( x , y | x o , y o ) = G ( x | x o ) d z o = 1 4 π ln [ ( x x o ) 2 + ( y y o ) 2 ]

This is a solution of the Laplace equation everywhere except at ( x o , y o ) where there is a line source of unit strength. The solution of the Poisson equation in two dimensions can be determined by convolution.

u ( x , y ) = G ( x , y | x o , y o ) ρ ( x o , y o ) d x o d y o

Assignment 7.2 derivation of the green's function

Derive the Green's function for the Poisson equation in 1-D, 2-D, and 3-D by transforming the coordinate system to cylindrical polar or spherical polar coordinate system for the 2-D and 3-D cases, respectively. Compare the results derived by convolution.

Method of images

Green's functions can also be determined for inhomogeneous boundary conditions (the boundary element method) but will not be discussed here. The Green's functions discussed above have an infinite domain. Homogeneous boundary conditions of the Dirichlet type ( u = 0 ) or Neumann type ( u / n = 0 ) along a plane(s) can be determined by the method of images.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Transport phenomena' conversation and receive update notifications?

Ask