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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

what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
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Source:  OpenStax, Transport phenomena. OpenStax CNX. May 24, 2010 Download for free at http://cnx.org/content/col11205/1.1
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