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Suppose one wished to find the solution to the Poisson equation in the semi-infinite domain, y > 0 with the specification of either u = 0 or u / n = 0 on the boundary, y = 0 . Denote as u 0 ( x , y , z ) the solution to the Poisson equation for a distribution of sources in the semi-infinite domain y > 0 . The solutions for the Dirichlet or Neumann boundary conditions at y = 0 are as follows.

u ( x , y , z ) = u 0 ( x , y , z ) - u 0 ( x , - y , z ) , for u = 0 a t y = 0 u ( x , y , z ) = u 0 ( x , y , z ) + u 0 ( x , - y , z ) , for d u / d y = 0 a t y = 0

The first function is an odd function of y and it vanishes at y = 0 . The second is an even function of y and its normal derivative vanishes at y = 0 .

An example of the method of images to satisfy either the Dirichlet or Neumann boundary conditions is illustrated in the following figure. The black curve is the response to a line sink at x = 1 . 5 . We desire to have either the function or the derivative at x = 0 to vanish. The red curve is a line sink at x = - 1 . 5 . The sum of the two functions is symmetric about x = 0 and has zero derivative there. The difference is anti-symmetric about x = 0 and vanishes at x = 0 .

Now suppose there is a second boundary that is parallel to the first, i.e. y = a that also has a Dirichlet or Neumann boundary condition. The domain of the Poisson equation is now 0 < y < a . Denote as u 1 the solution that satisfies the B C at y = 0 . A solution that satisfies the Dirichlet or Neumann boundary conditions at y = a are as follows.

u ( x , y , z ) = u 1 ( x , y , z ) - u 1 ( x , 2 a - y , z ) , for u = 0 at y = a u ( x , y , z ) = u 1 ( x , y , z ) + u 1 ( x , 2 a - y , z ) , for d u / d y = 0 at y = a

This solution satisfies the solution at y = a , but no longer satisfies the solution at y = 0 . Denote this solution as u 2 and find the solution to satisfy the B C at y = 0 . By continuing this operation, one obtains by induction a series solution that satisfies both boundary conditions. It may be more convenient to place the boundaries symmetric with respect to the axis in order to simplify the recursion formula.

Assignment 7.3

Calculate the solution for a unit line source at the origin of the x , y plane with zero flux boundary conditions at y = + 1 and y = - 1 . Prepare a contour plot of the solution for 0 < x < 5 . What is the limiting solution for large x ? Note: The boundary conditions are conditions on the derivative. Thus the solution is arbitrary by a constant.

Existence and uniqueness of the solution to the poisson equation

If the boundary conditions for Poisson equation are the Neumann boundary conditions, there are conditions for the existence to the solution and the solution is not unique. This is illustrated as follows.

2 u = - ρ in V , n u = f on S 2 u d V = - ρ d V n u d S = - ρ d V f d S = - ρ d V

This necessary condition for the existence of a solution is equivalent to the statement that the flux leaving the system must equal the sum of sources in the system. The solution to the Poisson equation with the Neumann boundary condition is arbitrary by a constant. If a constant is added to a solution, this new solution will still satisfy the Poisson equation and the Neumann boundary condition.

Green's function for the diffusion equation

We showed above how the solution to the Poisson equation with homogeneous boundary conditions could be obtained from the Green's function by convolution and method of images. Here we will obtain the Green's function for the diffusion equation for an infinite domain in one, two, or three dimensions. The Green's function is for the parabolic PDE

Questions & Answers

how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
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Joseph
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no can't
Lohitha
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currently
William
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Maira Reply
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Ali
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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
Google
da
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Bhagvanji
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Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
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ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
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Mahi
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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
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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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