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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

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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
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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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