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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
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Missy Reply
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Lale Reply
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Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
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Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
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Jyoti Reply
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Crow Reply
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RAW Reply
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I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
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What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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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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