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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

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Stoney Reply
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Introduction about quantum dots in nanotechnology
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s. Reply
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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
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s. Reply
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Do you know which machine is used to that process?
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s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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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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