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2 u - a 2 u t = - ρ ( x , t )

where the parameter a 2 represents the ratio of the storage capacity and the conductivity of the system and ρ is a known distribution of sources in space and time. The infinite domain Green's function g n ( x , t x o , t o ) is a solution of the following equation

2 g n ( x , t x o , t o ) - a 2 g n ( x , t x o , t o ) t = - δ ( x - x o ) δ ( t - t o )

The source term is an impulse in the spatial and time variables. The form of the Green's function for the infinite domain, for n dimensions, is (Morse and Feshbach, 1953)

g n ( x , t x o , t o ) = g n ( R , τ ) = 1 a 2 a 2 π τ n e - ( a 2 R 2 / 4 τ ) , τ > 0 0 , τ < 0 where τ = t - t o R = x - x o

This Green's function satisfies an important integral property that is valid for all values of n :

g n ( R , τ ) d V n = 1 a 2 , τ > 0

This expression is an expression of the conservation of heat energy. At a time t o at x o , a source of heat is introduced. The heat diffuses out through the medium, but in such a fashion that the total heat energy is unchanged.

The properties of this Green's function can be more easily seen be expressing it in a standard form

a 2 g n ( R , τ ) = 1 2 π ( 2 τ / a 2 ) n e - [ R 2 / 2 ( 2 τ / a 2 ) ] , τ > 0 0 , τ < 0

The normalized function a 2 g n for n = 3 represent the probability distribution of the location of a Brownian particle that was at x o at time t o . The cumulative probability is equal to unity.

The same normalized function for n = 1 , corresponds to the normal or Gaussian distribution with the standard deviation given by

σ = 2 τ a

Observe the Green's function in one, two, and three dimensions by executing greens.m and the function, greenf.m in the diffuse subdirectory of CENG501. You may wish to use the code as a template for future assignments.

Step Response Function The infinite domain Green's function is the impulse response function in space and time. The response for a distribution of sources in space or as an arbitrary function of time can be determined by convolution. In particular the response to a constant source for τ > 0 is the step response function . It has classical solutions in one and two dimensions. The unit step function or Heaviside function is the integral of the Dirac delta function.

- t δ ( t ' - t o ) d t ' = S ( t - t o ) = 1 , t - t o > 0 0 , t - t o < 0

The response function to a unit step in the source can be determined by integrating the Greens function or the impulse response function in time.

- t 2 g n - a 2 g n t d t ' = - δ ( x - x o ) - t δ ( t ' - t o ) d t ' 2 - t g n d t ' - a 2 - t g n d t ' t = - δ ( x - x o ) S t - t o 2 U n - a 2 U n t = - δ ( x - x o ) S t - t o where U n = - t g n d t '

In one dimension, the step response function that has a unit flux at x = 0 is (R. V. Churchill, Operational Mathematics , 1958) (note: source is 2 δ x - 0 S t - 0 )

U 1 flux = - 1 = 2 a 2 t π a 2 e - x 2 4 t / a 2 - a 2 x erfc x 4 t / a 2 , t > 0

For comparison, the function that has a value of unity at x = 0 (Dirichlet boundary condition) is

U 1 = 1 = erfc x 2 t / a 2 , t > 0

In two dimensions, the unit step response function for a continuous line source is (H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 1959)

U 2 = - a 2 4 π E i - R 2 4 t / a 2 , t > 0 R 2 = ( x - x o ) 2 + ( y - y o ) 2 - E i ( - x ) = x e - u u d u = exp int ( x ) , MATLAB function

For large times this function can be expressed as

U 2 approx . = a 2 4 π ln 4 t / a 2 R 2 - γ a 2 4 π , for 4 t a 2 R 2 > 100 γ = 0 . 5772 ...

In three dimensions, the unit step response function for a continuous point source is (H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 1959)

Questions & Answers

how can chip be made from sand
Eke Reply
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are nano particles real
Missy Reply
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
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where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
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.
yes that's correct
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?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
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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