lec13-3 - Green function for diffusion equation •...

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Unformatted text preview: Green function for diffusion equation • Consider the diffusion equation, ∂2u 1 ∂u =2 2 ∂x α ∂t • Let’s solve using the inverse Fourier transform, ∞ u (k , t )e ikx dx u (x , t ) = −∞ • Substitution in the differential equation allows us to easily integrate the time dependence... −k 2 u (k , t ) = 1 ∂ u (k , t ) α2 ∂ t 22 • We obtain by integration u (k , t ) = u (k , t = t )e −α k (t −t ) • We could also do it by separation of variables, practice it that way too.... Green function for diffusion equation, continued • Assume we have a point source at t = t , so that u (x , t = t ) = δ (x − x ) • We can then find u (k , t = t ) for the Fourier transform of the point source u (k , t = t ) = 1 2π ∞ δ (x − x )e −ikx dx = −∞ e −ikx 2π • Finally we find u (x , t ) for t > t from the inverse Fourier transform u (x , t ) = 1 2π ∞ e ik (x −x ) e −k 2 α2 (t −t ) dk −∞ • The integral can be done by “completing the squares” Green function for diffusion equation, continued • The result of the integral is actually the Green function G (x , x ; t , t ) G (x , x ; t , t ) = 1 [4πα2 (t 1/2 − t )] e −(x −x )2 /4α2 (t −t ) • Notice that the Green function only depends on x − x and t − t ∞ • We find that at all times, −∞ G (x , x ; t , t )dx = 1 • Then if we have a t = 0 distribution u (x , t = 0), we can find u (x , t ) just by doing the integral, ∞ u (x , t ) = G (x , x ; t , 0)u (x , t = 0)dx −∞ ...
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This note was uploaded on 07/30/2011 for the course PHZ 3113 taught by Professor Staff during the Spring '03 term at University of Central Florida.

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lec13-3 - Green function for diffusion equation •...

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