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Unformatted text preview: Notes on the fundamental solution of the diffusion equation Patrick De Leenheer * March 23, 2009 On p. 105 of our text it is claimed that the diffusion equation on R with the Dirac delta function as initial condition: u t = Du xx , u ( x, 0) = ( x ) has the following solution: u ( x,t ) = 1 2 Dt e- x 2 4 Dt , also called the fundamental solution . It is not shown how this solution is obtained, and these notes will outline a way to do it. Dilation Let u ( x,t ) be any solution of u t = u xx . Then given a parameter m 6 = 0, we see that u ( mx,m 2 t ) is also a solution (just plug in the latter function in the equation and see that it satisfies it, regardless of the value of m !). This suggests we might attempt to find solutions that depend on the ratio x 2 /t instead of on the pair ( x,t ). Therefore, we let u ( x,t ) be of the following form: u ( x,t ) = v x 2 t , for some appropriate function v , yet to be determined. In this case, u t =- x 2 t 2 v x 2 t and u x = 2 x t v x 2...
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- Summer '06