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i am trying to find the fundamental solution to

[tex]\frac{\partial c}{\partial t} = D\frac{\partial ^2 c}{\partial x^2}[/tex]

where c=c(x,t)

with initial condition being [tex]c(x,0)=\delta (x)[/tex]

where [tex]\delta (x)[/tex] is the dirac delta function.

i have the solution and working written out in front of me.

first off its got the laplace transform of [tex]\frac{\partial c}{\partial t}[/tex] as

[tex] u\hat c (x,u) -c(x,0)[/tex]

and the Fourier transform of [tex]\frac{\partial ^2 c}{\partial x^2}[/tex] as

[tex]-q^2 \tilde c (q,t) [/tex]

and then out of nowhere we get

[tex]\hat c (q,u) = \frac{c(q,0)}{u+Dq^2}[/tex]

once that bit of magic is done and a leap of faith is taken then i can see how the rest of it falls into place but can anyone explain the above steps to me?

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# Fourier/Laplace transform for PDE

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