# Fourier/Laplace transform for PDE

1. May 10, 2006

hello
i am trying to find the fundamental solution to
$$\frac{\partial c}{\partial t} = D\frac{\partial ^2 c}{\partial x^2}$$
where c=c(x,t)
with initial condition being $$c(x,0)=\delta (x)$$
where $$\delta (x)$$ 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 $$\frac{\partial c}{\partial t}$$ as
$$u\hat c (x,u) -c(x,0)$$
and the Fourier transform of $$\frac{\partial ^2 c}{\partial x^2}$$ as
$$-q^2 \tilde c (q,t)$$
and then out of nowhere we get
$$\hat c (q,u) = \frac{c(q,0)}{u+Dq^2}$$
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?

2. Jun 2, 2006

### JohanL

To get the fourier and laplace transform of the first derivative you put df/dx into the definitions, integrate by parts and use, in the fourier case, that f(x) must vanish as x goes to infinity in order for the the Fourier transform of f(x) to exist. And this you easily can generalize to higher derivatives.

If you set c(q,0)=1 you get the Green function for the diffusion equation in the transformed space. The Green function is very useful for solving more complicated problems. A general solution to the diffusion equation is then integrals over the Green function G(r,r';t,t') times the source term, the boundary conditions, the initial condition, respectively.