when solving the heat equation using the Separation of variables method, we arrive at a solution of the form:

u(x,t) = sum An sin( n pi x) * exp(-k (npi)^2 t)

If we would solve the same heat equation using the fourier transform the solution will look like:

u(x,t) = 1/sqrt(2pi) * integral f_hat(w) exp(-c^2 w^2 t + i wx) dw

Both solutions look very differnet from each other. What is the trick/ transform one solution into the other. thanks

Question

when solving the heat equation using the Separation of variables method, we arrive at a solution of the form: 

u(x,t) = sum An sin( n pi x) * exp(-k (npi)^2 t)

If we would solve the same heat equation using the fourier transform the solution will look like:

u(x,t) = 1/sqrt(2pi) * integral f_hat(w) exp(-c^2 w^2 t + i wx) dw

Both solutions look very differnet from each other. What is the trick/ transform one solution into the other. thanks