() Diffusion from a point source. (10 points) The solution to the one-dimensional diffusion equation, tc=Dx22c, is: c(x,t)=4Dtc0ex2/4Dt To derive this expression, follow the steps below: (a) Take the Fourier transform of the diffusion equation by transforming in the spatial variables to obtain a new differential equation for c~(k,t) (b) Solve the resulting differential equation for c~(k,t). Then compute the inverse Fourier transform to arrive at the solution in real space, c(x,t). (c) Show that the solution for an arbitrary initial concentration distribution c(x,t=0) can be written as an integral over the solution for a point source. In particular, consider an initial concentration problem of the form c(x,0)=c0 for x0, and find the resulting diffusive profile. (d) Formally derive the relation x2=2Dt