(a) In order to solve the diffusion equation

by the method of separation of variables, let

and show that

where k² is the separation constant and T₀ is a constant.

(b) Assuming that S depends only on the x coordinate, show that

where k can be either positive or negative, and that

where n₀(x) = n(x, 0) is the known initial density distribution.

(c) Using Fourier transform theory, show that

and, consequently, that

(d) Taking as initial condition

show that

where τ_D = x₀²/D is a characteristic time for diffusion to smooth out the density n.

(e) Generalize the problem for the three-dimensional case in Cartesian coordinates, when S = S(r).

Transcribed Image Text:

Ən (r, t) Ət = DV²n(r, t) =