The propagator for one-dimensional diffusion

Before we show the general solution of the three-dimensional diifusion Eq. (3) or Eq. (6), we first solve the diffusion equation for diffusion along one axis, e.g., the z axis parallel to the pulsed field gradient. The diffusion equation is then  

Equation (7) can be solved by separation of the variables. We write P as a product of two functions X and T  

Here the constants are arbitrary integration constants, which in general are different for different X. The propagator is then (setting t = A)  

(14) we took care of the fact that the differentiation in the diffusion Eq. (3) was with respect to the primed co-ordinates. Because X is an arbitrary constant, the general one-dimensional solution of the diffusion equation is  

Solution (15) is written in the form of a normal mode expansion with eigenvalues —X D and eigenfunctions represented by the spatial parts of Eq. (15). Of course, the above solution must fulfil the appropriate initial and boundary conditions. 

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