Fick's second law is the continuity equation defined below with the first law substituted appropriately

Fick's second law or the continuity equation can be solved for a variety of different initial and boundary conditions.   Each solution represents a different diffusion profile.   Two of the most common profiles are available as part of this calculator.  

Constant-Surface-Concentration Diffusion
The initial condition for this profile is a dopant concentration of zero inside the semiconductor substrate.   For this profile the dopant concentration at the surface of the semiconductor remains constant, and the concentration eventually goes to zero at some point in the semiconductor.   These conditions can be summarized mathematically as

Constant-Total-Dopant Diffusion
The initial condition for this profile is the same as the constant-surface profile in that there exists a zero impurity concentration in the semiconductor.   However, the boundary conditions for this profile are different.   For this profile, a constant or fixed amount of impurities or dopants is deposited on the surface of the semiconductor rather than maintaining a set surface concentration as in the previous profile.   These boundary conditions can be summarized mathematically as

Extrinsic vs. Intrinsic Diffusion
The profiles calculated here are based on intrinsic diffusion.   Intrinsic diffusion is represented by constant diffusivities, and doping concentrations being less than then intrinsic-carrier concentration at the diffusion temperature. The intrinsic-carrier concentration for silicon at 1000°C is 5 X 10¹⁸ cm^-3 and 5 X 10¹⁷ cm^-3 for gallium arsenide. If the dopant-impurity concentration exceeds the intrinsic-carrier concentration at the diffusion temperature, then the diffusion becomes extrinsic, and the profiles become more complicated.   This calculator does not verify whether the diffusion conditions inputted meet the constraint of intrinsic diffusion.   It is assumed the user is knowledgeable as to whether or not the conditions meet the intrinsic diffusion requirement.   This calculator is entirely theoretical and predicts the diffusion profiles mathematically; however, some diffusion profiles may not be experimentally possible depending on the impurity, the semiconductor, and other diffusion parameters such as temperature.

References
H.C. Casey, and G.L. Pearson, "Diffusion in Semiconductors," in J.H. Crawford, and L.M. Slifkin, Eds.,Point Defects in Solids, Vol. 2, Plenum, New York, 1975.
S.M. Sze, Semiconductor Devices: Physics and Technology, 2^nd Ed., Wiley, New York, 2002.