A general solution of the diffusion equation for semiinfinite geometries

The solution of the time-dependent diffusion equation in a semiinfinite planar, cylindrical, or spherical geometry with common initial and asymptotic boundary conditions is considered. It is shown that this boundary value problem may be described by a single equation which involves only a first order spatial derivative and a half order time derivative. The replacement is exact in the planar and spherical geometry cases but approximate in the cylindrical case. This replacement permits the solution of the original boundary value problem to be written for any boundary condition at the origin. It also leads to a simple relationship between the boundary flux and the boundary intensive variable, which does not require a calculation of the intensive variable at all positions and times.