Various anomalous d-dimensional diffusion problems in the presence of an absorbing boundary with radial symmetry were considered. The motion of particles was described by a fractional diffusion equation. Their mean-square displacement was given by ⟨r2⟩∝tγ(0<γ≤1), resulting in normal diffusive motion if γ=1 and sub-diffusive motion otherwise. For the sub-diffusive case in sufficiently high dimensions, divergent series appear when the concentration or survival probabilities were evaluated via the method of separation of variables. While the solution for normal diffusion problems is, at most, divergent as t→0, the emergence of such series in the long-time domain was a specific feature of sub-diffusion problems. A method was presented for regularizing such series and, in some cases, validating the procedure by using alternative techniques (Laplace transform method and numerical simulations). In the normal diffusion case, it was found that the signature of the initial condition on the approach to the steady state rapidly faded away and the solution approached a single (the main) decay mode in the long-time regime. In marked contrast, long-time memory of the initial condition was present in the sub-diffusive case as the spatial part Ψ1(r) describing the long-time decay of the solution to the steady state was determined by a weighted superposition of all spatial modes characteristic of the normal diffusion problem, the weight being dependent on the initial condition. Interestingly, Ψ1(r) turns out to be independent of the anomalous diffusion exponent γ.

Divergent Series and Memory of the Initial Condition in the Long-Time Solution of Some Anomalous Diffusion Problems. S.B.Yuste, R.Borrego, E.Abad: Physical Review E, 2010, 81[2], 021105