Analysis of the anomalous diffusion in comb structure with absorbing boundary conditions

The diffusion in comb structure is an important kind of anomalous diffusion with widespread applications. The special structure corresponds to a novel characteristic of anomalous diffusion, which is characterised by the Dirac delta function in the governing equation. By considering the memory characteristic, the fractional derivative is introduced into the constitutive relation, and a new fractional governing equation in the infinite regions is constructed. Instead of simply truncating for the infinite regions, the exact absorbing boundary conditions are deduced by using the (inverse) Laplace transform technique and the stability is analysed. To deal with the governing equation containing the Dirac function, the finite difference method is proposed and the term with the Dirac function is handled using an integration method. The stability and convergence of the numerical scheme are discussed in detail. A fast algorithm is presented that the normal L1-scheme is approximated via a sum-of-exponentials approximation. Three examples are conducted, in which the particle distributions and the mean square displacement for the anomalous diffusion in comb structure are discussed. The computational time between the normal numerical scheme and the fast numerical scheme is compared and the rationality and validity of absorbing boundary conditions are analysed. An important finding is that the distribution of the mean square displacement with the absorbing boundary conditions can match the exact one accurately, which demonstrates the effectiveness of the method.