Analysis of absorbing boundary conditions for the anomalous diffusion in comb model on unbounded domain by finite volume method

Abstract

The diffusion in the comb model is an important kind of anomalous diffusion which is described by a governing equation containing the Dirac delta function and the solution’s domain is infinite. Two key problems are solved to analyze the mass transfer mechanism in the comb model. One is to construct the appropriate and reasonable boundaries for treating the infinite boundaries. The exact absorbing boundary conditions with the Caputo’s fractional derivative are deduced by the Laplace transform technique. The other one is applying the finite volume method by integrating the governing equation over a governed volume to obtain the numerical solution with the advantage that the effects of the singular Dirac functions disappear. In addition, by introducing a source term, the exact solution is deduced and the comparison between the numerical solution and the exact solution is presented, which verifies the effectiveness and reasonableness of the numerical method. The particle distributions with different parameters are analyzed and discussed. By comparing with the traditional truncated boundary method, it is noteworthy that the mean square displacement for the absorbing boundary conditions has a good agreement with the exact expression.