A fractional governing equation is derived from the two-dimensional anomalous diffusion in the comb model with the Cattaneo model by using rigorous derivation. One strategy to deal with the unbounded region is to create acceptable truncation. We abandon the traditional method of approximating infinite boundaries by enormous value and use the artificial boundary method to construct absorbing boundary conditions using the Laplace transform. The absorbing boundary conditions with the Mittag-Leffler function are obtained, and the stability is demonstrated. We discretise the governing equation by using the finite difference method, and the accuracy of the numerical method is confirmed by comparing with the exact solution, which is generated by introducing a source term. The particle distributions and the mean square displacement under the absorbing boundary conditions are in good agreement with the exact expressions which are superior to the conventional direct truncation boundary conditions. Additionally, the particle distributions under different parameters are examined and explained graphically.