analytical formula). The total spectrum is their convolution, whose explicit form can be found in Eq. (B12). By the way, when we put Eq. (24) into Eq. (22), the τ$_{S}$ factor in the front of source term in Eq. (22) is cancelled by factor Γ$_{total}$ in Eq. (24). So, τ$_{eff}$ or Γ$_{eff}$ is still the free parameter in the calculations of positron propagation. Eq. (21) can be approximately solved analytically. The final positron flux observed at the Earth Φ$_{e}$ + ( r$_{⊙}$, E ) is factorized into the convolution of H $_{e}$( E$_{s}$, E ) which encodes the whole astrophysical information and the inject spectrum dN e $_{S}$/dE$_{s}$ where β$_{e}$ + is the velocity of the observed positron and κ = ρ$_{⊙}$/ ( τ$_{eff}$ m$_{D}$ ). The concrete method to calculate the halo functions for electron H $_{e}$( E$_{s}$, E ) and for anti-proton H ¯ $_{p}$( E ) can be found in Ref. [8, 39]. To compare the data, we should consider the cosmic background. It is believed that the astrophysical e ± source is mainly due to supernova explosions (primary e $^{−}$) and the interactions between the CR nuclei and light atoms in interstellar medium (secondary e $^{±}$). They are usually parameterized in the following form [40] where the unit is GeV − $^{1}$cm − $^{2}$s − $^{1}$sr − $^{1}$. Φ bk,pri e − , Φ bk,sec e − , and Φ bk,sec e + are the background fluxes for the primary electron, secondary electron, and secondary positron, respectively. Then the positron fraction observed by the PAMELA experiment and the normalized total elec- tron/positron fluxes observed by the FERMI and ATIC experiments are respectively given by where κ ≤ 1 is a parameter which includes the uncertainties of the primary background electron production. The PAMELA positron fraction data and FERMI electron/positron data can be fitted very well by choosing proper τ$_{eff}$ and κ . The primary DM S three-body decays into muons, which subsequently decay to electron/positron, give the hard positron energy spectrum that is necessary to explain the steep rise in positron fraction observed by the PAMELA experiment [17]. As for the FERMI experiment, its data show a rather flat spectrum up to 12 Φ$_{e}$ + Φ$_{e}$ + + Φ$_{e}$ − = Φ bk,sec e + + Φ DM e + κ Φ bk,pri e − + Φ bk,sec e + + Φ bk,sec e − + Φ DM e + + Φ DM e − , E 3 $_{e}$(Φ$_{e}$ + + Φ$_{e}$ − ) = E 3 $_{e}$( κ Φ bk,pri e − + Φ bk,sec e + + Φ bk,sec e − + Φ DM e + + Φ DM e − ) , (27) Φ bk,pri e − = 0 . 16 E − 1 . 1 1 + 11 E 0 . 9 + 3 . 2 E 2 . $_{15}$, Φ bk,sec e − = 0 . 7 E 0 . 7 1 + 110 E 1 . 5 + 600 E 2 . 9 + 580 E 4 . $_{2}$, Φ bk,sec e + = 4 . 5 E 0 . 7 1 + 650 E 2 . 3 + 1500 E 4 . $_{2}$, (26) Φ$_{e}$ + ( r$_{⊙}$, E ) = β$_{e}$ + 4 π κ τ$_{E}$ E 2 ∫ E$_{max}$ E H $_{e}$( E$_{s}$, E ) dE$_{s}$ dN e S dE$_{s}$ , (25)