For instance, the diffusion equation for positron is where ψ ( ⃗ x, E ) denotes the positron number density per unit energy. Diffusion coefficient K = K$_{0}$E δ is space independent. The third term describes the energy loss of positron through the synchrotron radiation and inverse Compton scattering, with loss ratio b ( E ) = E $^{2}$/τ$_{E}$ where τ$_{E}$ = 10 $^{16}$s . q ( ⃗ x, E ) is the positron source term, describing the positron number density injected at ⃗ x , per unit time and energy. For simplicity, it can be expressed as the product of an astrophysics factor and a particle physics factor where we introduce another index F to distinguish the different states from the I -th decay channel. Moreover, the Milky Way DM density profile ρ ( ⃗ x ) in the spherical coordinates is generically written as where r$_{⊙}$ = 8 . 5 kpc is the distance from the solar system to the Milky Way center, and the DM profile density in the solar system is set to be ρ$_{⊙}$ = 0 . 3 GeV cm − $^{3}$. According to the N − body simulation, it has three popular choices parameterized by ( α, β, γ ). We use the NFW profile with (1 , 3 , 1) [37], and choose the DM central core r$_{s}$ = 20 kpc. By the way, the injected source term is linearly proportional to the decay DM relic density. Therefore, the fraction of the DM S relic density r$_{DM}$ to the whole DM relic density should be taken into account. For simplicity, we absorb it into the redefinition of S lifetime by τ$_{S}$ → r$_{DM}$τ$_{S}$ without disturbing any other terms. The normalized energy spectrum distribution function dN e $_{I,F}$/dE = (2 /m$_{D}$ ) dN e $_{I,F}$/dx describes the positrons from the specific DM decay intermediate state ( I, F ), with ( I, F ) = ˜ µ$_{L}$, A $^{0}$... , and so on. And B$_{I,F}$ = B$_{I}$ is the corresponding branch ratio. In this paper, summing over all the channels (see Appendix, with a little bit changed notation), we obtain the total fragmentation function where dN/dx$_{I,F}$ and 1 /R$_{I,F}$ dN/dx I,F e respectively give the fragmentation functions in the rest frame of the DM S and particle ( I, F ). The former is analytically calculable, while the later is obtained through PYTHIA [38] simulation (except for muon decay, we shall use the dN S e dx = Γ$_{eff}$ Γ$_{total}$ ∑ I,F ( C 2 $_{I,F}$R$_{I,F}$ ) × 1 R$_{I,F}$ d ˜ N dx$_{I,F}$ ⊗ dN dx I,F e , (24) ρ ( r ) = ρ$_{⊙}$ ( r$_{⊙}$ r ) γ ( 1 + ( r$_{⊙}$/r$_{s}$ ) α 1 + ( r/r$_{s}$ ) α ) ( β − γ ) /α , (23) q ( ⃗ x, E ) = ( 1 τ$_{S}$ ∑ I,F dN e I,F dE B$_{I,F}$ ) × ρ ( ⃗ x ) m$_{D}$ , (22) 11 ∂ψ ∂t − ∇ · ( K ( ⃗ x, E ) ∇ ψ ) − ∂ ∂E ( b ( E ) ψ ) = q ( ⃗ x, E ) , (21)