FIG. 4: Parameters of fractional Bessel function. ν as a func- tion of α and ρ . FIG. 3: Legendre odd function. Notice the direct of the curves. FIG. 5: Fractional Bessel function. The fractional index α shows up in the exponentials of the fractional Bessel series; a small difference from 1 changes the profile largely. For a big α , ρ is small, so the change is suppressed. ∂u ∂t = 1 r α D α $_{r}$( r $^{α}$Dα r u ) + Γ $^{2}$( α + 1) r 2 α ∂ 2 ∂θ $_{2}$u. (26) T ′ + a $^{2}$k$^{2}$T = 0 , (27) ∂ $^{2}$Θ ∂θ 2 + ν $^{2}$Θ = 0 , (28) 1 Γ $^{2}$( α + 1) r $^{α}$Dα $_{r}$( r $^{α}$Dα r R ) + k 2 r 2 α Γ $^{2}$( α + 1) R − ν $^{2}$R = 0 . (29) c$_{0}$ [( Γ( αρ + 1) Γ( αρ − α + 1) ) 2 − ν $^{2}$Γ$^{2}$( α + 1) ] = 0 , (30) c$_{1}$ [( Γ( αρ + α + 1) Γ( αρ + 1) ) 2 − ν $^{2}$Γ$^{2}$( α + 1) ] = 0 , (31) [( Γ( αρ + 1) Γ( αρ − α + 1) ) 2 − ν $^{2}$Γ$^{2}$( α + 1) ] = 0 , (33) c$_{m}$ [( Γ( αρ + αm + 1) Γ( αρ + αm − α + 1) ) 2 − ν $^{2}$Γ$^{2}$( α +1) ] + c$_{m}$$_{−}$$_{2}$k 2 = 0 . (32) d$_{n}$ = d$_{n}$$_{−}$$_{1}$ 1 [( Γ( αρ + α · 2 n +1) Γ( αρ + α · 2 n − α +1) ) 2 − ν $^{2}$Γ$^{2}$( α + 1) ] , (35) R$_{ρ}$ ( r ) = r αρ ∞ ∑ n =0 ( − 1) $^{n}$d$_{n}$ k 2 $^{n}$rα · 2 $^{n}$, (34) and c$_{1}$ = 0. By the recursive relation, a solution is implied where d$_{0}$ = 1, and ρ satisfies Eq.(33). Eq.(33) in one-order calculus ( α = 1) is simply ρ = ± ν . We meshed in Fig. 4 the surface defined by the equation (33). After solving the equation with ν = 3 for ρ when α varies, we plotted in Fig. 5 R$_{ρ}$ ( r ) belonging to different values of α . To have a starting term, c$_{0}$ = 0, so The first two equations are simple. The third equation is a fractional generalization of the Bessel equation [11, 12]. It can be solved by fractional series expansion. Since Bessel equation is singular at r = 0. We must use such ansatz: R = r αρ ∑ ∞ m =0 c$_{m}$r $^{αm}$. Substitute it into the above equation, we get By separation of variables u ( t, r, θ ) = R ( r )Θ( θ ) T ( t ), it can be decomposed to Here △$_{2}$ is the two dimensional Laplacian operator. Us- ing the fractional cylindrical Laplacian operator defined above, this equation becomes 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 =1/4  =1  =1.3  Legendre function odd, lamda=12  0 0.5 1 1.5 2 0 1 2 3 4 −5 0 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 =1  =2/3  =1/2 =3/2 Fractional Bessel functions  =3, k=1 