3 IV. FRACTIONAL SPHERICAL EQUATION AND FRACTIONAL CYLINDRICAL EQUATION A. Fractional Laplacian equation In this section, we consider the Laplacian equation with the 3-d Laplacian operaror defined above and the fractional heat conduct equation in a 2-d space with cylindrical symmetry. With the Laplacian operator defined above, the Lapla- cian equation becomes This equation can be solved by separation of variables. Let u = R ( r )Θ( θ )Φ( φ ) and substitute, the result is The second equation can be solved by fractional series expansion. Let R = ∑ ∞ m = −∞ c$_{m}$r $^{mα}$, and substitute, the result is that the nonzero components are the ones with m satisfying Otherwise, c$_{m}$ =0. The first equation is a variation of the ordinary spher- ical harmonic equation. By further decomposition, it turns to The first equation is simple. The second equation can be transformed by changing variables x = cos θ and p ( x ) = Θ( θ ) to This is a variation of the ordinary associated Legendre equation [11, 12]. By setting α = 1, we recover the ordi- nary associated Legendre equation. When m = 0 it is the Legendre equation. Make the ansatz p ( x ) = ∑ ∞ n =0 c$_{n}$x n and substitute, we find that the terms with even powers of x and the terms with odd powers of x are independent, so we can write p ( x ) = The heat conduct equation in 2-d space is Let α = 1 and λ = l ( l +1), we will get the Legendre poly- nomials. We calculated numerically the functions with some different values of the parameters and displayed the results in Fig. 2 and Fig. 3. c$_{0}$ · p$_{even}$ ( x ) + c$_{1}$ · p$_{odd}$ ( x ). The relation of successive coefficients is △ α u = 1 r 2 $_{α}$D α $_{r}$( r 2 $^{α}$Dα r u ) + Γ $^{2}$( α + 1) r 2 $^{α}$sin $^{α}$θ ∂ ∂θ (sin $^{α}$θ ∂ ∂θ u ) + Γ $^{2}$( α + 1) r 2 $^{α}$sin 2 $^{α}$θ ∂ 2 ∂φ 2 u = 0 . (17) 1 Θsin $^{α}$θ d dθ ( sin $^{α}$θ d Θ dθ ) + 1 Φsin 2 $^{α}$θ d $^{2}$Φ dφ 2 = − λ, (18) 1 R D α r ( r 2 $^{α}$Dα r R ) = λ Γ $^{2}$( α + 1) . (19) Γ( mα + α + 1) Γ( mα − α + 1) = λ Γ $^{2}$( α + 1) . (20) d $^{2}$Φ dφ 2 + m $^{2}$Φ = 0 , ( m = 0 , 1 , 2 , 3 , ... ) (21) sin $^{α}$θ Θ d dθ ( sin $^{α}$θ d Θ dθ ) = m 2 − λ sin 2 $^{α}$θ. (22) c$_{n}$$_{+2}$ = c$_{n}$ n 2 + nα − λ ( n + 2)( n + 1) . (24) (1 − x $^{2}$) d $^{2}$p dx 2 − (1 + α ) x dp dx + [ λ − m 2 (1 − x $^{2}$) α ] p = 0 . (23) ∂u ∂t = a $^{2}$△$_{2}$ u. (25) B. Fractional cylindrical equation FIG. 2: Legendre even function. Since the fractional index α occurs as a multiplicative factor in the variation version of the Legendre equation, small difference from 1 will not make large changes to the profile. We calculated the function with other small difference α ’s, but the results are not shown for their curves are very close to each other. Notice the direct of the curves. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −6 −4 −2 0 2 4 6 =1 =4  lamda=12  =1/10 Legendre function even,