content_image/416126.jpg content_image/416127.jpg content_image/416133.jpg content_image/416128.jpg content_image/416129.jpg content_image/416130.jpg content_image/416131.jpg content_image/416137.jpg content_image/416132.jpg content_image/416134.jpg
## IV. FRACTIONAL SPHERICAL EQUATION AND FRACTIONAL CYLINDRICAL 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 

## A. Fractional Laplacian equation 

<img src='content_image/416126.jpg'>

This equation can be solved by separation of variables. Let u = R ( r )Θ( θ )Φ( φ ) and substitute, the result is 

<img src='content_image/416127.jpg'>

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 

<img src='content_image/416133.jpg'>

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 ) = 

<img src='content_image/416128.jpg'>

<img src='content_image/416129.jpg'>

<img src='content_image/416130.jpg'>

<img src='content_image/416131.jpg'>

<img src='content_image/416137.jpg'>

c$_{0}$ · p$_{even}$ ( x ) + c$_{1}$ · p$_{odd}$ ( x ). The relation of successive coefficients 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. 

The heat conduct equation in 2-d space is 

<img src='content_image/416132.jpg'>

## B. Fractional cylindrical equation 

<img src='content_image/416134.jpg'>

