content_image/376852.jpg content_image/376849.jpg
<img src='content_image/376852.jpg'>

the equation (45) can be translated to a fractional differ- ential equation 

<img src='content_image/376849.jpg'>

[1] I. Podlubny, Fractional Differential Equations , Academic Press, San Diego, 1999. 

[2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Application of Fractional Differential Equations , El- sevier, Amsterdam, 2006. 

[3] G. M. Viswanathan, T. M. Viswanathan, arXiv:0807.1563v1 [physics.flu-dyn] 10 Jul 2008 

[4] V. Gariychuk, B. Datsko, V. Meleshko, Physica A 387 (2008) 418-424 

[5] V. Gariychuk, B. Datsko, Physics Letters A 372 (2008) 4902-4904 

[6] Lei Song, Shiyun Xu, Jianying Yang, Commun Nonlinear Sci Numer Simulat 15 (2010) 616-628 

[7] Nick Laskin, Phys. Rev. E 66, 056108 (2002) 

When α = 1, this equation reduces to the ordinary Gauss hypergeometric equation. We draw curves of some exam- ple functions in Fig. 7. 

In this letter, we defined fractional vector calculus in the spherical coordinate framework. We discussed Lapla- cian equation and heat conduct equation in this kind of framework. Some special functions are generalized. 

The geometry induced by fractional operations is non- addictive and nonlocal. This kind of geometry is not Riemannian; since fractional calculus has found applica- tions in many areas of science and engineering, it is the effective geometry of such physics processes. It will be meaningful to investigate further this kind of geometry. 

For a complete fractional vector calculus, fractional vector integral is indispensable. But in this letter we re- strained ourselves from this direction. It will be interest- ing to discuss fractional vector integral in our framework and generalize Green’s, Stokes’ and Gauss’s theorem. 

Special functions show up in different areas of physics and engineering. A fractional generalization of these functions may find applications in similar situations with anomalous tailing behaviors and/or nonlocal properties. 

[8] Marcelo R. Ubriaco, Physics Letters A 373 (2009) 2516- 2519 

[9] Sami I. Muslih, Om P.Agrawal, Int J Theor Phys, DOI: 10.1007/s10773-009-0200-1 

[10] Vasily E. Tarasov, Annals of Physics 323 (2008) 2756- 2778 

[11] Zhenjun Yan, Mathematical Physics Equations , 2nd Edit., in Chinese, University of Science and Technology of China Press, Hefei, 2001. 

[12] Zhuxi Wang, Dunren Guo, Introduction to Special Func- tions , in Chinese, Peking University Press, Beijing, 2000. 

[13] http://en.wikipedia.org/wiki/Hypergeometric function 

[14] http://en.wikipedia.org/wiki/Confluent hypergeometric function 

## VI. SUMMARY 

