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# Fractional Vector Calculus and Fractional Special Function 

Institute of Theoretical Physics, Lanzhou University, Lanzhou, 730000, China 

Ming-Fan Li, ∗ Ji-Rong Ren, † and Tao Zhu ‡ 

Fractional vector calculus is discussed in the spherical coordinate framework. A variation of the Legendre equation and fractional Bessel equation are solved by series expansion and numerically. Finally, we generalize the hypergeometric functions. 

Fractional calculus is the calculus of differentiation and integration of non-integer orders [1, 2]. During last three decades, fractional calculus has gained much attention due to its demonstrated applications in various fields of science and engineering, such as anomalous diffusion [3], fractional dynamical systems [4–6], fractional quantum mechanics [7], fractional statistics and thermodynamics [8], to name a few. 

Fractional vector calculus (FVC) is important in de- scribing processes in fractal media, fractional electrody- namics and fractional hydrodynamics [9, 10]. But an effective FVC is still lacking. There are many problems in defining an effective FVC. One is that fractional in- tegral and fractional derivative are defined “half” , that is to say, they are defined only on the right or the left side of an initial point. And if we make fractional series expansion, functions are all expanded on the right or the left neighborhood of the initial point. We cannot across this cutting point. So if we want to describe the behavior near the initial point, we need define both the right and the left functions. The situation will become even more complicated when we deal with high dimensions. In this letter we will define FVC in spherical coordinate frame- work. Since in the spherical coordinate framework the radius is naturally bounded to the positive half, we need just one fractional derivative. 

Using this frame, we will discuss the Laplacian equa- tion [11] with fractional radius derivative in 3-d space and the heat conduct equation [11] in 2-d space with cylin- drical symmetry. As a result, the corresponding special functions will be generalized. Finally, we will generalize hypergeometric functions [12–14]. 

To start, let’s briefly recall some basic facts in frac- tional calculus [1, 2]. There are many ways to define fractional integral and fractional derivative. In this let- 

PACS numbers: 02.30.Gp, 45.10.Hj, 02.40.Ky 

## I. INTRODUCTION 

## II. FRACTIONAL CALCULUS 

ter we will use Riemann-Liouville fractional integral and Caputo fractional derivative. 

Let f ( x ) be a function defined on the right side of a . Let α be a positive real. The Riemann-Liouville frac- tional integral is defined by 

The integral has a memory kernel. 

Let A ≡ [ α ] + 1. The Caputo fractional derivative is defined by 

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They have the following properties: 

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These properties are just fractional generalizations of 

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For a ‘good’ function, one can define its fractional Tay- lor series 

Explicitly, 

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