Higher-order fractional Green and Gauss formulas. (English) Zbl 1390.26008

Cheng, Jinfa; Dai, Weizhong

J. Math. Anal. Appl. 462, No. 1, 157-171 (2018).

Summary: Green’s formula and Gauss’s formula are two important formulas in vector calculus. In 2008, V. E. Tarasov [Ann. Phys. 323, No. 11, 2756–2778 (2008; Zbl 1180.78003)] developed the fractional Green and Gauss formulas and also suggested two possible extensions of his fractional vector formulas. The first possible extension is to prove his fractional integral theorems for a general form of domains and boundaries, such as elementary regions. The second one is to generalize the formulations of fractional integral theorems for \(\alpha > 1\). The purpose of this article is to follow the above two interesting suggestions and present the higher-order fractional Green and Gauss formulas that are the extensions of fractional integral theorems obtained by Tarasov. In particular, the obtained formulas can be reduced to the classical Green and Gauss formulas when \(\alpha = 1\) and the fractional Green and Gauss formulas in [loc. cit.] when \(0 < \alpha \leq 1\), respectively.

MSC:

26A33	Fractional derivatives and integrals
26B20	Integral formulas of real functions of several variables (Stokes, Gauss, Green, etc.)

Keywords:

Green’s formula; Gauss’s formula; Caputo’s fractional derivative

Citations:

Zbl 1180.78003

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