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

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.)

Citations:

Zbl 1180.78003
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References:

[1] Aifantis, E. C., Gradient nanomechanics: applications to deformation, fracture, and diffusion in nanopolycrystals, Metall. Mater. Trans. A, 42, 2985-2998, (2011)
[2] Cheng, J. F., Theory of fractional difference equations, (2011), Xiamen University Press Xiamen, China, (in Chinese)
[3] Diethelm, K.; Ford, N. J., Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 229-248, (2002) · Zbl 1014.34003
[4] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and applications of fractional differential equations, North-Holland Mathematical Studies, vol. 204, (2006), Elsevier Amsterdam · Zbl 1092.45003
[5] Kiryakova, V., Generalized fractional calculus and applications, (1979), Pitman Press San Francisco
[6] Miller, K. S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), John Wiley and Sons New York · Zbl 0789.26002
[7] Odibat, Z. M.; Shawagfeh, N. T., Generalized Taylor’s formula, Appl. Math. Comput., 186, 286-293, (2007) · Zbl 1122.26006
[8] Oldham, K. B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004
[9] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[10] Ross, B., A brief history and exposition of the fundamental theory of fractional calculus, Lecture Notes in Math., 457, 1-36, (1975)
[11] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives: theory and applications, (1993), Gordon and Breach Science Philadelphia · Zbl 0818.26003
[12] Tarasov, V. E., Fractional vector calculus and fractional Maxwell’s equations, Ann. Physics, 323, 2756-2778, (2008) · Zbl 1180.78003
[13] Tarasov, V. E., Fractional dynamics: applications of fractional calculus to dynamics of particles, fields and media, (2011), Springer New York
[14] Tarasov, V. E., Toward lattice fractional vector calculus, J. Phys. A: Math. Theor., 47, (2014) · Zbl 1308.26011
[15] Tarasov, V. E.; Aifantis, E. C., Non-standard extensions of gradient elasticity: fractional non-locality, memory and fractality, Commun. Nonlinear Sci. Numer. Simul., 22, 197-227, (2015) · Zbl 1398.35280
[16] Trujillo, J. J.; Rivero, M.; Bonilla, B., On a Riemann-Liouville generalized Taylor’s formula, J. Math. Anal., 231, 255-265, (1999) · Zbl 0931.26004
[17] Zaslavsky, G. M., Hamiltonian chaos and fractional dynamics, (2005), Oxford, London · Zbl 1080.37082
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