A Truncated $\mathcal{V}$-Fractional Derivative in $\mathbb{R}^n$

Abstract

Using the six parameters truncated Mittag-Leffler function, we introduce a convenient truncated function to define the so-called truncated V-fractional derivative type. In this sense, we propose the derivative of a vector valued function and define the V-fractional Jacobian matrix whose properties allow us to say that: the multivariable truncated V-fractional derivative type, as proposed here, generalizes the truncated V-fractional derivative type and can bee extended to obtain a truncated V-fractional partial derivative type. As applications, we discuss and prove the order change associated with two indices  of two truncated V-fractional partial derivative type and propose the truncated V-fractional Green theorem.

Keywords

• Truncated $\mathcal{V}$-fractional derivative,multivariable truncated $\mathcal{V}$-fractional derivative,truncated $\mathcal{V}$-fractional partial derivative,truncated $\mathcal{V}$-fractional Jacobian matrix

References

1. Atangana, A., Baleanu, D., Alsaedi, A., New properties of conformable derivative, Open Mathematics, 13(1)(2015), 1–10.
2. Baleanu, D., Machado, J. A. T., Luo, A. C., Fractional Dynamics and Control, Springer, New York, 2011.
3. Diethelm, K., Fractional differential equations. Theory and numerical treatment. Scriptum, Institute of Computational Mathematics, Technical University of Braunschweig, 2003.
4. Gözütok, N. Y., Gözütok, U., Multi-variable conformable fractional calculus, Filomat, 32(1)(2018), 45–53.
5. Katugampola, U. N., A new fractional derivative with classical properties, arXiv preprint arXiv:1410.6535.
6. Khalil, R., Al Horani, M., Yousef, A., Sababheh, M., A new definition of fractional derivative, J. Comput. and Appl. Math., 264(2014), 65–70.
7. Kilbas, A. A., Srivastava, H. M., Trujillo, J. J., Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
8. Monje, C. A., Chen, Y., Vinagre, B. M., Xue, D., Feliu-Batlle, V., Fractional-order systems and controls: fundamentals and applications, Springer-Verlag, London, 2010.

9. Vanterler da C. Sousa, J., Capelas de Oliveira, E., M-fractional derivative with classical properties, arXiv:1704.08186 [math.CA], (2017).
10. Vanterler da C. Sousa, J., Capelas de Oliveira, E., Mittag-Leffer functions and the truncated V-fractional derivative, Mediterr. J. Math., 4(6)(2017) 244.
11. Vanterler da C. Sousa, J., Capelas de Oliveira, E., A new truncated M-fractional derivative unifying some fractional derivatives with classical properties, Inter. J. Anal. and Appl., 16(1)(2018), 83–96.
12. Stewart, J., Calculus, Cengage Learning, Boston, 2015.