Conformable Derivatives and Integrals for the Functions of Two Variables

Year 2021, Volume: 9 Issue: 1, 49 - 59, 28.04.2021

Muhammet Bozkurt Abdullah Akkurt , Hüseyin Yildirim

https://izlik.org/JA58LZ59EA

Abstract

In this paper, we introduce conformable derivatives and integrals for the functions of two variables. This class of new fractional operators includes many definitions in the literature, such as Riemann-Liouville Fractional Derivatives and Integrals [6,7], Conformable Calculus [8,9], etc. In addition, some basic definitions and theorems have been obtained for these operators.

Keywords

conformable integrals , fractional derivatives , fractional integrals

References

• [1] A. Akkurt, M. E. Yıldırım, and H. Yıldırım, A new Generalized fractional derivative and integral, Konuralp Journal of Mathematics, Volume 5 No. 2 pp. 248–259 (2017).
• [2] M. Z. Sarıkaya, A. Akkurt, H. Budak, M. E. Yıldırım, and H. Yıldırım, Hermite-Hadamard’s inequalities for conformable fractional integrals. An Interna-tional Journal of Optimization and Control: Theories & Applications (IJOCTA), 9(1), 49-59 (2019). doi:http://dx.doi.org/10.11121/ijocta.01.2019.00559.
• [3] R. Almeida, M. Guzowska, and T. Odzijewicz, A remark on local fractional calculus and ordinary derivatives, Open Mathematics, 14(1), 1122-1124 (2016). doi: https://doi.org/10.1515/math-2016-0104
• [4] D. R. Anderson, Taylor’s Formula and Integral Inequalities for Conformable Fractional Derivatives. In: Pardalos P., Rassias T. (eds) Contributions in Mathematics and Engineering. Springer, 2016, Cham
• [5] U. N. Katugampola, A new fractional derivative with classical properties, arXiv:1410.6535v1 [math.CA] 2014
• [6] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Eqnarrays, in: Math. Studies., North-Holland, New York, 2006.
• [7] S. G. Samko, A. A. Kilbas amd O. I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
• [8] T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics 279 (2015) 57-66.
• [9] R. Khalil, M. Al horani, A. Yousef, M. Sababheh, A new de nition of fractional derivative, Journal of Computational Apllied Mathematics, 264 (2014), 65-70.
• [10] F. Jarad, E. Ugurlu, T. Abdeljawad and D. Baleanu, On a new class of fractional operators, Advances in Difference Eqnarrays, 2017, 2017:247, https://doi.org/10.1186/s13662-017-1306-z
• [11] D. P. Mourya, Fractional integrals of the functions of two variables. Proc. Indian Acad. Sci. 72, 173–184 (1970). https://doi.org/10.1007/BF03049707.
• [12] M. Z. Sarikaya , On the Hermite-Hadamard-type inequalities for co-ordinated convex function via fractional integrals, Integral Transforms and Special Functions, 25(2), 2014, pp:134-147.
• [13] M.Z. Sarikaya, H. Budak and F. Usta, On generalized the conformable fractional calculus, TWMS J. App. Eng. Math. V.9, N.4, 2019, pp. 792-799.
• [14] F. Jarad, E. Ugurlu˘ and T. Abdeljawad, On a new class of fractional operators. Adv Differ Equ 2017, 247 (2017). https://doi.org/10.1186/s13662-017-1306-z.