A NEW GENERALIZED FRACTIONAL DERIVATIVE AND INTEGRAL

Year 2017, Volume: 5 Issue: 2, 248 - 259, 15.10.2017

Abdullah Akkurt , Merve Esra Yıldırım Hüseyin Yıldırım

Abstract

In this article, we introduce a new general definition of fractional derivative and fractional integral, which depends on an unknown kernel. By using these definitions, we obtain the basic properties of fractional integral and fractional derivative such as Product Rule, Quotient Rule, Chain Rule, Roll's Theorem and Mean Value Theorem. We give some examples.

Keywords

Fractional Calculus , Fractional derivative , Conformable Fractional Integral

References

• [1] A. Akkurt, M.E. Yıldırım and H. Yıldırım , On Some Integral Inequalities for Conformable Fractional Integrals, Asian Journal of Mathematics and Computer Research, 15(3): 205-212, 2017.
• [2] R. Almeida, M. Guzowska and T. Odzijewicz, A remark on local fractional calculus and ordinary derivatives, arXiv preprint arXiv:1612.00214.
• [3] A. Kilbas, H. Srivastava, J. Trujillo, Theory and Applications of Fractional Differential Equations, in: Math. Studies., North-Holland, New York, 2006.
• [4] U. Katumgapola, A new fractional derivative with classical properties, preprint.
• [5] U.N. Katugampola, New Approach to a generalized fractional integral, Appl. Math. Comput. 218(3), (2011), 860-865.
• [6] T. Abdeljawad, On conformable fractional calculus, Journal of Computational and Applied Mathematics 279 (2015) 57-66.
• [7] 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.
• [8] O.S. Iyiola and E.R. Nwaeze, Some new results on the new conformable fractional calculus with application using D'Alambert approach, Progr. Fract. Differ. Appl., 2(2), 115-122, 2016.
• [9] M. Abu Hammad, R. Khalil, Conformable fractional heat differential equations, International Journal of Differential Equations and Applications 13(3), 2014, 177-183.
• [10] M. Abu Hammad, R. Khalil, Abel's formula and wronskian for conformable fractional differential equations, International Journal of Differential Equations and Applications 13(3), 2014, 177-183.
• [11] Samko, S.G.; Kilbas, A.A.; Marichev, O.I.: Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993.
• [12] J. Vanterler da C. Sousa, E. Capelas de Oliveira, A new truncated M-fractional derivative unifying some fractional derivatives with classical properties, arXiv:1704.08187.
• [13] J. Vanterler da C. Sousa, E. Capelas de Oliveira , M -fractional derivative with classical properties, arXiv:1704.08186.
• [14] M. Z. Sarikaya, H. Budak and F.Usta, On generalized the conformable fractional calculus, RGMIA Research Report Collection, 19 (2016), Article 121, 7 pp.