Fractional Order Integration: A New Perspective based on Karcı’s Fractional Order Derivative

Abstract

There are many methods/definitions for fractional order derivatives, and naturally, there are many definitions for fractional order integrals based on these definitions. In this paper, a new definition for fractional order integral was emphasized based on the definition for fractional order derivative made by Karcı.

Keywords

• Fractional Calculus
• Integration
• Fractional Order Derivatives

References

1. Baron, M.E.,”The Origin of the Infinitesimal Calculus”, New York, 1969.
2. Bataineh, A.S., Alomari, A.K., Noorani, M.S.M., Hashim, I., Nazar, R.,”Series Solutions of Systems of Nonlinear Fractional Differential Equations”, Acta Applied Mathematics, 105:189-198,2009.
3. Das, S.,”Functional Fractional Calculus”, Springer-Verlag Berlin Heidelberg, 2011.
4. Diethelm, K., Ford, N.J., Freed, A.D., Luchko, Y.,”Algorithms fort he Fractional Calculus: A Selection of Numerical Methdos, Computer Methods in Applied Mechanics and Engineering”, 194:743-773,2005.
5. Goldenbaum, U., Jesseph, D.,”Infinitesimal Differences: Controversies between Leibniz and his Contemporaries”, New York, 2008. He, J.-H., Elagan,S.K., Li,Z.B., “Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus”, Physics Letters A, 376:257-259, 2012.
6. Karcı, A., “Kesirli Türev için Yapılan Tanımlamaların Eksiklikleri ve Yeni Yaklaşım”, TOK-2013 Turkish Automatic Control National Meeting and Exhibition, Malatya, Turkey, 2013a.
7. Karcı, A.,”A New Approach for Fractional Order Derivative and Its Applications”, Universal Journal of Engineering Sciences, 1:110-117, 2013b.
8. Karcı, A., “Properties of Fractional Order Derivatives for Groups of Relations/Functions”, Universal Journal of Engineering Sciences, vol: 3, pp: 39-45, 2015a.

9. Karcı, A.,”The Linear, Nonlinear and Partial Differential Equations are not Fractional Order Differential Equations”, Universal Journal of Engineering Sciences, vol: 3, pp: 46-51, 2015b.
10. Karcı, A.,“Generalized Fractional Order Derivatives for Products and Quotients”, Science Innovation, vol:3, pp:58-62, 2015c.
11. Karcı, A.,“Chain Rule for Fractional Order Derivatives”, Science Innovation, vol:3, 63-67, 2015d.
12. Karcı,A., “The Properties of New Approach of Fractional Order Derivative”, Journal of the Faculty of Engineering and Architecture of Gazi University, Vol.30, pp: 487-501, 2015e.
13. Karcı, A.,“Fractional order entropy New perspectives”, Optik - International Journal for Light and Electron Optics, Vol:127, no:20, pp:9172-9177, 2016.
14. Karcı, A.,“Malatya Functions: Symmetric Functions Obtained by Applying Fractional Order Derivative to Karcı Entropy”, Anatolian Science-Journal of Computer Sciences, Vol:2, Issue: 2, pp:1-8, 2017.
15. Kilbas, A.A., Srivastava, H.M., Trujillo,J.J.”Theory and Applications of Fractional Differential Equations”, Elsevier, 2006.
16. L'Hôpital, G. 1696. Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes ("Infinitesimal calculus with applications to curved lines"), Paris,1696.
17. L'Hôpital, G. 1715. Analyse des infinement petits, Paris,1715.
18. Leibniz, G.F. 1695. Correspondence with l‘Hospital, 1695.
19. Li, C., Chen,A., Ye,J.,”Numerical Approaches to Fractional Calculus and Fractional Ordinary Differential Equation”, Journal of Computational Physics, 230:3352-3368,2011.
20. Mandelbrot, B.B., van Ness, J.W.,”Fractional Brownian motion, fractional noise and applications”, SIAM Rev. 10: 422,1968.
21. Mirevski, S.P., Boyadjiev, L., Scherer, R.,”On the Riemann-Liouville Fractional Calculus, g-Jacobi Functions and F.Gauss Functions”, Applied Mathematics and Computation, 187:315-325, 2007.
22. Newton, I.,”Philosophiæ Naturalis Principia Mathematica”,1687.
23. Schiavone, S.E., Lamb, W.,”A Fractional Power Approach to Fractional Calculus”, Journal of Mathematical Analysis and Applications, 149:377-401,1990.