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

Year 2021, Volume: 6 Issue: 2, 102 - 105, 01.06.2021

Ali Karci

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

• Baron, M.E.,”The Origin of the Infinitesimal Calculus”, New York, 1969.
• 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.
• Das, S.,”Functional Fractional Calculus”, Springer-Verlag Berlin Heidelberg, 2011.
• 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.
• 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.
• Karcı, A., “Kesirli Türev için Yapılan Tanımlamaların Eksiklikleri ve Yeni Yaklaşım”, TOK-2013 Turkish Automatic Control National Meeting and Exhibition, Malatya, Turkey, 2013a.
• Karcı, A.,”A New Approach for Fractional Order Derivative and Its Applications”, Universal Journal of Engineering Sciences, 1:110-117, 2013b.
• Karcı, A., “Properties of Fractional Order Derivatives for Groups of Relations/Functions”, Universal Journal of Engineering Sciences, vol: 3, pp: 39-45, 2015a.
• 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.
• Karcı, A.,“Generalized Fractional Order Derivatives for Products and Quotients”, Science Innovation, vol:3, pp:58-62, 2015c.
• Karcı, A.,“Chain Rule for Fractional Order Derivatives”, Science Innovation, vol:3, 63-67, 2015d.
• 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.
• Karcı, A.,“Fractional order entropy New perspectives”, Optik - International Journal for Light and Electron Optics, Vol:127, no:20, pp:9172-9177, 2016.
• 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.
• Kilbas, A.A., Srivastava, H.M., Trujillo,J.J.”Theory and Applications of Fractional Differential Equations”, Elsevier, 2006.
• L'Hôpital, G. 1696. Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes ("Infinitesimal calculus with applications to curved lines"), Paris,1696.
• L'Hôpital, G. 1715. Analyse des infinement petits, Paris,1715.
• Leibniz, G.F. 1695. Correspondence with l‘Hospital, 1695.
• Li, C., Chen,A., Ye,J.,”Numerical Approaches to Fractional Calculus and Fractional Ordinary Differential Equation”, Journal of Computational Physics, 230:3352-3368,2011.
• Mandelbrot, B.B., van Ness, J.W.,”Fractional Brownian motion, fractional noise and applications”, SIAM Rev. 10: 422,1968.
• 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.
• Newton, I.,”Philosophiæ Naturalis Principia Mathematica”,1687.
• Schiavone, S.E., Lamb, W.,”A Fractional Power Approach to Fractional Calculus”, Journal of Mathematical Analysis and Applications, 149:377-401,1990.