# zbMATH — the first resource for mathematics

A note on Caputo’s derivative operator interpretation in economy. (English) Zbl 1437.91125
Summary: We propound the economic idea in terms of fractional derivatives, which involves the modified Caputo’s fractional derivative operator. The suggested economic interpretation is based on a generalization of average count and marginal value of economic indicators. We use the concepts of $$T$$-indicators which analyses the economic performance with the presence of memory. The reaction of economic agents due to recurrence identical alteration is minimized by using the modified Caputo’s derivative operator of order $$\lambda$$ instead of integer order derivative $$n$$. The two sides of Caputo’s derivative are expressed by a brief time-line. The degree of attenuation is further depressed by involving the modified Caputo’s operator.

##### MSC:
 91B02 Fundamental topics (basic mathematics, methodology; applicable to economics in general) 26A33 Fractional derivatives and integrals
Full Text:
##### References:
 [1] Sen, M., Introduction to Fractional-Order Operators and Their Engineering Applications, (2014) [2] Machado, J. A. T.; Galhano, A. M. S. F.; Alexandra, M. S. F.; Trujillo, J. J., On development of fractional calculus during the last fifty years, Scientometrics, 98, 1, 577-582, (2014) [3] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional Integrals and Derivatives, Theory and Applications, (1993), Pennsylvania, Pa, USA: Gordon and Breach, Pennsylvania, Pa, USA · Zbl 0818.26003 [4] Kiryakova, V. S., Generalized Fractional Calculus and Applications, (1993), CRC press [5] Mendes, R. V., Introduction to Fractional Calculus (based on lectures by R. Gorenflo, F. Mainardi and I. Podlubny), (2008) [6] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, (2006) · Zbl 1092.45003 [7] Nonnenmacher, T. F.; Metzler, R., Applications of Fractional Calculus Ideas to Biology, (1998), World Scientific [8] Ibrahim, R. W.; Darus, M., Infective disease processes based on fractional differential equation, Proceedings of the 3rd International Conference on Mathematical Sciences, ICMS 2013 [9] Scalas, E.; Gorenflo, R.; Mainardi, F., Fractional calculus and continuous-time finance, Physica A: Statistical Mechanics and its Applications, 284, 1–4, 376-384, (2000) [10] Laskin, N., Fractional market dynamics, Physica A: Statistical Mechanics and its Applications, 287, 3-4, 482-492, (2000) [11] Mainardi, F.; Raberto, M.; Gorenflo, R.; Scalas, E., Fractional calculus and continuous-time finance. II: The waiting-time distribution, Physica A: Statistical Mechanics and its Applications, 287, 3-4, 468-481, (2000) [12] Tarasova, V. V.; Tarasov, V. E., Marginal utility for economic processes with memory, Almanah Sovremennoj Nauki i Obrazovaniya [Almanac of Modern Science and Education], 7, (2016) · Zbl 1438.91059 [13] Ibrahim, R. W.; Darus, M., Differential operator generalized by fractional derivatives, Miskolc Mathematical Notes, 12, 2, 167-184, (2011) · Zbl 1265.30062 [14] Darus, M.; Ibrahim, R. W., On classes of analytic functions containing generalization of integral operator, Journal of the Indonesian Mathematical Society, 17, 1, 29-38, (2011) · Zbl 1234.30009 [15] Salah, J.; Darus, M., A note on generalized Mittag-Leffler function and applications, Far East Journal of Mathematical Sciences (FJMS), 48, 1, 33-46, (2011) · Zbl 1211.30029 [16] Salah, J., Fekete-szegö problems involving certain integral operator, International Journal of Mathematics Trends and Technology, 7, 1, 54-60, (2014) [17] Tarasov, V. E., Interpretation of fractional derivatives as reconstruction from sequence of integer derivatives, Fundamenta Informaticae, 151, 1-4, 431-442, (2017) · Zbl 1376.26009 [18] Tarasova, V. V.; Tarasov, V. E., Economic Interpretation of Fractional Derivatives, Progress in Fractional Differentiation and Applications, 3, 1, 1-7, (2017) [20] Hairer, E.; Wanner, G., L’analyse au fil de l’histoire, 10, (2001), Springer Science & Business Media [21] Glaschke, P., Tautochrone and brachistochrone shape solutions for rocking rigid bodies, (2016) [24] Tarasova, V. V.; Tarasov, V. E., On applicability of point price elasticity of demand to exchange trading on us dollar, Scientific Perspective, 6, 6-11, (2016) [25] Tarasov, V. E.; Tarasova, V. V., Long and short memory in economics: fractional-order difference and differentiation, IRA-International Journal of Management and Social Sciences, 5, 2, 327-334, (2016) [26] Li, C.; Qian, D.; Chen, Y. Q., On Riemann-Liouville and Caputo derivatives, Discrete Dynamics in Nature and Society, 2011, (2011) · Zbl 1213.26008 [27] Luchko, Y.; Trujillo, J. J., Caputo-type modification of the Erdélyi-Kober fractional derivative, Fractional Calculus and Applied Analysis, 10, 3, 249-267, (2007) · Zbl 1152.26304 [28] Kiryakova, V.; Luchko, Y., Riemann-Liouville and caputo type multiple Erdélyi-Kober operators, Open Physics, 11, 10, 1314-1336, (2013) [29] Oldham, K.; Spanier, J., The Fractional Calculus Theory and Applications of Differentiation and Integration to Arbitrary Order, 111, (1974), Elsevier · Zbl 0292.26011 [30] Salah, J., A note on the modified caputo’s fractional calculus derivative operator, International Journal of Pure and Applied Mathematics, 109, 3, 665-67, (2016) · Zbl 1355.33013 [31] Salah, J.; Darus, M., A subclass of uniformly convex functions associated with a fractional calculus operator involving Caputo’s fractional differentiation, Acta Universitatis Apulensis, 24, 295-306, (2010) · Zbl 1224.30078 [32] Caputo, M., Linear models of dissipation whose Q is almost frequency independent-II, The Geophysical Journal of the Royal Astronomical Society, 13, 5, 529-539, (1967) [34] Kahana, M. J.; Adler, M., Note on the Power Law of Forgetting, (2017) [35] Tarasova, V. V.; E Tarasov, V., Fractional Dynamics of Natural Growth And Memory Effect in Economics, European Research, 12, 23, 30-37, (2016)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.