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A Review Of Definitions For Fractional Derivatives And Integral

Abstract

This paper presents a review of definitions of fractional order derivatives and integrals that appear in mathematics, physics, and engineering. © 2014 Edmundo Capelas de Oliveira and José António Tenreiro Machado.2014Dzherbashyan, M.M., Nersesian, A.B., About application of some integro-differential operators (1958) Doklady Akademii Nauk (Proceedings of the Russian Academy of Sciences), 121 (2), pp. 210-213Dzherbashyan, M.M., Nersesian, A.B., The criterion of the expansion of the functions to Dirichlet series (1958) Izvestiya Akademii Nauk Armyanskoi SSR: Seriya Fiziko-Matematicheskih Nauk, 11 (5), pp. 85-108Miller, K.S., Ross, B., (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations, , New York, NY, USA John Wiley & Sons MR1219954Oldham, K.B., Spanier, J., (1974) The Fractional Calculus: Theory and Application of Differentiation and Integration to Arbitrary Order, , New York, NY, USA Academic Press MR0361633Machado, J.T., Kiryakova, V., Mainardi, F., Recent history of fractional calculus (2011) Communications in Nonlinear Science and Numerical Simulation, 16 (3), pp. 1140-1153. , 10.1016/j.cnsns.2010.05.027 MR2736622 ZBL1221.26002Riemann, B., (1876) Versuch Einer Allgemeinen Auffassung der Integration und Differentiation. Gesammelte Mathematische Werke und Wissenschaftlicher Nachlass, , Teubner, Leipzig Dover, New York, NY, USA, 1953Grünwald, A.K., Über "begrenzte" derivationen und deren anwendung (1867) Zeitschrift für Mathematik und Physik, 12, pp. 441-480Letnikov, A.V., Theory of differentiation with an arbitrary index (1868) Sbornik: Mathematics, 3, pp. 1-66Weyl, H., Bemerkungen zum begriff des differentialquotienten gebrochener ordung vierteljahresschr (1917) Naturforschende Gesellschaft in Zürich, 62, pp. 296-302Riesz, M., L'intégrale de Riemann-Liouville et le problème de Cauchy (1949) Acta Mathematica, 81 (1), pp. 1-222. , MR0030102 10.1007/BF02395016 ZBL0033.27601Riesz, M., L'intégrale de Riemann-Liouville et le problème de Cauchy pour l'équation des ondes (1939) Bulletin de la Société Mathématique de France, 67, pp. 153-170. , MR1505102Caputo, M., Linear models of dissipation whose q is almost frequency independent-ii (1967) Geophysical Journal of the Royal Astronomical Society, 13 (5), pp. 529-539. , 10.1111/j.1365-246X.1967.tb02303.xFigueiredo Camargo, R., Chiacchio, A.O., Capelas De Oliveira, E., Differentiation to fractional orders and the fractional telegraph equation (2008) Journal of Mathematical Physics, 49 (3). , 033505 10.1063/1.2890375 MR2406802 ZBL1153.81330Caponetto, R., Dongola, G., Fortuna, L., Petras, I., (2010) Fractional Order Systems: Modeling and Control Applications, , Singapore World ScientificDavison, M., Essex, C., Fractional differential equations and initial value problems (1998) The Mathematical Scientist, 23 (2), pp. 108-116. , MR1738325 ZBL0919.34005Jumarie, G., On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion (2005) Applied Mathematics Letters, 18 (7), pp. 817-826. , DOI 10.1016/j.aml.2004.09.012, PII S089396590400401XJumarie, G., An approach to differential geometry of fractional order via modified Riemann-Liouville derivative (2012) Acta Mathematica Sinica, 28 (9), pp. 1741-1768. , 10.1007/s10114-012-0507-3 MR2956177 ZBL1266.26013Jumarie, G., On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling (2013) Central European Journal of Physics, 11 (6), pp. 617-633. , 2-s2.0-84879595941 10.2478/s11534-013-0256-7Kilbas, A.A., Srivastava, H.M., Trujillo, J.J., (2006) Theory and Applications of Fractional Differential Equations, 204. , Amsterdam, The Netherlands Elsevier North-Holland Mathematics Studies MR2218073Monje, C.A., Chen, Y., Vinagre, B.M., Xue, D., Feliu, V., (2010) Fractional-Order Systems and Controls: Fundamentals and Applications, , London, UK Springer 10.1007/978-1-84996-335-0 MR3012798Podlubny, I., (1999) Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution, 198. , San Diego, Calif, USA Academic Press Mathematics in Science and Engineering MR1658022Grigoletto, E.C., De Oliveira, E.C., Fractional versions of the fundamental theorem of calculus (2013) Applied Mathematics, 4, pp. 23-33Hardy, G.H., Fractional versions of the fundamental theorem of calculus (2013) The Journal of the London Mathematical Society i, 20 (1), pp. 48-57Naqvi, Q.A., Abbas, M., Complex and higher order fractional curl operator in electromagnetics (2004) Optics Communications, 241 (4-6), pp. 349-355. , 2-s2.0-7544233166 10.1016/j.optcom.2004.07.028Odibat, Z.M., Shawagfeh, N.T., Generalized Taylor's formula (2007) Applied Mathematics and Computation, 186 (1), pp. 286-293. , DOI 10.1016/j.amc.2006.07.102, PII S0096300306009398Capelas De Oliveira, E., Vaz Jr., J., Tunneling in fractional quantum mechanics (2011) Journal of Physics A: Mathematical and Theoretical, 44 (18). , 185303 10.1088/1751-8113/44/18/185303 MR2788727 ZBL1215.81115Pedro, H.T.C., Kobayashi, M.H., Pereira, J.M.C., Coimbra, C.F.M., Variable order modeling of diffusive-convective effects on the oscillatory flow past a sphere (2008) Journal of Vibration and Control, 14 (9-10), pp. 1659-1672. , 10.1177/1077546307087397 MR2462970 ZBL1229.76099Ross, B., Samko, S.G., Love, E.R., Functions that have no first order derivative might have fractional derivatives of all orders less than one (1994) Real Analysis Exchange, 20 (2), pp. 140-157. , MR1313679Tarasov, V.E., Fractional vector calculus and fractional Maxwell's equations (2008) Annals of Physics, 323 (3), pp. 2756-2778. , 10.1016/j.aop.2008.04.005 MR2463217 ZBL1180.78003Tarasov, V.E., (2011) Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, , Heidelberg, Germany Springer Nonlinear Physical Science 10.1007/978-3-642-14003-7 MR2796453Trujillo, J.J., Rivero, M., Bonilla, B., On a Riemann-Liouville generalized Taylor's formula (1999) Journal of Mathematical Analysis and Applications, 231 (1), pp. 255-265. , 10.1006/jmaa.1998.6224 MR1670860 ZBL0931.26004Valerio, D., Trujillo, J.J., Rivero, M., Machado, J.T., Baleanu, D., Fractional calculus: A survey of useful formulas (2013) The European Physical Journal Special Topics, 222 (8), pp. 1827-1846. , 10.1140/epjst/e2013-01967-yAnastassiou, G.A., (2009) Fractional Differentiation Inequalities, , Dordrecht, The Netherlands Springer 10.1007/978-0-387-98128-4 MR2513750Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J., (2012) Fractional Calculus: Models and Numerical Methods, 3. , Singapore World Scientific Series on Complexity, Nonlinearity and Chaos 10.1142/9789814355216 MR2894576Kiryakova, V., (1994) Generalized Fractional Calculus and Applications, 301. , Harlow, NY, USA Longman Scientific & Technical MR1265940Mainardi, F., (2010) Fractional Calculus and Waves in Linear Viscoelasticity, , London, UK Imperial College Press 10.1142/9781848163300 MR2676137 ZBL1221.26013Mathai, A.M., Haubold, H.J., (2008) Special Functions for Applied Scientists, , New York, NY, USA SpringerValerio, D., Da Costa, J.S., (2012) An Introduction to Fractional Control, , Stevenage, UK IETYang, X.J., (2011) Local Fractional Functional Analysis and Its Applications, , Hong Kong Asian Academic Publisher LimitedYang, X.J., (2012) Advanced Local Fractional Calculus and Its Applications, , New York, NY, USA World ScienceDorrego, G.A., Cerutti, R.A., The k -fractional Hilfer derivative (2013) International Journal of Mathematical Analysis, 7 (11), pp. 543-550. , MR3004317Samko, S.G., Kilbas, A.A., Marichev, O.I., (1993) Fractional Integrals and Derivatives: Theory and Applications, , Amsterdam, The Netherlands Gordon and Breach Science Publishers MR1347689Coimbra, C.F.M., Mechanics with variable-order differential operators (2003) Annalen der Physik, 12 (11-12), pp. 692-703. , 10.1002/andp.200310032 MR2020716 ZBL1103.26301Ramirez, L.E.S., Coimbra, C.F.M., On the selection and meaning of variable order operators for dynamic modeling (2010) International Journal of Differential Equations, 2010, p. 16. , 10.1155/2010/846107 846107 MR2575291 ZBL1207.34011Ramirez, L.E.S., Coimbra, C.F.M., On the variable order dynamics of the nonlinear wake caused by a sedimenting particle (2011) Physica D: Nonlinear Phenomena, 240 (13), pp. 1111-1118. , 10.1016/j.physd.2011.04.001 MR2812362 ZBL1219.76054Osler, T.J., Leibniz rule for fractional derivatives generalized and an application to infinite series (1970) SIAM Journal on Applied Mathematics, 18, pp. 658-674. , MR0260942 10.1137/0118059 ZBL0201.44102Cossar, J., A theorem on Cesàro summability (1941) Journal of the London Mathematical Society i, 16, pp. 56-68. , MR0005154 ZBL0028.39302Bosanquet, L.S., Note on convexity theorems (1943) Journal of the London Mathematical Society i, 18 (4), pp. 146-148. , MR0010599 ZBL0061.12403Lupas, A.A., Some differential subordinations using Ruscheweyh derivative and Salagean operator (2013) Advances in Difference Equations, , 10.1186/1687-1847-2013-150Meerschaert, M.M., Mortensen, J., Wheatcraft, S.W., Fractional vector calculus for fractional advection-dispersion (2006) Physica A: Statistical Mechanics and its Applications, 367, pp. 181-190. , DOI 10.1016/j.physa.2005.11.015, PII S0378437105012070Ruscheweyh, S., New criteria for univalent functions (1975) Proceedings of the American Mathematical Society, 49 (1), pp. 109-115. , MR0367176 10.1090/S0002-9939-1975-0367176-1 ZBL0303.3000

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