A generalization of truncated M-fractional derivative and applications to fractional differential equations

[1]Andrews L.C., (1985), Special Functions for Engineers and Applied Mathematicians, Macmillan Publishing Company, New York.AndrewsL.C.1985Special Functions for Engineers and Applied MathematiciansMacmillan Publishing CompanyNew YorkSearch in Google Scholar

[2]P. Agarwal, (2012), Further results on fractional calculus of Saigo operators, Applications and Applied Mathematics, 7 (2), pp. 585–594.AgarwalP.2012Further results on fractional calculus of Saigo operatorsApplications and Applied Mathematics72585594Search in Google Scholar

[3]P. Agarwal, J. Choi, R. B. Paris, (2015), Extended Riemann-Liouville fractional derivative operator and its applications. Journal of Nonlinear Science and Applications, 8 (5), pp.451–466.AgarwalP.ChoiJ.ParisR. B.2015Extended Riemann-Liouville fractional derivative operator and its applicationsJournal of Nonlinear Science and Applications8545146610.22436/jnsa.008.05.01Search in Google Scholar

[4]A. Atangana, D. Baleanu, (2016), New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, Thermal Science, 20 (2), pp. 763–769.AtanganaA.BaleanuD.2016New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer modelThermal Science20276376910.2298/TSCI160111018ASearch in Google Scholar

[5]A. Atangana, J. F. Gómez-Aguilar, (2017), Hyperchaotic behavior obtained via a nonlocal operator with exponential decay and Mittag-Leffler laws, Chaos, Solitons and Fractals, 102, pp. 285–294.AtanganaA.Gómez-AguilarJ. F.2017Hyperchaotic behavior obtained via a nonlocal operator with exponential decay and Mittag-Leffler lawsChaos, Solitons and Fractals10228529410.1016/j.chaos.2017.03.022Search in Google Scholar

[6]M. Caputo, M. Fabrizio, (2015), A new definition of fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2), pp. 73–85.CaputoM.FabrizioM.2015A new definition of fractional derivative without singular kernelProgress in Fractional Differentiation and Applications127385Search in Google Scholar

[7]Y. Çenesiz, (2015), A. Kurt, The solutions of time and space conformable fractional heat equations with conformable Fourier Transform, Acta Univ. Sapientiae, Mathematica, 7, pp. 130–140.ÇenesizY.KurtA.2015The solutions of time and space conformable fractional heat equations with conformable Fourier TransformActa Univ. Sapientiae, Mathematica713014010.1515/ausm-2015-0009Search in Google Scholar

[8]V.B.L. Chaurasia and D. Kumar, (2010), On the solutions of generalized fractional kinetic equations, Adv. Studies Theor. Phys, 4 (16), pp. 773–780.ChaurasiaV.B.L.KumarD.2010On the solutions of generalized fractional kinetic equationsAdv. Studies Theor. Phys416773780Search in Google Scholar

[9]A. Chouhan and A. M. Khan, (2015), Unified integrals associated with H-functions and M-series, Journal of Fractional Calculus and Applications, 6 (2), pp. 11–16.ChouhanA.KhanA. M.2015Unified integrals associated with H-functions and M-seriesJournal of Fractional Calculus and Applications621116Search in Google Scholar

[10]A. Chouhan and S. Sarswat, (2012), Certain properties of fractional calculus operators associated with M-series, Scientia: Series A: Mathematical Sciences, 22, pp. 25–30.ChouhanA.SarswatS.2012Certain properties of fractional calculus operators associated with M-seriesScientia: Series A: Mathematical Sciences222530Search in Google Scholar

[11]A. Çetinkaya, İ. O. Kıymaz, P. Agarwal, R. Agarwal, (2018), A comparative study on generating function relations for generalized hypergeometric functions via generalized fractional operators, Advances in Difference Equations, 2018:156.ÇetinkayaA.Kıymazİ. O.AgarwalP.AgarwalR.2018A comparative study on generating function relations for generalized hypergeometric functions via generalized fractional operatorsAdvances in Difference Equations201815610.1186/s13662-018-1612-0Search in Google Scholar

[12]R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, (2014), Mittag-Leffler Functions, Related Topic and Applications, Springer, Berlin.GorenfloR.KilbasA. A.MainardiF.RogosinS. V.2014Mittag-Leffler Functions, Related Topic and ApplicationsSpringerBerlin10.1007/978-3-662-43930-2Search in Google Scholar

[13]J. F. Gómez-Aguilar, A. Atangana, (2017), New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, The European Physical Journal Plus, 132:13, pp. 1–21.Gómez-AguilarJ. F.AtanganaA.2017New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applicationsThe European Physical Journal Plus1321312110.1140/epjp/i2017-11293-3Search in Google Scholar

[14]A. Faraj, T. Salim, S. Sadek and J. Ismail, (2014), A Generalization of M-Series and Integral Operator Associated with Fractional Calculus, Asian Journal of Fuzzy and Applied Mathematics, 2 (5), pp. 142–155.FarajA.SalimT.SadekS.IsmailJ.2014A Generalization of M-Series and Integral Operator Associated with Fractional CalculusAsian Journal of Fuzzy and Applied Mathematics25142155Search in Google Scholar

[15]O. S. Iyiola, E. R. Nwaeze, (2016), Some new results on the new conformable fractional calculus with application using D’Alambert approach, Progr. Fract. Differ. Appl., 2 (2), pp. 115–122.IyiolaO. S.NwaezeE. R.2016Some new results on the new conformable fractional calculus with application using D’Alambert approachProgr. Fract. Differ. Appl.2211512210.18576/pfda/020204Search in Google Scholar

[16]U.R. Katugampola, (2014), A new fractional derivative with classical properties, arXiv:1410.6535v2.KatugampolaU.R.2014A new fractional derivative with classical propertiesarXiv:1410.6535v2.Search in Google Scholar

[17]R. Khalil, M.A. Horani, A. Yousef and M. Sababheh, (2014), A new definition of fractional derivative, Jour. of Comp. and App. Math., 264, pp. 65–70.KhalilR.HoraniM.A.YousefA.SababhehM.2014A new definition of fractional derivativeJour. of Comp. and App. Math.264657010.1016/j.cam.2014.01.002Search in Google Scholar

[18]İ. O. Kıymaz, A. Çetinkaya, P. Agarwal, (2016), An extension of Caputo fractional derivative operator and its applications, Journal of Nonlinear Science and Applications, 9 (6), pp. 3611–3621.Kıymazİ. O.ÇetinkayaA.AgarwalP.2016An extension of Caputo fractional derivative operator and its applicationsJournal of Nonlinear Science and Applications963611362110.22436/jnsa.009.06.14Search in Google Scholar

[19]İ. O. Kıymaz, A. Çetinkaya, P. Agarwal, S. Jain, (2017), On a New Extension of Caputo Fractional Derivative Operator, Advances in Real and Complex Analysis with Applications, Birkhäuser Basel, pp. 261–275.Kıymazİ. O.ÇetinkayaA.AgarwalP.JainS.2017On a New Extension of Caputo Fractional Derivative OperatorAdvances in Real and Complex Analysis with ApplicationsBirkhäuserBasel26127510.1007/978-981-10-4337-6_11Search in Google Scholar

[20]A.A. Kilbas, H.M. Srivastava, J.J. Trujilli, (2006), Theory and Applications of Fractional Differential Equations, El-sever, Amsterdam, The Netherlands.KilbasA.A.SrivastavaH.M.TrujilliJ.J.2006Theory and Applications of Fractional Differential EquationsEl-sever, AmsterdamThe NetherlandsSearch in Google Scholar

[21]D. Kumar and R.K. Saxena, (2015), Generalized fractional calculus of the M-Series involving F3 hypergeometric function, Sohag J. Math, 2 (1), pp. 17–22.KumarD.SaxenaR.K.2015Generalized fractional calculus of the M-Series involving F3 hypergeometric functionSohag J. Math211722Search in Google Scholar

[22]J. Losada, J. J. Nieto, (2015), Properties of a new fractional derivative without singular kernel, Progress in Fractional Differentiation and Applications, 1 (2), pp. 87–92.LosadaJ.NietoJ. J.2015Properties of a new fractional derivative without singular kernelProgress in Fractional Differentiation and Applications128792Search in Google Scholar

[23]G.M. Mittag-Leffler, (1903), Sur la nouvelle fonction E(x), C. R. Acad. Sci. Paris, (Ser. II) 137, pp. 554–558.Mittag-LefflerG.M.1903Sur la nouvelle fonction E(x)C. R. Acad. Sci. Paris, (Ser. II)137554558Search in Google Scholar

[24]T.R. Prabhakar, (1971), A singular integral equation with a generalized Mittag-Leffler function in the kernel, Yokohama Math. J., 19, pp. 7–15.PrabhakarT.R.1971A singular integral equation with a generalized Mittag-Leffler function in the kernelYokohama Math. J.19715Search in Google Scholar

[25]M. Sharma, (2008), Fractional integration and fractional differentiation of the M-series, Fract. Calc. Appl. Anal., 11:2, pp. 187–191.SharmaM.2008Fractional integration and fractional differentiation of the M-seriesFract. Calc. Appl. Anal.112187191Search in Google Scholar

[26]M. Sharma and R. Jain, (2009), A note on a generalized M-series as a special function of fractional calculus, Fract. Calc. Appl. Anal., 12:4, pp. 449–452.SharmaM.JainR.2009A note on a generalized M-series as a special function of fractional calculusFract. Calc. Appl. Anal.124449452Search in Google Scholar

[27]J.V.C. Sousa and E.C. de Oliveira, (2017), On the local M-derivative, arXiv:1704.08186v3.SousaJ.V.C.de OliveiraE.C.2017On the local M-derivativearXiv:1704.08186v3.Search in Google Scholar

[28]J.V.C. Sousa and E.C. de Oliveira, (2017), Mittag-Leffler Functions and the Truncated 𝒱-fractional Derivative, Mediterr. J. Math., 14:244.SousaJ.V.C.de OliveiraE.C.2017Mittag-Leffler Functions and the Truncated 𝒱-fractional DerivativeMediterr. J. Math.1424410.1007/s00009-017-1046-zSearch in Google Scholar

[29]J.V.C. Sousa and E.C. de Oliviera, (2018), A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Inter. of Jour. Analy. and Appl., 16 (1), pp. 83–96.SousaJ.V.C.de OlivieraE.C.2018A new truncated M-fractional derivative type unifying some fractional derivative types with classical propertiesInter. of Jour. Analy. and Appl.1618396Search in Google Scholar

[30]J.V.C. Sousa and E.C. de Oliviera, (2018), Two new fractional derivatives of variable order with non-singular kernel and fractional differential equation, Computational and Applied Mathematics, 37 (4), pp. 5375–5394.SousaJ.V.C.de OlivieraE.C.2018Two new fractional derivatives of variable order with non-singular kernel and fractional differential equationComputational and Applied Mathematics3745375539410.1007/s40314-018-0639-xSearch in Google Scholar

[31]J.V.C. Sousa and E.C. de Oliviera, (2018), A Truncated 𝒱-Fractional Derivative in Rⁿ, Turk. J. Math. Comput. Sci., 8, pp. 49–64.SousaJ.V.C.de OlivieraE.C.2018A Truncated 𝒱-Fractional Derivative in RⁿTurk. J. Math. Comput. Sci.84964Search in Google Scholar

[32]A. Wiman, (1905), Uber den Fundamental satz in der Theorie de Funktionen E(x), Acta Math., 29, pp. 191–201.WimanA.1905Uber den Fundamental satz in der Theorie de Funktionen E(x)Acta Math.2919120110.1007/BF02403202Search in Google Scholar

[33]X. J. Yang, H. M. Srivastava, J. T. Machado, (2016), A new fractional derivative without singular kernel: Application to the modeling of the steady heat flow, Thermal Science, 20 (2), pp. 753–756.YangX. J.SrivastavaH. M.MachadoJ. T.2016A new fractional derivative without singular kernel: Application to the modeling of the steady heat flowThermal Science20275375610.2298/TSCI151224222YSearch in Google Scholar