Operational calculus for the Riemann–Liouville fractional derivative with respect to a function and its applications

[1] A. Ahmadova, I.T. Huseynov, A. Fernandez, N.I. Mahmudov, Trivariate Mittag-Leffler functions used to solve multi-order systems of fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 97C (2021), # 105735.10.1016/j.cnsns.2021.105735Suche in Google Scholar

[2] R. Almeida, M. Jleli, B. Samet, A numerical study of fractional relaxation-oscillation equations involving ψ-Caputo fractional derivative. Revista de la Real Acad. de Cienc. Exactas, Fís. y Naturales: Ser. A. Matemáticas 113, No 3 (2019), 1873–1891.10.1007/s13398-018-0590-0Suche in Google Scholar

[3] D. Baleanu, A. Fernandez, A generalisation of the Malgrange–Ehrenpreis theorem to find fundamental solutions to fractional PDEs. Electron. J. Qual. Theory Differ. Equ. 2017 (2017), No 15, 1–12.10.14232/ejqtde.2017.1.15Suche in Google Scholar

[4] D. Baleanu, A. Fernandez, On fractional operators and their classifications. Mathematics 7, No 9, (2019), # 830.10.3390/math7090830Suche in Google Scholar

[5] M.A. Al-Bassam, Y. F. Luchko, On generalized fractional calculus and its application to the solution of integro-differential equations. J. Fract. Calc. 7 (1995), 69–88.Suche in Google Scholar

[6] I.H. Dimovski, Operational calculus for a class of differential operators. C.R. Acad. Bulgare Sci. 19, No 12 (1966), 1111–1114.Suche in Google Scholar

[7] I.H. Dimovski, On an operational calculus for a differential operator. C.R. Acad. Bulgare Sci. 21, No 6 (1968), 513–516.Suche in Google Scholar

[8] I.H. Dimovski, Convolutional Calculus. Bulgarian Academy of Sciences, Sofia (1982); 2nd Ed., Kluwer Academic Publisher, Dordrecht (1990).10.1007/978-94-009-0527-6Suche in Google Scholar

[9] I.H. Dimovski, The commutant of the Riemann-Liouville operator of fractional integration. Fract. Calc. Appl. Anal. 12, No 4 (2009), 443–448; at http://www.math.bas.bg/complan/fcaa/volume12/fcaa124/Dimovski_fcaa_12_4.pdf.Suche in Google Scholar

[10] S. Dugowson, Les différentielles métaphysiques: histoire et philosophie de la généralisation de l’ordre de dérivation. PhD Thesis, Université Paris Nord (1994).Suche in Google Scholar

[11] H.M. Fahad, M. ur Rehman, A. Fernandez, On Laplace transforms with respect to functions and their applications to fractional differential equations. Preprint: arXiv:1907.04541 (2020).10.1002/mma.7772Suche in Google Scholar

[12] A. Fernandez, D. Baleanu, A.S. Fokas, Solving PDEs of fractional order using the unified transform method. Appl. Math. Comput. 339 (2018), 738–749.10.1016/j.amc.2018.07.061Suche in Google Scholar

[13] A. Fernandez, C. Kürt, M.A. Özarslan, A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators. Comput. Appl. Math. 39 (2020), No 200.10.1007/s40314-020-01224-5Suche in Google Scholar

[14] H.G. Flegg, Mikusinski’s operational calculus. Int. J. Math. Educ. Sci. Technol. 5, No 2 (1974), 131–137.10.1080/0020739740050201Suche in Google Scholar

[15] R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications. Springer, Berlin (2014); 2nd Ed. (2020).10.1007/978-3-662-43930-2Suche in Google Scholar

[16] R. Gorenflo, Y. F. Luchko, Operational method for solving generalized Abel integral equations of second kind. Integr. Transf. Spec. Funct. 5 (1997), 47–58.10.1080/10652469708819125Suche in Google Scholar

[17] R. Gorenflo, Yu.F. Luchko, H.M. Srivastava, Operational method for solving integral equations with Gauss–s hypergeometric function as a kernel. Internal. J. Math. & Statist. Sci. 6 (1997), 179–200.Suche in Google Scholar

[18] S.B. Hadid, Yu.F. Luchko, An operational method for solving fractional differential equations of an arbitrary real order. Panamer. Math. J. 6 (1996), 57–73.Suche in Google Scholar

[19] L.A.-M. Hanna, M. Al-Kandari, Yu.F. Luchko, Operational method for solving fractional differential equations with the left-and right-hand sided Erdélyi-Kober fractional derivatives. Fract. Calc. Appl. Anal. 23, No 1 (2020), 103-–125; 10.1515/fca-2020-0004; https://www.degruyter.com/journal/key/FCA/23/1/html.Suche in Google Scholar

[20] L.A-M. Hanna, Yu.F. Luchko, Operational calculus for the Caputo-type fractional Erdélyi-Kober derivative and its applications. Integr. Transf. Spec. Funct. 25 (2014), 359–373.10.1080/10652469.2013.856901Suche in Google Scholar

[21] R. Hilfer, Yu.F. Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann-Liouville fractional derivatives. Fract. Calc. Appl. Anal. 12, No 3 (2009), 299–318; at http://www.math.bas.bg/complan/fcaa/volume12/fcaa123/Hilfer_Luchko_Tomovski_FCAA_12_3.pdf.Suche in Google Scholar

[22] R. Hilfer, Yu. Luchko, Desiderata for fractional derivatives and integrals. Mathematics 7, No 2 (2019), # 149.10.3390/math7020149Suche in Google Scholar

[23] I.T. Huseynov, A. Ahmadova, G.O. Ojo, N.I. Mahmudov, A natural extension of Mittag-Leffler function associated with a triple infinite series. Preprint: 2011.03999 (2020).Suche in Google Scholar

[24] F. Jarad, T. Abdeljawad, Generalized fractional derivatives and Laplace transform. Discrete Contin. Dyn. Syst. Ser. S (2019), 1775–1786.10.3934/dcdss.2020039Suche in Google Scholar

[25] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations. Elsevier 204, North-Holland (2006).10.1016/S0304-0208(06)80001-0Suche in Google Scholar

[26] Yu.F. Luchko, Operational method in fractional calculus. Fract. Calc. Appl. Anal. 2, No 4 (1999), 463–488.Suche in Google Scholar

[27] Yu.F. Luchko, R. Gorenflo, An operational method for solving fractional differential equations. Acta Math. Vietnamica 24 (1999), 207–234.Suche in Google Scholar

[28] Yu.F. Luchko, H.M. Srivastava, The exact solution of certain differential equations of fractional order by using operational calculus. Comput. Math. Appl. 29 (1995), 73–85.10.1016/0898-1221(95)00031-SSuche in Google Scholar

[29] Yu.F. Luchko, S. Yakubovich, An operational method for solving some classes of integro-differential equations. Differential Equations 30 (1994), 247–256.Suche in Google Scholar

[30] J.A. Tenreiro Machado, V. Kiryakova, The chronicles of fractional calculus. Fract. Calc. Appl. Anal. 20, No 2 (2017), 307–336; 10.1515/fca-2017-0017; https://www.degruyter.com/journal/key/FCA/20/2/html.Suche in Google Scholar

[31] J. Mikusiński, Operational Calculus. Pergamon Press, Oxford (1959).Suche in Google Scholar

[32] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley, New York (1993).Suche in Google Scholar

[33] K.B. Oldham, J. Spanier, The Fractional Calculus. Academic Press, New York-London (1974).Suche in Google Scholar

[34] T.J. Osler, Leibniz rule for fractional derivatives generalized and an application to infinite series. SIAM J. Appl. Math. 18, No 3 (1970), 658–674.10.1137/0118059Suche in Google Scholar

[35] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego (1998).Suche in Google Scholar

[36] J.E. Restrepo, M. Ruzhansky, D. Suragan, Explicit solutions for linear variable–coefficient fractional differential equations with respect to functions. Appl. Math. Comput. 403 (2021), # 126177.10.1016/j.amc.2021.126177Suche in Google Scholar

[37] J.E. Restrepo, D. Suragan, Oscillatory solutions of fractional integro-differential equations. Math. Methods Appl. Sci. 43, No 15 (2020), 9080–9089.10.1002/mma.6602Suche in Google Scholar

[38] S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publ., N. York-London (1993).Suche in Google Scholar

[39] R.K. Saxena, S.L. Kalla, R. Saxena, Multivariate analogue of generalised Mittag-Leffler function. Integr. Transf. Spec. Funct. 22, No 7 (2011), 533–548.10.1080/10652469.2010.533474Suche in Google Scholar

[40] H.G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y.Q. Chen, A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlin. Sci. Numer. Simul. 64 (2018), 213–231.10.1016/j.cnsns.2018.04.019Suche in Google Scholar

[41] S.B. Yakubovich, Yu.F. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions. Kluwer Academic Publishers, Dordrecht-London (1994).10.1007/978-94-011-1196-6Suche in Google Scholar