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Nested Functions of Type Supertrigonometric and Superhyperbolic via Mittag-Leffler Functions
Authors: A. H. Ansari, S. Maksimović, L. Guran, M. Zhou
Keywords: supertrigonometric and superhyperbolic functions, Mittag-Leffler functions, fractional differential equations
Abstract:
Abstract: In this paper nested functions of type supertrigonometric and superhyperbolic are introduced and their properties are determined. Using these functions some classes of fractional differential equations are solved.
References:
[1] A. H. ANSARI, X.L. LIU, V. N. MISHRA, On Mittag-Leffler function and beyond, Nonlinear Sci. Lett. A, vol.8, 2 (2017), 187-199.
[2] A. B. ANTONEVICH, E. V. KUZMINA, On Generalized Solutions of Some Differential Equations with Singular Coefficients, Sci. Pub. State Univ. Novi Pazar, Ser A: Appl. Math. Inform. Mech. vol. 14, 1 (2022), 1-11.
[3] R.L. BAGLEY, The initial value problem for fractional order differential equations with constant coefficients, Air Force Institute of Technology Report, AFIT-TR-EN-88-1, 1988.
[4] G. DATTOLI, K. GORSKA, A. HORZELA, S. LICCIARDI, R.M. PIDATELLA, Comments on the properties of Mittag-Leffler function, Eur. Phys. J. Spec. Top. vol. 226, 16–18 (2017), 3427–3443.
[5] A. ERDELYI, W. MAGNUS, F. OBERHETTINGER, F.G. TRICOMI, Higher transcendental functions, vol. 3, McGraw-Hill, New York, NY, 1955.
[6] R. GORENFLO, A.A. KILBAS, F. MAINARDI, S.V. ROGOSIN, Mittag-Leffler Functions, related topics and applications, vol. 2. Springer, Berlin, 2014.
[7] R. GORENFLO, A. A. KILBAS, F.MAINARDI, S. ROGOSIN, Mittag-Leffler functions, related topics and applications, Second Edition, Springer-Verlag GmbH Germany, part of Springer Nature, 2020.
[8] H.J. HAUBOLD, A.M. MATHAI, R.K. SAXENA, Mittag-Leffler functions and their applications, J. Appl. Math. (2011), 1–51.
[9] P. HUMBERT, Quelques resultats relatifs ´ a la fonction de Mittag-Leffler ` C. R. Hebd. Seances Acad. Sci. vol. 236, 15 (1953), 1467–1468.
[10] W. KOEPF, Hypergeometric summation an algorithmic approach to summation and special function identities, Second Edition, Springer-Verlag London, 2014.
[11] F. MAINARDI, Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models, Copyright 2010 by Imperial College Press.
[12] K.S. MILLER, B. ROSS, An introduction to the fractional calculus and fractional differential equations, John Wiley & Sons. Inc., New York, 1993.
[13] G. M. MITTAG-LEFFLER, Une generalisation de l’integrale de Laplace-Abel, Comptes Rendus de l’Academie des Sciences Serie II, vol. 137, (1903), 537–539.
[14] G. M. MITTAG-LEFFLER, Sur la nouvelle fonction Eα(z), Comptes Rendus de l’Academie des Sciences, vol. 137, (1903), 554–558.
[15] Y.L. LUKE, Special functions and their approximations, Vol. 1, Ed. Academic Press, 1969.
[16] K. OLDHAM, J. SPANIER, The fractional calculus, theory and applications of differentiation and integration of arbitrary order, Academic Press, USA, 1974.
[17] I. PODLUBNY, The Laplace transform method for linear differential equations of the fractional order. arXiv: Functional Analysis, 1997.
[18] X.J. YANG, Theory and applications of special functions for scientists and engineers, Springer; 1st ed. , 2021.
[19] X.J. YANG, An introduction to hypergeometric, supertrigonometric and superhyperbolic functions, Academic Press Elsevier, 2021.
[20] A. WIMAN, Uber den fundamental satz in der theorie der funcktionen Ea(x) Acta Math. 29, (1905), 191–201.