On the fractional calculus of multivariate Mittag-Leffler functions

References

• B. Acay, M. Inç, Electrical circuits RC LC, and RLC under generalized type non-local singular fractional operator, Fractal Fractional 5(1) (2021), 9.
   Web of Science ®Google Scholar
• B. Adcock and J.M. Cardenas, Near-optimal sampling strategies for multivariate function approximation on general domains, SIAM J. Math. Data Sci. 2(3) (2020), pp. 607–630.
   Google Scholar
• 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.
   Web of Science ®Google Scholar
• A. Ahmadova, N.I. Mahmudov, Langevin differential equations with general fractional orders and their applications to electric circuit theory, J. Comput. Appl. Math. 388 (2021), 113299.
   Web of Science ®Google Scholar
• R. Almeida, A Caputo fractional derivative of a function with respect to another function, Commun. Nonlinear Sci. Numer. Simul. 44 (2017), pp. 460–481.
   Web of Science ®Google Scholar
• D. Baleanu and A. Fernandez, On some new properties of fractional derivatives with Mittag–Leffler kernel, Commun. Nonlinear Sci. Numer. Simulation 59 (2018), pp. 444–462.
   Web of Science ®Google Scholar
• D. Baleanu, A. Fernandez, On fractional operators and their classifications, Math. 7(9) (2019), 830.
   Web of Science ®Google Scholar
• L. Boyadjiev, H.-J. Dobner, S.L. Kalla, A fractional integro-differential equation of Volterra type, Math. Comput. Model. 28(10) (1998), pp. 103–113.
   Google Scholar
• G. Dattoli, L. Giannessi, L. Mezi and A. Torre, FEL time-evolution operator, Nuclear Instrum Methods Phys Res A304 (1991), pp. 541–544.
   Web of Science ®Google Scholar
• G. Dattoli, S. Lorenzutta, G. Maino and A. Torre, Analytical treatment of the high-gain free electron laser equation, Radiat Phys Chem 48(1) (1996), pp. 29–40.
   Web of Science ®Google Scholar
• A. Erdelyi, An integral equation involving Legendre functions, J. Soc. Ind. Appl. Math. 12(1) (1964), pp. 15–30.
   Google Scholar
• H.M. Fahad, A. Fernandez, M. ur Rehman, M. Siddiqi, Tempered and Hadamard-type fractional calculus with respect to functions, Mediterr. J. Math. 2020. arXiv:1907.04551.
   Web of Science ®Google Scholar
• A. Fernandez, D. Baleanu, H.M. Srivastava, Series representations for models of fractional calculus involving generalised Mittag-Leffler functions, Commun. Nonlinear Sci. Numer. Simul. 67 (2019), pp. 517–527.
   Web of Science ®Google Scholar
• A. Fernandez, C. Kürt, M.A. Özarslan, A naturally emerging bivariate Mittag-Leffler function and associated fractional-calculus operators, Comput. Appl. Math. 39 (2020), 200.
   Web of Science ®Google Scholar
• A. Fernandez, M.A. Özarslan and D. Baleanu, On fractional calculus with general analytic kernels, Appl. Math. Comput. 354 (2019), pp. 248–265.
   Web of Science ®Google Scholar
• R. Garra, E. Orsingher, F. Polito, A note on Hadamard fractional differential equations with varying coefficients and their applications in probability, Math. 6(1) (2018), 4.
   Web of Science ®Google Scholar
• S. Geva and J. Sitte, A constructive method for multivariate function approximation by multilayer perceptrons, IEEE Trans. Neural Netw. 3(4) (1992), pp. 621–624.
   PubMed Web of Science ®Google Scholar
• R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler Functions Related Topics and Applications, Springer, Berlin, 2016.
   Google Scholar
• R. Hilfer ed., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
   Google Scholar
• J. Hristov ed., The Craft of Fractional Modelling in Science and Engineering, MDPI, Basel, 2018.
   Google Scholar
• I.T. Huseynov, A. Ahmadova, A. Fernandez, N.I. Mahmudov, Explicit analytic solutions of incommensurate fractional differential equation systems, Appl. Math. Comput. 390C (2021), 125590.
   Google Scholar
• 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, arXiv:2011.03999.
   Google Scholar
• A.A. Kilbas, M. Saigo, R.K. Saxena, Solution of Volterra integro-differential equations with generalized Mittag-Leffler function in the kernels, Int. Transf. Spec. Funct. 15(1) (2004), pp. 31–49.
   Web of Science ®Google Scholar
• A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
   Google Scholar
• V. Kiryakova, The special functions of fractional calculus as generalized fractional calculus operators of some basic functions, Comput. Math. Appl. 59(3) (2010), pp. 1128–1141.
   Web of Science ®Google Scholar
• R.C. Koeller, Applications of fractional calculus to the theory of viscoelasticity, J. Appl. Mech. 51 (1984), pp. 299–307.
   Web of Science ®Google Scholar
• R.L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, Connecticut, 2006.
   Google Scholar
• F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, 2010.
   Google Scholar
• F. Mainardi and R. Gorenflo, Fractional calculus and special functions, Lecture Notes Math. Phys. Univ. Bologna: Bologna, Italy (2013), pp. 1–64.
   Google Scholar
• K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.
   Google Scholar
• K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, San Diego, 1974.
   Google Scholar
• T.J. Osler, Leibniz rule for fractional derivatives generalised and an application to infinite series, SIAM J. Appl. Math. 18 (1970), pp. 658–674.
   Web of Science ®Google Scholar
• T.J. Osler, The fractional derivative of a composite function, SIAM J. Math. Anal. 1 (1970), pp. 288–93.
   Google Scholar
• T.J. Osler, Fractional derivatives and Leibniz rule, Am. Math. Mon. 78 (1971), pp. 645–649.
   Web of Science ®Google Scholar
• M.A. Özarslan, On a singular integral equation including a set of multivariate polynomials suggested by Laguerre polynomials, Appl. Math. Comput. 229 (2014), pp. 350–358.
   Web of Science ®Google Scholar
• M.A. Özarslan and C. Kürt, Bivariate Mittag-Leffler functions arising in the solutions of convolution integral equation with 2D-Laguerre-Konhauser polynomials in the kernel, Appl. Math. Comput. 347 (2019), pp. 631–644.
   Web of Science ®Google Scholar
• I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
   Google Scholar
• T.R. Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the kernel, Yokohama Math. J. 19 (1971), pp. 7–15.
   Google Scholar
• S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Taylor & Francis, London, 2002.[orig. ed. in Russian; Nauka i Tekhnika, Minsk, 1987].
   Google Scholar
• R.K. Saxena and S.L. Kalla, Solution of Volterra-type integro-differential equations with a generalized Lauricella confluent hypergoemetric function in the kernels, Int. J. Math. Math. Sci. 8 (2005) (2005), pp. 1155–1170.
   Google Scholar
• R.K. Saxena, S.L. Kalla and R. Saxena, Multivariate analogue of generalised Mittag-Leffler function, Int. Transf. Spec. Funct. 22(7) (2011), pp. 533–548.
   Web of Science ®Google Scholar
• Y. Shin and D. Xiu, A randomized algorithm for multivariate function approximation, SIAM J. Sci. Comput. 39(3) (2017), pp. A983–A1002.
   Web of Science ®Google Scholar
• J.V. da Sousa and E.C. de Oliveira, On the Ψ-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul. 60 (2018), pp. 72–91.
   Web of Science ®Google Scholar
• H.M. Srivastava, A. Fernandez, D. Baleanu, Some new fractional-calculus connections between Mittag-Leffler functions, Math. 7(6) (2019), 485.
   Web of Science ®Google Scholar
• H.M. Srivastava, P. Harjule and R. Jain, A general fractional differential equation associated with an integral operator with the H-function in the kernel, Russ. J. Math. Phys. 22(1) (2015), pp. 112–126.
   Web of Science ®Google Scholar
• H.M. Srivastava, R.K. Raina and X.J. Yang, Special functions in fractional calculus and related fractional differintegral equations, 2019.doi:https://doi.org/10.1142/8936.
   Google Scholar
• H.M. Srivastava and Ž. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel, Appl. Math. Comput. 211(1) (2009), pp. 198–210.
   Web of Science ®Google Scholar
• N. Su, The fractional boussinesq equation of groundwater flow and its applications, J. Hydrol. 547 (2017), pp. 403–412.
   Web of Science ®Google Scholar