An operational calculus formulation of fractional calculus with general analytic kernels

Noosheza Rani; Arran Fernandez; Noosheza Rani; Arran Fernandez

doi:10.3934/era.2022216

Electronic Research Archive

2022, Volume 30, Issue 12: 4238-4255. doi: 10.3934/era.2022216

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An operational calculus formulation of fractional calculus with general analytic kernels

Noosheza Rani ,
Arran Fernandez ^,

Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Famagusta, Northern Cyprus, via Mersin-10, Turkey

Academic Editor: Jean-Daniel Djida

Received: 26 May 2022 Revised: 05 August 2022 Accepted: 31 August 2022 Published: 26 September 2022

Fractional calculus with analytic kernels provides a general setting of integral and derivative operators that can be connected to Riemann–Liouville fractional calculus via convergent infinite series. We interpret these operators from an algebraic viewpoint, using Mikusiński's operational calculus, and utilise this algebraic formalism to solve some fractional differential equations.
- fractional integrals,
- fractional derivatives,
- operational calculus,
- fractional differential equations,
- Mikusiński's operational calculus
Citation: Noosheza Rani, Arran Fernandez. An operational calculus formulation of fractional calculus with general analytic kernels[J]. Electronic Research Archive, 2022, 30(12): 4238-4255. doi: 10.3934/era.2022216

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Abstract

Fractional calculus with analytic kernels provides a general setting of integral and derivative operators that can be connected to Riemann–Liouville fractional calculus via convergent infinite series. We interpret these operators from an algebraic viewpoint, using Mikusiński's operational calculus, and utilise this algebraic formalism to solve some fractional differential equations.

References

[1]	K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, New York, 1993.
[2]	K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
[3]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, 1993.
[4]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006.
[5]	R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. https://doi.org/10.1142/3779
[6]	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. Nonlinear Sci. Numer. Simul., 64 (2018), 213–231. https://doi.org/10.1016/j.cnsns.2018.04.019 doi: 10.1016/j.cnsns.2018.04.019
[7]	D. Baleanu, A. Fernandez, On fractional operators and their classifications, Mathematics, 7 (2019), 830. https://doi.org/10.3390/math7090830 doi: 10.3390/math7090830
[8]	A. N. Kochubei, General fractional calculus, evolution equations, and renewal processes, Integr. Equations Oper. Theory, 71 (2011), 583–600. https://doi.org/10.1007/s00020-011-1918-8 doi: 10.1007/s00020-011-1918-8
[9]	Y. Luchko, General fractional integrals and derivatives of arbitrary order, Symmetry, 13 (2021), 755. https://doi.org/10.3390/sym13050755 doi: 10.3390/sym13050755
[10]	A. Fernandez, M. A. Özarslan, D. Baleanu, On fractional calculus with general analytic kernels, Appl. Math. Comput., 354 (2019), 248–265. https://doi.org/10.1016/j.amc.2019.02.045 doi: 10.1016/j.amc.2019.02.045
[11]	M. Jleli, M. Kirane, B. Samet, A derivative concept with respect to an arbitrary kernel and applications to fractional calculus, Math. Methods Appl. Sci., 42 (2019), 137–160. https://doi.org/10.1002/mma.5329 doi: 10.1002/mma.5329
[12]	D. Zhao, M. Luo, Representations of acting processes and memory effects: General fractional derivative and its application to theory of heat conduction with finite wave speeds, Appl. Math. Comput., 346 (2019), 531–544. https://doi.org/10.1016/j.amc.2018.10.037 doi: 10.1016/j.amc.2018.10.037
[13]	L. A. Pipes, The operational calculus Ⅰ, J. Appl. Phys., 10 (1939), 172. https://doi.org/10.1063/1.1707292 doi: 10.1063/1.1707292
[14]	J. Mikusiński, Operational Calculus, Pergamon Press, Oxford, 1959.
[15]	H. G. Flegg, Mikusinski's operational calculus, Int. J. Math. Educ. Sci. Tech., 5 (1974), 131–137. https://doi.org/10.1080/0020739740050201 doi: 10.1080/0020739740050201
[16]	M. Gutterman, An operational method in partial differential equations, SIAM J. Appl. Math., 17 (1969), 468–493. https://doi.org/10.1137/0117046 doi: 10.1137/0117046
[17]	Y. Luchko, Operational method in fractional calculus, Fractional Calc. Appl. Anal., 2 (1999), 463–488.
[18]	S. B. Hadid, Y. F. Luchko, An operational method for solving fractional differential equations of an arbitrary real order, Panam. Math. J., 6 (1996), 57–73.
[19]	Y. Luchko, R. Gorenflo, An operational method for solving fractional differential equations, Acta Math. Vietnam., 24 (1999), 207–234.
[20]	R. Hilfer, Y. F. Luchko, Z. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann–Liouville fractional derivatives, Fract. Calc. Appl. Anal., 12 (2009), 299–318.
[21]	Y. Luchko, S. Yakubovich, An operational method for solving some classes of integro-differential equations, Differ. Uravn., 30 (1994), 269–280.
[22]	S. Yakubovich, Y. Luchko, The Hypergeometric Approach to Integral Transforms and Convolutions, Kluwer Academic Publishers, Amsterdam, 1994. https://doi.org/10.1007/978-94-011-1196-6
[23]	N. Rani, A. Fernandez, Mikusinski's operational calculus for Prabhakar fractional calculus, Integr. Transf. Spec. Funct., (2022), 1–21. https://doi.org/10.1080/10652469.2022.2057970 doi: 10.1080/10652469.2022.2057970
[24]	N. Rani, A. Fernandez, Solving Prabhakar differential equations using Mikusinski's operational calculus, Comput. Appl. Math., 41 (2022), 107. https://doi.org/10.1007/s40314-022-01794-6 doi: 10.1007/s40314-022-01794-6
[25]	Y. Luchko, Operational calculus for the general fractional derivative and its applications, Fract. Calculus Appl. Anal., 24 (2021), 338–375. https://doi.org/10.1515/fca-2021-0016 doi: 10.1515/fca-2021-0016
[26]	Y. Luchko, Fractional differential equations with the general fractional derivatives of arbitrary order in the Riemann–Liouville sense, Mathematics, 10 (2022), 849. https://doi.org/10.3390/math10060849 doi: 10.3390/math10060849
[27]	H. M. Fahad, A. Fernandez, Operational calculus for Riemann–Liouville fractional calculus with respect to functions and the associated fractional differential equation, Fract. Calculus Appl. Anal., 24 (2021), 518–540. https://doi.org/10.1515/fca-2021-0023 doi: 10.1515/fca-2021-0023
[28]	H. M. Fahad, A. Fernandez, Operational calculus for Caputo fractional calculus with respect to functions and the associated fractional differential equation, Appl. Math. Comput., 409 (2021), 126400. https://doi.org/10.1016/j.amc.2021.126400 doi: 10.1016/j.amc.2021.126400
[29]	A. Fernandez, D. Baleanu, H. M. Srivastava, Series representations for fractional-calculus operators involving generalised Mittag-Leffler functions, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 517–527. https://doi.org/10.1016/j.cnsns.2018.07.035 doi: 10.1016/j.cnsns.2018.07.035
[30]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1998.
[31]	A. Fernandez, Tables of composition properties of fractional integrals and derivatives, preprint.
[32]	I. Dimovski, Operational calculus for a class of differential operators, CR Acad. Bulg. Sci., 19 (1966), 1111–1114.
[33]	N. Sonine, Sur la généralisation d'une formule d'Abel, Acta Math., 4 (1884), 171–176. https://doi.org/10.1007/BF02418416 doi: 10.1007/BF02418416
[34]	J. Wick, Über eine Integralgleichung vom Abelschen Typ, Angew. Math., 48 (1968), T39–T41.

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