Mellin transform for fractional integrals with general analytic kernel

Maliha Rashid; Amna Kalsoom; Maria Sager; Mustafa Inc; Dumitru Baleanu; Ali S. Alshomrani; Maliha Rashid; Amna Kalsoom; Maria Sager; Mustafa Inc; Dumitru Baleanu; Ali S. Alshomrani

doi:10.3934/math.2022524

AIMS Mathematics

2022, Volume 7, Issue 5: 9443-9462. doi: 10.3934/math.2022524

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Mellin transform for fractional integrals with general analytic kernel

1.
Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan
2.
Department of Computer Engineering, Biruni University, Istanbul, Turkey
3.
Department of Mathematics, Firat University, Elazig 23119, Turkey
4.
Department of Medical Research, China Medical University, Taichung, Taiwan
5.
Department of Mathematics, Çankaya University, Balgat 06530, Ankara, Turkey
6.
Institute of Space Sciences, P.O. Box, MG-23, R 76900, Magurele-Bucharest, Romania
7.
Faculty of Science, Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia

Received: 27 November 2021 Revised: 21 January 2022 Accepted: 10 February 2022 Published: 14 March 2022
MSC : Primary: 58F15, 58F17; Secondary: 53C35

Many different operators of fractional calculus have been proposed, which can be organized in some general classes of operators. According to this study, the class of fractional integrals and derivatives can be classified into two main categories, that is, with and without general analytical kernel (introduced in 2019). In this article, we define the Mellin transform for fractional differential operator with general analytic kernel in both Riemann-Liouville and Caputo derivatives of order $ \varsigma\ge0 $ and $ \varrho $ be a fixed parameter. We will also establish relation between Mellin transform with Laplace and Fourier transforms.
- Mellin transform,
- fractional integrals,
- Caputo fractional derivative,
- Laplace and Fourier transforms
Citation: Maliha Rashid, Amna Kalsoom, Maria Sager, Mustafa Inc, Dumitru Baleanu, Ali S. Alshomrani. Mellin transform for fractional integrals with general analytic kernel[J]. AIMS Mathematics, 2022, 7(5): 9443-9462. doi: 10.3934/math.2022524

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Abstract

Many different operators of fractional calculus have been proposed, which can be organized in some general classes of operators. According to this study, the class of fractional integrals and derivatives can be classified into two main categories, that is, with and without general analytical kernel (introduced in 2019). In this article, we define the Mellin transform for fractional differential operator with general analytic kernel in both Riemann-Liouville and Caputo derivatives of order $ \varsigma\ge0 $ and $ \varrho $ be a fixed parameter. We will also establish relation between Mellin transform with Laplace and Fourier transforms.

References

[1]	I. Podlubny, Fractional differential equations, Academic Press, Cambridge, 1999.
[2]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integrals and derivatives, Vol. 1, Switzerland: Gordon and breach science publishers, 1993.
[3]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley, New York, 1993.
[4]	B. Riemann, Versuch einer allgemeinen auffassung der integration und differentiation, In: Gessamelte mathematische werke, Druck und Verlag: Leipzig, Germany, 1876.
[5]	L. G. Romero, L. Luque, k-Weyl fractional derivative, integral and integral transform, Int. J. Contemp. Math. Sci., 8 (2013), 263–270.
[6]	G. H. Hardy, J. E. Littlewood, Some properties of fractional integrals I, Math. Z., 27 (1928), 565–606.
[7]	G. H. Hardy, J. E. Littlewood, Some properties of fractional integrals II, Math. Z., 34 (1932), 403–439.
[8]	M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl., 1 (2015), 1–13.
[9]	S. Dugowson, Les differentielles metaphysiques: Histoire et philosophie de la generalisation de l'ordre de derivation, Ph.D. Thesis, Universite Paris Nord, Paris, France, 1994.
[10]	R. Hilfer, Threefold introduction to fractional derivatives, Anomalous transport: Foundations and applications, Wiley-VCH Verlag, 2008, 17–73. http://dx.doi.org/10.1002/9783527622979
[11]	D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus: Models and numerical methods, 2 Eds., World Scientific, 2017.
[12]	S. F. Lacroix, Traité du calcul différentiel et du calcul intégral, volumn 1 (French Edition), Chez JBM Duprat, Libraire pour les Mathématiques, quai des Augustins, 1797.
[13]	N. Sonine, Sur la differentiation a indice quelconque, Mat. Sb., 6 (1872), 1–38.
[14]	A. Atangana, On the new fractional derivative and application to nonlinear Fisher's reaction-diffusion equation, Appl. Math. Comput., 273 (2016), 948–956. https://doi.org/10.1016/j.amc.2015.10.021 doi: 10.1016/j.amc.2015.10.021
[15]	J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular Kernel, Progr. Fract. Differ. Appl., 1 (2015), 87–92.
[16]	D. Li, W. Sun, C. Wu, A novel numerical approach to time-fractional parabolic equations with nonsmooth solutions, Numer. Math.: Theor., Meth. Appl., 14 (2021), 355–376. https://doi.org/10.4208/nmtma.OA-2020-0129 doi: 10.4208/nmtma.OA-2020-0129
[17]	H. Qin, D. Li, Z. Zhang, A novel scheme to capture the initial dramatic evolutions of nonlinear subdiffusion equations, J. Sci. Comput., 89 (2021), 1–20. https://doi.org/10.1007/s10915-021-01672-z doi: 10.1007/s10915-021-01672-z
[18]	A. Fernandez, M. A. Ozarslan, 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
[19]	N. Zhou, H. Li, D. Wang, S. Pan, Z. Zhou, Image compression and encryption scheme based on 2D compressive sensing and fractional Mellin transform, Opt. Commun., 343 (2015), 10–21. https://doi.org/10.1016/j.optcom.2014.12.084 doi: 10.1016/j.optcom.2014.12.084
[20]	M. Wang, Y. Pousset, P. Carré, C. Perrine, N. Zhou, J. Wu, Optical image encryption scheme based on apertured fractional Mellin transform, Opt. Laser Technol., 124 (2020), 106001. https://doi.org/10.1016/j.optlastec.2019.106001 doi: 10.1016/j.optlastec.2019.106001
[21]	A. Kiliçman, M. Omran, Note on fractional Mellin transform and applications, SpringerPlus, 5 (2016), 1–8. https://doi.org/10.1186/s40064-016-1711-x doi: 10.1186/s40064-016-1711-x
[22]	L. Sörnmo, P. Laguna, Bioelectrical signal processing in cardiac and neurological applications, Vol. 8, Academic Press, 2005.
[23]	S. Das, K. Maharatna, Fractional dynamical model for the generation of ECG like signals from filtered coupled Van-der Pol oscillators, Comput. Meth. Prog. Bio., 112 (2013), 490–507. https://doi.org/10.1016/j.cmpb.2013.08.012 doi: 10.1016/j.cmpb.2013.08.012
[24]	Z. B. Vosika, G. M. Lazovic, G. N. Misevic, J. B. Simic-Krstic, Fractional calculus model of electrical impedance applied to human skin, PloS One, 8 (2013), e59483. https://doi.org/10.1371/journal.pone.0059483 doi: 10.1371/journal.pone.0059483
[25]	A. D. Poularikas, The transforms and applications handbook, CRC Press, 1996.
[26]	B. Davies, Integral transforms and their applications, Springer, 2001.
[27]	H. Bateman, Tables of integral transforms (Volumes 1 & 2), McGraw-Hill, New York, 1954.
[28]	M. Erdélyi, T. Oberhettinger, Tables of integral transforms, McGraw-Hill, New York, 1954.
[29]	I. N. Sneddon, The use of integral transforms, McGraw-Hill, New York, 1972.
[30]	P. L. Butzer, A. A. Kilbas, J. J. Trujillo, Mellin transform analysis and integration by parts for Hadamard-type fractional integrals, J. Math. Anal. Appl., 270 (2002), 1–15. https://doi.org/10.1016/S0022-247X(02)00066-5 doi: 10.1016/S0022-247X(02)00066-5
[31]	A. Kilicman, Distributions theory and neutrix calculus, Universiti Putra Malaysia Press, Serdang, 2006.
[32]	F. Gerardi, Application of Mellin and Hankel transforms to networks with time-varying parameters, IRE Trans. Circuit Theory, 6 (1959), 197–208. https://doi.org/10.1109/TCT.1959.1086540 doi: 10.1109/TCT.1959.1086540
[33]	P. Flajolet, X. Gourdon, P. Dumas, Mellin transforms and asymptotics harmonic sums, Theor. Comput. Sci., 144 (1995), 3–58. https://doi.org/10.1016/0304-3975(95)00002-E doi: 10.1016/0304-3975(95)00002-E
[34]	I. Dimovski, Operational calculus for a class of differential operators, CR Acad. Bulg. Sci., 19 (1966), 1111–1114.
[35]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Vol. 204, Elsevier, 2006.
[36]	Y. Luchko, Operational method in fractional calculus, Fract. Calc. Appl. Anal., 2 (1999), 463–488.
[37]	I. H. Dimovski, V. S. Kiryakova, Transform methods and special functions varna 96 second international workshop proceedings, 1996.
[38]	M. Caputo, Elasticita e dissipazione, Zani-Chelli, Bologna, 1969.
[39]	C. M. S. Oumarou, H. M. Fahad, J. D. Djida, A. Fernandez, On fractional calculus with analytic kernels with respect to functions, Comput. Appl. Math., 40 (2021), 1–24. https://doi.org/10.1007/s40314-021-01622-3 doi: 10.1007/s40314-021-01622-3
[40]	J. Twamley, G. J. Milburn, The quantum Mellin transform, New J. Phys., 8 (2006), 328.
[41]	A. Makarov, S. Postovalov, A. Ermakov, Research of digital Algorithms implementing integrated Mellin transform for signal processing in automated control systems, In: 2019 International Multi-Conference on Industrial Engineering and Modern Technologies (FarEastCon), 2019, 1–6. 10.1109/FarEastCon.2019.8934434
[42]	V. Namias, The fractional order Fourier transform and its application to quantum mechanics, IMA J. Appl. Math., 25 (1980), 241–265. https://doi.org/10.1093/imamat/25.3.241 doi: 10.1093/imamat/25.3.241
[43]	A. Torre, Linear and radial canonical transforms of fractional order, Comput. Appl. Math., 153 (2003), 477–486. https://doi.org/10.1016/S0377-0427(02)00637-4 doi: 10.1016/S0377-0427(02)00637-4
[44]	P. R. Deshmukh, A. S. Gudadhe, Analytical study of a special case of complex canonical transform, Global J. Math. Sci., 2 (2010), 261–270.
[45]	L. Debnath, D. Bhatta, Integral transforms and their applications, 2 Eds., Chapman and Hall/CRC, 2006.

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