×

Riemann-Liouville derivative over the space of integrable distributions. (English) Zbl 1516.26004

Summary: In this paper, we generalize the Riemann-Liouville differential and integral operators on the space of Henstock-Kurzweil integrable distributions, \(D_{HK}\). We obtain new fundamental properties of the fractional derivatives and integrals, a general version of the fundamental theorem of fractional calculus, semigroup property for the Riemann-Liouville integral operators and relations between the Riemann-Liouville integral and differential operators. Also, we achieve a generalized characterization of the solution for the Abel integral equation. Finally, we show relations for the Fourier transform of fractional derivative and integral. These results are based on the properties of the distributional Henstock-Kurzweil integral and convolution.

MSC:

26A39 Denjoy and Perron integrals, other special integrals
26A33 Fractional derivatives and integrals
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] A. Alexiewicz, Linear functionals on Denjoy-integrable functions, Colloquium Math., 1, 289-293 (1948) · Zbl 0037.35603 · doi:10.4064/cm-1-4-289-293
[2] A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Phys. A., 505, 688-706 (2018) · Zbl 1514.34009 · doi:10.1016/j.physa.2018.03.056
[3] A. Atangana and J. F. Gómez-Aguilar, Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena, Eur. Phys. J. Plus, 133 (2018), 22 pp.
[4] A. Atangana; J. F. Gómez-Aguilar, Fractional derivatives with no-index law property: Application to chaos and statistics, Chaos Solitons Fractals, 114, 516-535 (2018) · Zbl 1415.34010 · doi:10.1016/j.chaos.2018.07.033
[5] A. Atangana; J. F. Gómez-Aguilar, Numerical approximation of Riemann-Liouville definition of fractional derivative: From Riemann-Liouville to Atangana-Baleanu, Numer. Methods Partial Differential Equations, 34, 1502-1523 (2018) · Zbl 1417.65113 · doi:10.1002/num.22195
[6] A. Atangana; S. Qureshi, Modeling attractors of chaotic dynamical systems with fractal-fractional operators, Chaos Solitons Fractals, 123, 320-337 (2019) · Zbl 1448.65268 · doi:10.1016/j.chaos.2019.04.020
[7] A. Atangana; A. Shafiq, Differential and integral operators with constant fractional order and variable fractional dimension, Chaos Solitons Fractals, 127, 226-243 (2019) · Zbl 1448.34011 · doi:10.1016/j.chaos.2019.06.014
[8] R. G. Bartle, A Modern Theory of Integration, Graduate Studies in Mathematics, 32. American Mathematical Society, Providence, RI, 2001. · Zbl 0968.26001
[9] D. A. Benson, The Fractional Advection-Dispersion Equation: Development and Application, Ph.D. thesis, University of Nevada in Reno, 1998.
[10] B. Bongiorno, Relatively weakly compact sets in the Denjoy space, J. Math. Study, 27, 37-44 (1994) · Zbl 1045.26502
[11] B. Bongiorno; T. V. Panchapagesan, On the Alexiewicz topology of the Denjoy space, Real Anal. Exchange, 21, 604-614 (1995/96) · Zbl 0879.26028 · doi:10.2307/44152670
[12] K. Diethelm, The Analysis of Fractional Differential Equations. An Application-oriented Exposition Using Differential Operators of Caputo Type, Lecture Notes in Mathematics, 2004. Springer-Verlag, Berlin, 2010. · Zbl 1215.34001
[13] J. J. Duistermaat and J. A. C. Kolk, Distributions. Theory and Applications, Birkhäuser Boston, Inc., Boston, MA, 2010. · Zbl 1213.46001
[14] W. G. Glöckle; T. F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophys. J., 68, 46-53 (1995) · doi:10.1016/s0006-3495(95)80157-8
[15] W. G. Glöckle; T. F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophys. J., 68, 46-53 (1995) · doi:10.1016/s0006-3495(95)80157-8
[16] J. F. Gómez-Aguilar and A. Atangana, New insight in fractional differentiation: Power, exponential decay and Mittag-Leffler laws and applications, Eur. Phys. J. Plus, 132 (2017), 21 pp. · Zbl 1400.35066
[17] J. F. Gómez-Aguilar, H. Yépez-Martínez, R. F. Escobar-Jiménez, V. H. Olivares-Peregrino, J. M. Reyes and I. O. Sosa, Series solution for the time-fractional coupled mKdV equation using the homotopy analysis method, Math. Probl. Eng., 2016 (2016), Art. ID 7047126, 8 pp. · Zbl 1400.35066
[18] R. A. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics, 4, American Mathematical Society, Providence, RI, 1994. · Zbl 1028.35001
[19] L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, \(2^{nd}\) edition, Classics in Mathematics, Springer-Verlag, Berlin, 2003. · Zbl 1092.45003
[20] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006. · Zbl 1072.26005
[21] D. S. Kurtz and C. W. Swartz, Theories of Integration. The Integrals of Riemann, Lebesgue, Henstock-Kurzweil, and Mcshane, Series in Real Analysis, 9. World Scientific Publishing Co., Inc., River Edge, N.J., 2004. · Zbl 1335.46033 · doi:10.1515/fca-2015-0013
[22] C. K. Li, Several results of fractional derivatives in \(D'(R_+)\), Fract. Calc. Appl. Anal., 18, 192-207 (2015) · Zbl 0525.65005 · doi:10.1515/fca-2015-0013
[23] R. Marks; M. Hall, Differintegral interpolation from a bandlimited signal’s samples, IEEE Trans. Acoust., Speech, Signal Processing, 29, 872-877 (1981) · Zbl 0486.26005 · doi:10.1109/tassp.1981.1163636
[24] R. M. McLeod, The Generalized Riemann Integral, Carus Math. Monographs, 20. Mathematical Association of America, Washington, D.C., 1980. · Zbl 0266.26008 · doi:10.1080/00029890.1973.11993291
[25] E. J. McShane, A unified theory of integration, Amer. Math. Monthly, 80, 349-359 (1973) · Zbl 1437.28001 · doi:10.1080/00029890.1973.11993291
[26] G. A. Monteiro, A. Slavík and M. Tvrdý, Kurzweil-Stieltjes Integral. Theory and Applications, Series in Real Analysis, 15. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019.
[27] V. F. Morales-Delgado, M. A. Taneco-Hernández and J. F. Gómez-Aguilar, On the solutions of fractional order of evolution equations, Eur. Phys. J. Plus, 132 (2017), 14 pp.
[28] W. Rudin, Functional Analysis, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. · Zbl 0079.13302 · doi:10.1512/iumj.1958.7.57009
[29] W. Rudin, Representation of functions by convolutions, J. Math. Mech., 7, 103-115 (1958) · Zbl 0818.26003 · doi:10.1512/iumj.1958.7.57009
[30] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. · Zbl 0781.34003
[31] Š. Schwabik, Generalized Ordinary Differential Equations, Series in Real Analysis, 5. World Scientific Publishing Co., Inc., River Edge, N.J., 1992. · Zbl 1192.46039
[32] E. Talvila, Convolutions with the continuous primitive integral, Abstr. Appl. Anal., 2009 (2009), Art. ID 307404, 18 pp. · Zbl 1037.42007 · doi:10.1215/ijm/1258138475
[33] E. Talvila, Henstock-Kurzweil Fourier transforms, Illinois J. Math., 46, 1207-1226 (2002) · Zbl 1152.37324 · doi:10.1215/ijm/1258138475
[34] E. Talvila, The distributional Denjoy integral, Real Anal. Exchange, 33, 51-82 (2008) · Zbl 1154.26011 · doi:10.14321/realanalexch.33.1.0051
[35] A. A. Tateishi, H. V. Ribeiro and E. K. Lenzi, The role of fractional time-derivative operators on anomalous diffusion, Front. Phys., 5 (2017), 9 pp. · Zbl 1362.45009 · doi:10.1007/s00605-015-0853-1
[36] G. J. Ye; W. Liu, The distributional Henstock-Kurzweil integral and applications, Monatsh. Math., 181, 975-989 (2016) · Zbl 1374.26021 · doi:10.1007/s00605-015-0853-1
[37] G. J. Ye; W. Liu, The distributional Henstock-Kurzweil integral and applications: A survey, J. Math. Study, 49, 433-448 (2016) · Zbl 1374.26021 · doi:10.4208/jms.v49n4.16.06
[38] H. Yépez-Martínez; J. F. Gómez-Aguilar; I. O. Sosa; J. M. Reyes; J. Torres-Jiménez, The Feng’s first integral method applied to the nonlinear mKdV space-time fractional partial differential equation, Rev. Mex. Fís., 62, 310-316 (2016) · Zbl 1336.34001
[39] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, N.J., 2014. · Zbl 1336.34001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.