Riemann-Liouville derivative over the space of integrable distributions

  • Received: 01 December 2019 Revised: 01 February 2020
  • Primary: 26A33, 26A39; Secondary: 46F12, 47G20

  • 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.

    Citation: María Guadalupe Morales, Zuzana Došlá, Francisco J. Mendoza. Riemann-Liouville derivative over the space of integrable distributions[J]. Electronic Research Archive, 2020, 28(2): 567-587. doi: 10.3934/era.2020030

    Related Papers:

  • 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.



    加载中


    [1] Linear functionals on Denjoy-integrable functions. Colloquium Math. (1948) 1: 289-293.
    [2] Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties. Phys. A. (2018) 505: 688-706.
    [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. doi: 10.1140/epjp/i2018-12021-3
    [4] Fractional derivatives with no-index law property: Application to chaos and statistics. Chaos Solitons Fractals (2018) 114: 516-535.
    [5] Numerical approximation of Riemann-Liouville definition of fractional derivative: From Riemann-Liouville to Atangana-Baleanu. Numer. Methods Partial Differential Equations (2018) 34: 1502-1523.
    [6] Modeling attractors of chaotic dynamical systems with fractal-fractional operators. Chaos Solitons Fractals (2019) 123: 320-337.
    [7] Differential and integral operators with constant fractional order and variable fractional dimension. Chaos Solitons Fractals (2019) 127: 226-243.
    [8] R. G. Bartle, A Modern Theory of Integration, Graduate Studies in Mathematics, 32. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/032
    [9] D. A. Benson, The Fractional Advection-Dispersion Equation: Development and Application, Ph.D. thesis, University of Nevada in Reno, 1998.
    [10] Relatively weakly compact sets in the Denjoy space. J. Math. Study (1994) 27: 37-44.
    [11] On the Alexiewicz topology of the Denjoy space. Real Anal. Exchange (1995/96) 21: 604-614.
    [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. doi: 10.1007/978-3-642-14574-2
    [13] J. J. Duistermaat and J. A. C. Kolk, Distributions. Theory and Applications, Birkhäuser Boston, Inc., Boston, MA, 2010. doi: 10.1007/978-0-8176-4675-2
    [14] (1964) Generalized Functions, Volume 1: Properties and Operations.Academic Press.
    [15] A fractional calculus approach to self-similar protein dynamics. Biophys. J. (1995) 68: 46-53.
    [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. doi: 10.1140/epjp/i2017-11293-3
    [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. doi: 10.1155/2016/7047126
    [18] R. A. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics, 4, American Mathematical Society, Providence, RI, 1994. doi: 10.1090/gsm/004
    [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. doi: 10.1007/978-3-642-96750-4
    [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. doi: 10.1016/s0304-0208(06)x8001-5
    [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. doi: 10.1142/5538
    [22] Several results of fractional derivatives in $D'(R_+)$. Fract. Calc. Appl. Anal. (2015) 18: 192-207.
    [23] Differintegral interpolation from a bandlimited signal's samples. IEEE Trans. Acoust., Speech, Signal Processing (1981) 29: 872-877.
    [24] R. M. McLeod, The Generalized Riemann Integral, Carus Math. Monographs, 20. Mathematical Association of America, Washington, D.C., 1980. doi: 10.5948/upo9781614440208
    [25] A unified theory of integration. Amer. Math. Monthly (1973) 80: 349-359.
    [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. doi: 10.1142/9432
    [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. doi: 10.1140/epjp/i2017-11341-0
    [28] W. Rudin, Functional Analysis, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973.
    [29] Representation of functions by convolutions. J. Math. Mech. (1958) 7: 103-115.
    [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.
    [31] Š. Schwabik, Generalized Ordinary Differential Equations, Series in Real Analysis, 5. World Scientific Publishing Co., Inc., River Edge, N.J., 1992. doi: 10.1142/1875
    [32] E. Talvila, Convolutions with the continuous primitive integral, Abstr. Appl. Anal., 2009 (2009), Art. ID 307404, 18 pp. doi: 10.1155/2009/307404
    [33] Henstock-Kurzweil Fourier transforms. Illinois J. Math. (2002) 46: 1207-1226.
    [34] The distributional Denjoy integral. Real Anal. Exchange (2008) 33: 51-82.
    [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. doi: 10.3389/fphy.2017.00052
    [36] The distributional Henstock-Kurzweil integral and applications. Monatsh. Math. (2016) 181: 975-989.
    [37] The distributional Henstock-Kurzweil integral and applications: A survey. J. Math. Study (2016) 49: 433-448.
    [38] The Feng's first integral method applied to the nonlinear mKdV space-time fractional partial differential equation. Rev. Mex. Fís. (2016) 62: 310-316.
    [39] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, N.J., 2014. doi: 10.1142/9069
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1739) PDF downloads(264) Cited by(3)

Article outline

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog