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June  2020, 28(2): 567-587. doi: 10.3934/era.2020030

Riemann-Liouville derivative over the space of integrable distributions

María Guadalupe Morales 1,, , Zuzana Došlá 1, and  Francisco J. Mendoza 2,

1. 

Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2,611 37 Brno, Czech Republic
2. 

Facultad de Ciencias Físico-Matemáticas, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y 18 Sur S/N, Puebla, Puebla, 72570, México

* Corresponding author: María Guadalupe Morales.

Received  December 2019 Revised  February 2020 Published  April 2020

Fund Project: The second author is supported by Grant GA 17-03224S of the Czech Science Foundation. The third author acknowledges partial support by SNI-CONACYT and VIEP-BUAP

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.

Keywords: Distributional Henstock-Kurzweil integral, convolution, Henstock-Kurzweil integrable distribution, Riemann-Liouville fractional differential operator, Riemann-Liouville fractional integral operator.
Mathematics Subject Classification: Primary: 26A33, 26A39; Secondary: 46F12, 47G20.
Citation: María Guadalupe Morales, Zuzana Došlá, Francisco J. Mendoza. Riemann-Liouville derivative over the space of integrable distributions. Electronic Research Archive, 2020, 28 (2) : 567-587. doi: 10.3934/era.2020030
References:
[1]

A. Alexiewicz, Linear functionals on Denjoy-integrable functions, Colloquium Math., 1 (1948), 289-293.  doi: 10.4064/cm-1-4-289-293.  Google Scholar

[2]

A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Phys. A., 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.  Google Scholar

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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.  Google Scholar

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A. Atangana and J. F. Gómez-Aguilar, Fractional derivatives with no-index law property: Application to chaos and statistics, Chaos Solitons Fractals, 114 (2018), 516-535.  doi: 10.1016/j.chaos.2018.07.033.  Google Scholar

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A. Atangana and 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 (2018), 1502-1523.  doi: 10.1002/num.22195.  Google Scholar

[6]

A. Atangana and S. Qureshi, Modeling attractors of chaotic dynamical systems with fractal-fractional operators, Chaos Solitons Fractals, 123 (2019), 320-337.  doi: 10.1016/j.chaos.2019.04.020.  Google Scholar

[7]

A. Atangana and A. Shafiq, Differential and integral operators with constant fractional order and variable fractional dimension, Chaos Solitons Fractals, 127 (2019), 226-243.  doi: 10.1016/j.chaos.2019.06.014.  Google Scholar

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R. G. Bartle, A Modern Theory of Integration, Graduate Studies in Mathematics, 32. American Mathematical Society, Providence, RI, 2001. doi: 10.1090/gsm/032.  Google Scholar

[9]

D. A. Benson, The Fractional Advection-Dispersion Equation: Development and Application, Ph.D. thesis, University of Nevada in Reno, 1998. Google Scholar

[10]

B. Bongiorno, Relatively weakly compact sets in the Denjoy space, J. Math. Study, 27 (1994), 37-44.   Google Scholar

[11]

B. Bongiorno and T. V. Panchapagesan, On the Alexiewicz topology of the Denjoy space, Real Anal. Exchange, 21 (1995/96), 604-614.  doi: 10.2307/44152670.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14] I. M. Gel'fand and G. E. Shilov, Generalized Functions, Volume 1: Properties and Operations, Academic Press, New York-London, 1964.  doi: 10.1090/chel/377.  Google Scholar
[15]

W. G. Glöckle and T. F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophys. J., 68 (1995), 46-53.  doi: 10.1016/s0006-3495(95)80157-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

C. K. Li, Several results of fractional derivatives in $D'(R_+)$, Fract. Calc. Appl. Anal., 18 (2015), 192-207.  doi: 10.1515/fca-2015-0013.  Google Scholar

[23]

R. Marks and M. Hall, Differintegral interpolation from a bandlimited signal's samples, IEEE Trans. Acoust., Speech, Signal Processing, 29 (1981), 872-877.  doi: 10.1109/tassp.1981.1163636.  Google Scholar

[24]

R. M. McLeod, The Generalized Riemann Integral, Carus Math. Monographs, 20. Mathematical Association of America, Washington, D.C., 1980. doi: 10.5948/upo9781614440208.  Google Scholar

[25]

E. J. McShane, A unified theory of integration, Amer. Math. Monthly, 80 (1973), 349-359.  doi: 10.1080/00029890.1973.11993291.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

W. Rudin, Functional Analysis, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973.  Google Scholar

[29]

W. Rudin, Representation of functions by convolutions, J. Math. Mech., 7 (1958), 103-115.  doi: 10.1512/iumj.1958.7.57009 .  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

E. Talvila, Convolutions with the continuous primitive integral, Abstr. Appl. Anal., 2009 (2009), Art. ID 307404, 18 pp. doi: 10.1155/2009/307404.  Google Scholar

[33]

E. Talvila, Henstock-Kurzweil Fourier transforms, Illinois J. Math., 46 (2002), 1207-1226.  doi: 10.1215/ijm/1258138475.  Google Scholar

[34]

E. Talvila, The distributional Denjoy integral, Real Anal. Exchange, 33 (2008), 51-82.  doi: 10.14321/realanalexch.33.1.0051.  Google Scholar

[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.  Google Scholar

[36]

G. J. Ye and W. Liu, The distributional Henstock-Kurzweil integral and applications, Monatsh. Math., 181 (2016), 975-989.  doi: 10.1007/s00605-015-0853-1.  Google Scholar

[37]

G. J. Ye and W. Liu, The distributional Henstock-Kurzweil integral and applications: A survey, J. Math. Study, 49 (2016), 433-448.  doi: 10.4208/jms.v49n4.16.06.  Google Scholar

[38]

H. Yépez-Martínez, J. F. Gómez-Aguilar, I. O. Sosa, J. M. Reyes and J. Torres-Jiménez, The Feng's first integral method applied to the nonlinear mKdV space-time fractional partial differential equation, Rev. Mex. Fís., 62 (2016), 310-316.   Google Scholar

[39]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, N.J., 2014. doi: 10.1142/9069.  Google Scholar

show all references

References:
[1]

A. Alexiewicz, Linear functionals on Denjoy-integrable functions, Colloquium Math., 1 (1948), 289-293.  doi: 10.4064/cm-1-4-289-293.  Google Scholar

[2]

A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Phys. A., 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.  Google Scholar

[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.  Google Scholar

[4]

A. Atangana and J. F. Gómez-Aguilar, Fractional derivatives with no-index law property: Application to chaos and statistics, Chaos Solitons Fractals, 114 (2018), 516-535.  doi: 10.1016/j.chaos.2018.07.033.  Google Scholar

[5]

A. Atangana and 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 (2018), 1502-1523.  doi: 10.1002/num.22195.  Google Scholar

[6]

A. Atangana and S. Qureshi, Modeling attractors of chaotic dynamical systems with fractal-fractional operators, Chaos Solitons Fractals, 123 (2019), 320-337.  doi: 10.1016/j.chaos.2019.04.020.  Google Scholar

[7]

A. Atangana and A. Shafiq, Differential and integral operators with constant fractional order and variable fractional dimension, Chaos Solitons Fractals, 127 (2019), 226-243.  doi: 10.1016/j.chaos.2019.06.014.  Google Scholar

[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.  Google Scholar

[9]

D. A. Benson, The Fractional Advection-Dispersion Equation: Development and Application, Ph.D. thesis, University of Nevada in Reno, 1998. Google Scholar

[10]

B. Bongiorno, Relatively weakly compact sets in the Denjoy space, J. Math. Study, 27 (1994), 37-44.   Google Scholar

[11]

B. Bongiorno and T. V. Panchapagesan, On the Alexiewicz topology of the Denjoy space, Real Anal. Exchange, 21 (1995/96), 604-614.  doi: 10.2307/44152670.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14] I. M. Gel'fand and G. E. Shilov, Generalized Functions, Volume 1: Properties and Operations, Academic Press, New York-London, 1964.  doi: 10.1090/chel/377.  Google Scholar
[15]

W. G. Glöckle and T. F. Nonnenmacher, A fractional calculus approach to self-similar protein dynamics, Biophys. J., 68 (1995), 46-53.  doi: 10.1016/s0006-3495(95)80157-8.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

C. K. Li, Several results of fractional derivatives in $D'(R_+)$, Fract. Calc. Appl. Anal., 18 (2015), 192-207.  doi: 10.1515/fca-2015-0013.  Google Scholar

[23]

R. Marks and M. Hall, Differintegral interpolation from a bandlimited signal's samples, IEEE Trans. Acoust., Speech, Signal Processing, 29 (1981), 872-877.  doi: 10.1109/tassp.1981.1163636.  Google Scholar

[24]

R. M. McLeod, The Generalized Riemann Integral, Carus Math. Monographs, 20. Mathematical Association of America, Washington, D.C., 1980. doi: 10.5948/upo9781614440208.  Google Scholar

[25]

E. J. McShane, A unified theory of integration, Amer. Math. Monthly, 80 (1973), 349-359.  doi: 10.1080/00029890.1973.11993291.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

W. Rudin, Functional Analysis, McGraw-Hill Series in Higher Mathematics. McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973.  Google Scholar

[29]

W. Rudin, Representation of functions by convolutions, J. Math. Mech., 7 (1958), 103-115.  doi: 10.1512/iumj.1958.7.57009 .  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

E. Talvila, Convolutions with the continuous primitive integral, Abstr. Appl. Anal., 2009 (2009), Art. ID 307404, 18 pp. doi: 10.1155/2009/307404.  Google Scholar

[33]

E. Talvila, Henstock-Kurzweil Fourier transforms, Illinois J. Math., 46 (2002), 1207-1226.  doi: 10.1215/ijm/1258138475.  Google Scholar

[34]

E. Talvila, The distributional Denjoy integral, Real Anal. Exchange, 33 (2008), 51-82.  doi: 10.14321/realanalexch.33.1.0051.  Google Scholar

[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.  Google Scholar

[36]

G. J. Ye and W. Liu, The distributional Henstock-Kurzweil integral and applications, Monatsh. Math., 181 (2016), 975-989.  doi: 10.1007/s00605-015-0853-1.  Google Scholar

[37]

G. J. Ye and W. Liu, The distributional Henstock-Kurzweil integral and applications: A survey, J. Math. Study, 49 (2016), 433-448.  doi: 10.4208/jms.v49n4.16.06.  Google Scholar

[38]

H. Yépez-Martínez, J. F. Gómez-Aguilar, I. O. Sosa, J. M. Reyes and J. Torres-Jiménez, The Feng's first integral method applied to the nonlinear mKdV space-time fractional partial differential equation, Rev. Mex. Fís., 62 (2016), 310-316.   Google Scholar

[39]

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific Publishing Co. Pte. Ltd., Hackensack, N.J., 2014. doi: 10.1142/9069.  Google Scholar

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