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Year 2021, Volume: 5 Issue: 2, 203 - 214, 30.06.2021

Rachid Belgacem Ahmed Bokhari Boualem Sadaoui

https://doi.org/10.31197/atnaa.828468
Cited By: 1

Abstract

References

  • [1] Om.P. Agrawal, Fractional optimal control of a distributed system using eigenfunctions, ASME J. Comput. Nonlinear Dyn.3(2) (2008), 021204 (6 pages).
  • [2] F.B.M. Belgacem, A.A. Karaballi, S.L. Kalla, Analytical investigations of the Sumudu transform and applications to integral production equations, Mathematical Problems in Engineering, 3, (2003), 103-118.
  • [3] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus: Models and Numerical Methods, Series on Complexity, Non linearity and Chaos, World Scientific Publishing, Boston, Mass, USA, 2012.
  • [4] F.B.M. Belgacem, A.A. Karaballi, Sumudu transform fundamental properties investigations and applications, Journal of Applied Mathematics and Stochastic Analysis, (2006); 2006:Article ID 91083:1-23.
  • [5] R. Belgacem, D. Baleanu, A. Bokhari, Shehu Transform and Applications to Caputo-Fractional Differential Equations, Int.J. Anal. Appl. 6, (2019), 917-927.
  • [6] A. Bokhari, D. Baleanu, R. Belgacem, Application of Shehu transform to Atangana-Baleanu derivatives, J. Math. Comput.SCI-JM, 20, (2019), 101-107.
  • [7] D. Brockmann, I.M. Sokolov IM, Levy flights in external force fields: from model to equations, Chem. Phys. 284, (2002), 409?421. https://doi.org/10.1016/S0301-0104(02)00671-7.
  • [8] M. Caputo, Linear model of dissipation whose Q is almost frequency independent-II, Geophysical Journal of the Royal Astronomical Society, 13, (1967), 529-539.
  • [9] A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, McGraw-Hill, New York. 1955.
  • [10] R. Garra, R. Goreno, F. Polito, Z. Tomovski, Hilfer-Prabhakar Derivative and Some Applications, Appl. Math. Comput.1, 242, (2014), 576-589.
  • [11] R. Garra, R. Garrappa, The Prabhakar or three parameter Mittag-Leffer function: theory and application. Communications in Nonlinear Science and Numerical Simulation, 56 (2018), 314-329.
  • [12] V. Gill, J. Singh, Y. Singh, Analytical solution of generalized space-time fractional advection-dispersion equation via coupling of Sumudu and Fourier transforms, Front Phys. 2019. https://doi:10.3389/fphy.2018.00151.
  • [13] I.U. Haq, Z. Ullah , Natural decomposition method and coupled systems of nonlinear fractional order partial differential equations, Results in Nonlinear Analysis, 3, (2020), 35-44.
  • [14] R. Hilfer, Threefold introduction to fractional derivatives, Anomalous transport, foundations and application Wiley-VCH, Weinheim, Germany, (2008), 17-73.
  • [15] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 2006.
  • [16] F. Liu, P. Zhuang, K. Burrage, Numerical methods and analysis for a class of fractional advection-dispersion models, Computers and Mathematics with Applications, 64 (2012), 2990-3007.
  • [17] F. Mainardi, Fractional Calculus and Waves in Linear Visco-elasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.
  • [18] K.S. Miller, B. Ross, An Introduction to the Fractional Integrals and Derivatives, Theory and Applications. New York: Willey, 1993.
  • [19] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic, New York, 1974.
  • [20] S.K. Panchal, P.V. Dole, A.D. Khandagale, k-Hilfer-Prabhakar Fractional Derivatives and Applications, Indian journal of mathematics, 59, (2017), 367-383.
  • [21] S.K. Panchal, A.D. Khandagale, P.V. Dole, Sumudu Transform of Hilfer-Prabhakar Fractional Derivatives and with Applications, Proceeding of National Conference on Recent Trends in Mathematics, 1, (2017), 60-66.
  • [22] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego. 1999.
  • [23] T.R. Prabhakar,A singular integral equation with a generalized Mittag-Lefler function in the kernel, Yokohama Math. J.19 (1971), 7-15.
  • [24] S.G. Samko, A.A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives-Theory and Applications, Amsterdam:Gordon and Breach Science Publishers, 1993.
  • [25] S. Maitama, W. Zhao, New Integral Transform: Shehu Transform a Generalization of Sumudu and Laplace Transform for Solving Differential Equations, Int. J. Anal. Appl. 17, (2019), 167-190.
  • [26] Y. Singh, V. Gill, S. Kundu, D. Kumar, On the Elzaki transform and its applications in fractional free electron laser equation, Acta Univ. Sapientiae Mathematica, 11, (2019), 419-429.
  • [27] A. Wiman, Über den fundamental satz in der teorie der funktionen E α (x), Acta Math. 29, (1905) , 191-201.
  • [28] D. Ziane, R. Belgacem, A. Bokhari, A new modified Adomian decomposition method for non linear partial differential equations, Open J. Math. Anal. 3, (2019), 81-90.

Shehu Transform of Hilfer-Prabhakar Fractional Derivatives and Applications on some Cauchy Type Problems

Year 2021, Volume: 5 Issue: 2, 203 - 214, 30.06.2021

Rachid Belgacem Ahmed Bokhari Boualem Sadaoui

https://doi.org/10.31197/atnaa.828468
Cited By: 1

Abstract

In this paper, we are interested on the Shehu transform of both Prabhakar and Hilfer–Prabhakar fractional derivative and its regularized version. These results are presented in terms of Mittag-Leffler type function and also utilized to obtain the solutions of some Cauchy type problems, such as Space-time Fractional Advection-Dispersion equation and Generalized fractional Free Electron Laser (FEL) equation, at which Hilfer-Prabhakar fractional derivative of fractional order and its regularized version are involved.

Keywords

Hilfer–Prabhakar derivatives Prabhakar integrals Mittag–Leffler functions Shehu transform

References

  • [1] Om.P. Agrawal, Fractional optimal control of a distributed system using eigenfunctions, ASME J. Comput. Nonlinear Dyn.3(2) (2008), 021204 (6 pages).
  • [2] F.B.M. Belgacem, A.A. Karaballi, S.L. Kalla, Analytical investigations of the Sumudu transform and applications to integral production equations, Mathematical Problems in Engineering, 3, (2003), 103-118.
  • [3] D. Baleanu, K. Diethelm, E. Scalas, J.J. Trujillo, Fractional Calculus: Models and Numerical Methods, Series on Complexity, Non linearity and Chaos, World Scientific Publishing, Boston, Mass, USA, 2012.
  • [4] F.B.M. Belgacem, A.A. Karaballi, Sumudu transform fundamental properties investigations and applications, Journal of Applied Mathematics and Stochastic Analysis, (2006); 2006:Article ID 91083:1-23.
  • [5] R. Belgacem, D. Baleanu, A. Bokhari, Shehu Transform and Applications to Caputo-Fractional Differential Equations, Int.J. Anal. Appl. 6, (2019), 917-927.
  • [6] A. Bokhari, D. Baleanu, R. Belgacem, Application of Shehu transform to Atangana-Baleanu derivatives, J. Math. Comput.SCI-JM, 20, (2019), 101-107.
  • [7] D. Brockmann, I.M. Sokolov IM, Levy flights in external force fields: from model to equations, Chem. Phys. 284, (2002), 409?421. https://doi.org/10.1016/S0301-0104(02)00671-7.
  • [8] M. Caputo, Linear model of dissipation whose Q is almost frequency independent-II, Geophysical Journal of the Royal Astronomical Society, 13, (1967), 529-539.
  • [9] A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, McGraw-Hill, New York. 1955.
  • [10] R. Garra, R. Goreno, F. Polito, Z. Tomovski, Hilfer-Prabhakar Derivative and Some Applications, Appl. Math. Comput.1, 242, (2014), 576-589.
  • [11] R. Garra, R. Garrappa, The Prabhakar or three parameter Mittag-Leffer function: theory and application. Communications in Nonlinear Science and Numerical Simulation, 56 (2018), 314-329.
  • [12] V. Gill, J. Singh, Y. Singh, Analytical solution of generalized space-time fractional advection-dispersion equation via coupling of Sumudu and Fourier transforms, Front Phys. 2019. https://doi:10.3389/fphy.2018.00151.
  • [13] I.U. Haq, Z. Ullah , Natural decomposition method and coupled systems of nonlinear fractional order partial differential equations, Results in Nonlinear Analysis, 3, (2020), 35-44.
  • [14] R. Hilfer, Threefold introduction to fractional derivatives, Anomalous transport, foundations and application Wiley-VCH, Weinheim, Germany, (2008), 17-73.
  • [15] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 2006.
  • [16] F. Liu, P. Zhuang, K. Burrage, Numerical methods and analysis for a class of fractional advection-dispersion models, Computers and Mathematics with Applications, 64 (2012), 2990-3007.
  • [17] F. Mainardi, Fractional Calculus and Waves in Linear Visco-elasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.
  • [18] K.S. Miller, B. Ross, An Introduction to the Fractional Integrals and Derivatives, Theory and Applications. New York: Willey, 1993.
  • [19] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic, New York, 1974.
  • [20] S.K. Panchal, P.V. Dole, A.D. Khandagale, k-Hilfer-Prabhakar Fractional Derivatives and Applications, Indian journal of mathematics, 59, (2017), 367-383.
  • [21] S.K. Panchal, A.D. Khandagale, P.V. Dole, Sumudu Transform of Hilfer-Prabhakar Fractional Derivatives and with Applications, Proceeding of National Conference on Recent Trends in Mathematics, 1, (2017), 60-66.
  • [22] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego. 1999.
  • [23] T.R. Prabhakar,A singular integral equation with a generalized Mittag-Lefler function in the kernel, Yokohama Math. J.19 (1971), 7-15.
  • [24] S.G. Samko, A.A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives-Theory and Applications, Amsterdam:Gordon and Breach Science Publishers, 1993.
  • [25] S. Maitama, W. Zhao, New Integral Transform: Shehu Transform a Generalization of Sumudu and Laplace Transform for Solving Differential Equations, Int. J. Anal. Appl. 17, (2019), 167-190.
  • [26] Y. Singh, V. Gill, S. Kundu, D. Kumar, On the Elzaki transform and its applications in fractional free electron laser equation, Acta Univ. Sapientiae Mathematica, 11, (2019), 419-429.
  • [27] A. Wiman, Über den fundamental satz in der teorie der funktionen E α (x), Acta Math. 29, (1905) , 191-201.
  • [28] D. Ziane, R. Belgacem, A. Bokhari, A new modified Adomian decomposition method for non linear partial differential equations, Open J. Math. Anal. 3, (2019), 81-90.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Rachid Belgacem This is me 0000-0002-1697-4075

Ahmed Bokhari This is me 0000-0002-0402-5542

Boualem Sadaoui 0000-0002-4127-701X

Publication Date June 30, 2021
Published in Issue Year 2021 Volume: 5 Issue: 2

Cite

Cited By

Operational calculus for Hilfer-Prabhakar operator Applications to inverse problems
Physica Scripta
https://doi.org/10.1088/1402-4896/acf170