Research Article
BibTex RIS Cite

Generalized Shehu Transform to $\Psi$-Hilfer-Prabhakar Fractional Derivative and its Regularized Version

Year 2022, Volume: 6 Issue: 3, 364 - 379, 30.09.2022

Sachın Magar Ahmed Hamoud Amol Khandagale Kirtiwant Ghadle

https://doi.org/10.31197/atnaa.1032207

Abstract

In this manuscript, athours interested on the generalized Shehu transform of $\Psi$-Riemann-Liouville, $\Psi$-Caputo, $\Psi$-Hilfer fractional derivatives. Moreover, $\Psi$-Prabhakar, $\Psi$-Hilfer-Prabhakar fractional derivatives and its regularized version presented in terms of the $\Psi$-Mittag-Leffler type function.
They are also utilised to solve several Cauchy type problems involving $\Psi$-Hilfer-Prabhakar fractional derivatives and their regularised form, such as the space-time fractional advection-dispersion equation and the generalized fractional free-electron laser (FEL) equation.

Keywords

$\Psi$-Prabhakar integral $\Psi$-Hilfer-Prabhakar derivative $\Psi$-Mittag-Leffler function $\Psi$-Shehu transform

References

  • [1] R.A. Almeida, Caputo fractional derivative of a function with respect to another function, Communications in Nonlinear Science and Numerical Simulation, 44 (2017) 460-481. https://doi.org/10.1016/j.cnsns.2016.09.006
  • [2] R. Belgacem, D. Baleanu and A. Bokharia, Shehu transform and applications to Caputo-fractional differential equations, International Journal of Analysis and Applications, 6 (2019) 917-927.
  • [3] A. Bokharia, D. Baleanu and R. Belgacema, Application of Shehu transform to Atangana-Baleanu derivatives, J. Math. Computer Sci., 20 (2020) 101-107. http://dx.doi.org/10.22436/jmcs.020.02.03
  • [4] R. Belgacem, D. Baleanu and A. Bokhari, Shehu transform and applications to Caputo-fractional differential equations, Int. J. Anal. Appl. 6 (2019) 917-927.
  • [5] D. Brockmann and I.M. Sokolov IM, Levy lights in external force fields: from model to equations, Chem. Phys. 284 (2002) 409-421.
  • [6] L. Debnath and D. Bhatta, Integral Transforms and Their Applications, Chapman and Hall /CRC, Taylor and Francis Group, New York, 2007.
  • [7] R. Garra and R. Garrappa, The Prabhakar or Three Parameter Mittag-Leffler function: Theory and application., Commu- nications in Nonlinear Science and Numerical Simulation, 56 (2018) 314-329. https://doi.org/10.1016/j.cnsns.2017.08.018
  • [8] K.P. Ghadle, S.K. Magar and P.V. Dole, A new Sumudu type integral transform an its applications: Progress in Fractional Di?erentiation and Applications, 7(3) (2021) 145-152. http://dx.doi.org/10.18576/pfda/070302
  • [9] R. Garra, R. Goreno, F. Polito and Z. Tomovski, Hilfer-Prabhakar derivative and some applications, Applied Mathematics and Computation, 242(1) (2014) 576-589.
  • [10] V. Gill, J. Singh and Y. Singh, Analytical solution of generalized space-time fractional advection-dispersion equation via coupling of Sumudu and Fourier transforms, Frontiers in Physics, (2019). https://doi:10.3389/fphy.2018.00151.
  • [11] A.A. Hamoud, Existence and uniqueness of solutions for fractional neutral Volterra-Fredholm integro differential equations, Advances in the Theory of Nonlinear Analysis and its Application, 4 (2020) 321-331. https://doi.org/10.31197/atnaa.799854
  • [12] A.A. Hamoud, N.M. Mohammed and K.P. Ghadle, Existence and uniqueness results for Volterra-Fredholm integro differential equations, Advances in the Theory of Nonlinear Analysis and its Application, 4(4) (2020) 361-372. https://doi.org/10.31197/atnaa.703984
  • [13] A.A. Hamoud and K.P. Ghadle, Some new existence, uniqueness and convergence results for fractional Volterra-Fredholm integro-differential equations, Journal of Applied and Computational Mechanics, 5(1) (2019) 58-69.
  • [14] A.A. Hamoud, N.M. Mohammed and K.P. Ghadle, Existence, uniqueness and stability results for nonlocal fractional nonlinear Volterra-Fredholm integro differential equations, Discontinuity, Nonlinearity, and Complexity, 11(2) (2022) 343- 352.
  • [15] A.A. Hamoud, A.A. Sharif, K.P. Ghadle, Existence, uniqueness and stability results of fractional Volterra-Fredholm integro di?erential equations of ψ-Hilfer type, Discontinuity, Nonlinearity, and Complexity, 10(03) (2021), 535-545.
  • [16] F. Jarad and T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete and Continuous Dynamical Systems Series S, 2019 (2019) 1775-1786. https://doi.org/10.3934/dcdss.2020039
  • [17] A. Khalouta and A. Kadem, A New combination method for solving nonlinear Liouville-Caputo and Caputo-Fabrizio time-fractional Reaction-Diffusion-Convection equations, Malaysian Journal of Mathematical Sciences, 15(2) (2021) 199- 215.
  • [18] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam, London, New York, Elsevier (North-Holland) Science Publishers, (North-Holland Mathematical Studies), 204, 2006.
  • [19] A.A. Kilbas, M. Saigo and R.K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms and Special Functions, 15 (2004) 31-49. https://doi.org/10.1080/10652460310001600717
  • [20] S.K. Magar, P.V. Dole and K.P. Ghadle, Prabhakar and Hilfer-Prabhakar fractional derivatives in the settiong of Ψ- fractional calculus and its applications, Kragujevac Journal of Mathematics, 48(4) (2024) 515-533.
  • [21] S. Maitama and W. Zhao, New integral transform: Shehu transform a generalization of Sumudu and Laplace transforms for solving differential equations , International Journal of Analysis and Applications, 17 (2019) 167-190.
  • [22] R.R. Nigmatullin, A.A. Khamzin and D. Baleanu, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms and Special Functions, 39 (2004) 2983-2992. https://doi.org/10.1080/10652460310001600717
  • [23] T.R. Prabhakar, A singular equation with a generalized Mittag-Leffler function in the kernel, Yokohama Mathematical Journal, 19 (1971) 7-15.
  • [24] B. Rachid, B. Ahmed and S. Boualem, Shehu transform of Hilfer-Prabhakar fractional derivatives and applications on some Cauchy type problems, Advances in the Theory of Nonlinear Analysis and its Application, 5(2) (2021) 203-214.
  • [25] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, New York, Gordon and Breach, 1993.
  • [26] R.K. Saxena and M. Saigo, Certain properties of fractional calculus operators associated with generalized Mittag-Leffler function, Fractional Calculus and Applied Analysis, 8 (2005) 1-14.
  • [27] J.V.D. Sousa and de E.C. Oliveira, On the Ψ-Hilfer fractional derivative, Communications in Nonlinear Science and Numerical Simulation, 60 (2018) 72-91. https://doi.org/10.1016/j.cnsns.2018.01.0
  • [28] T.G. Thange and A.R. Gade, Fractional Shehu transform and its applications, South East Asian J. of Mathematics and Mathematical Sciences, 17(2) (2021) 1-14.

Year 2022, Volume: 6 Issue: 3, 364 - 379, 30.09.2022

Sachın Magar Ahmed Hamoud Amol Khandagale Kirtiwant Ghadle

https://doi.org/10.31197/atnaa.1032207

Abstract

References

  • [1] R.A. Almeida, Caputo fractional derivative of a function with respect to another function, Communications in Nonlinear Science and Numerical Simulation, 44 (2017) 460-481. https://doi.org/10.1016/j.cnsns.2016.09.006
  • [2] R. Belgacem, D. Baleanu and A. Bokharia, Shehu transform and applications to Caputo-fractional differential equations, International Journal of Analysis and Applications, 6 (2019) 917-927.
  • [3] A. Bokharia, D. Baleanu and R. Belgacema, Application of Shehu transform to Atangana-Baleanu derivatives, J. Math. Computer Sci., 20 (2020) 101-107. http://dx.doi.org/10.22436/jmcs.020.02.03
  • [4] R. Belgacem, D. Baleanu and A. Bokhari, Shehu transform and applications to Caputo-fractional differential equations, Int. J. Anal. Appl. 6 (2019) 917-927.
  • [5] D. Brockmann and I.M. Sokolov IM, Levy lights in external force fields: from model to equations, Chem. Phys. 284 (2002) 409-421.
  • [6] L. Debnath and D. Bhatta, Integral Transforms and Their Applications, Chapman and Hall /CRC, Taylor and Francis Group, New York, 2007.
  • [7] R. Garra and R. Garrappa, The Prabhakar or Three Parameter Mittag-Leffler function: Theory and application., Commu- nications in Nonlinear Science and Numerical Simulation, 56 (2018) 314-329. https://doi.org/10.1016/j.cnsns.2017.08.018
  • [8] K.P. Ghadle, S.K. Magar and P.V. Dole, A new Sumudu type integral transform an its applications: Progress in Fractional Di?erentiation and Applications, 7(3) (2021) 145-152. http://dx.doi.org/10.18576/pfda/070302
  • [9] R. Garra, R. Goreno, F. Polito and Z. Tomovski, Hilfer-Prabhakar derivative and some applications, Applied Mathematics and Computation, 242(1) (2014) 576-589.
  • [10] V. Gill, J. Singh and Y. Singh, Analytical solution of generalized space-time fractional advection-dispersion equation via coupling of Sumudu and Fourier transforms, Frontiers in Physics, (2019). https://doi:10.3389/fphy.2018.00151.
  • [11] A.A. Hamoud, Existence and uniqueness of solutions for fractional neutral Volterra-Fredholm integro differential equations, Advances in the Theory of Nonlinear Analysis and its Application, 4 (2020) 321-331. https://doi.org/10.31197/atnaa.799854
  • [12] A.A. Hamoud, N.M. Mohammed and K.P. Ghadle, Existence and uniqueness results for Volterra-Fredholm integro differential equations, Advances in the Theory of Nonlinear Analysis and its Application, 4(4) (2020) 361-372. https://doi.org/10.31197/atnaa.703984
  • [13] A.A. Hamoud and K.P. Ghadle, Some new existence, uniqueness and convergence results for fractional Volterra-Fredholm integro-differential equations, Journal of Applied and Computational Mechanics, 5(1) (2019) 58-69.
  • [14] A.A. Hamoud, N.M. Mohammed and K.P. Ghadle, Existence, uniqueness and stability results for nonlocal fractional nonlinear Volterra-Fredholm integro differential equations, Discontinuity, Nonlinearity, and Complexity, 11(2) (2022) 343- 352.
  • [15] A.A. Hamoud, A.A. Sharif, K.P. Ghadle, Existence, uniqueness and stability results of fractional Volterra-Fredholm integro di?erential equations of ψ-Hilfer type, Discontinuity, Nonlinearity, and Complexity, 10(03) (2021), 535-545.
  • [16] F. Jarad and T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete and Continuous Dynamical Systems Series S, 2019 (2019) 1775-1786. https://doi.org/10.3934/dcdss.2020039
  • [17] A. Khalouta and A. Kadem, A New combination method for solving nonlinear Liouville-Caputo and Caputo-Fabrizio time-fractional Reaction-Diffusion-Convection equations, Malaysian Journal of Mathematical Sciences, 15(2) (2021) 199- 215.
  • [18] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam, London, New York, Elsevier (North-Holland) Science Publishers, (North-Holland Mathematical Studies), 204, 2006.
  • [19] A.A. Kilbas, M. Saigo and R.K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms and Special Functions, 15 (2004) 31-49. https://doi.org/10.1080/10652460310001600717
  • [20] S.K. Magar, P.V. Dole and K.P. Ghadle, Prabhakar and Hilfer-Prabhakar fractional derivatives in the settiong of Ψ- fractional calculus and its applications, Kragujevac Journal of Mathematics, 48(4) (2024) 515-533.
  • [21] S. Maitama and W. Zhao, New integral transform: Shehu transform a generalization of Sumudu and Laplace transforms for solving differential equations , International Journal of Analysis and Applications, 17 (2019) 167-190.
  • [22] R.R. Nigmatullin, A.A. Khamzin and D. Baleanu, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms and Special Functions, 39 (2004) 2983-2992. https://doi.org/10.1080/10652460310001600717
  • [23] T.R. Prabhakar, A singular equation with a generalized Mittag-Leffler function in the kernel, Yokohama Mathematical Journal, 19 (1971) 7-15.
  • [24] B. Rachid, B. Ahmed and S. Boualem, Shehu transform of Hilfer-Prabhakar fractional derivatives and applications on some Cauchy type problems, Advances in the Theory of Nonlinear Analysis and its Application, 5(2) (2021) 203-214.
  • [25] S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, New York, Gordon and Breach, 1993.
  • [26] R.K. Saxena and M. Saigo, Certain properties of fractional calculus operators associated with generalized Mittag-Leffler function, Fractional Calculus and Applied Analysis, 8 (2005) 1-14.
  • [27] J.V.D. Sousa and de E.C. Oliveira, On the Ψ-Hilfer fractional derivative, Communications in Nonlinear Science and Numerical Simulation, 60 (2018) 72-91. https://doi.org/10.1016/j.cnsns.2018.01.0
  • [28] T.G. Thange and A.R. Gade, Fractional Shehu transform and its applications, South East Asian J. of Mathematics and Mathematical Sciences, 17(2) (2021) 1-14.
There are 28 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Sachın Magar 0000-0002-8036-356X

Ahmed Hamoud 0000-0002-8877-7337

Amol Khandagale This is me 0000-0002-8028-4335

Kirtiwant Ghadle 0000-0003-3205-5498

Publication Date September 30, 2022
Published in Issue Year 2022 Volume: 6 Issue: 3

Cite