Generalized Shehu Transform to Ψ -Hilfer-Prabhakar Fractional Derivative and its Regularized Version
Keywords:
Ψ-Prabhakar integral, Ψ-Hilfer-Prabhakar derivative, Ψ-Mittag-Leffler function, Ψ-Shehu transformAbstract
In this manuscript, athours interested on the generalized Shehu transform of Ψ-Riemann-Liouville, Ψ-Caputo, Ψ-Hilfer fractional derivatives. Moreover, Ψ-Prabhakar, Ψ-Hilfer-Prabhakar fractional derivatives and its regularized version presented in terms of the Ψ-Mittag-Leffler type function.
They are also utilised to solve several Cauchy type problems involving Ψ-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.
References
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
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.
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
R. Belgacem, D. Baleanu and A. Bokhari, Shehu transform and applications to Caputo-fractional differential equations, Int. J. Anal. Appl. 6 (2019) 917-927.
D. Brockmann and I.M. Sokolov IM, Levy lights in external force fields: from model to equations, Chem. Phys. 284 (2002) 409-421.
L. Debnath and D. Bhatta, Integral Transforms and Their Applications, Chapman and Hall /CRC, Taylor and Francis Group, New York, 2007.
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
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
R. Garra, R. Goreno, F. Polito and Z. Tomovski, Hilfer-Prabhakar derivative and some applications, Applied Mathematics and Computation, 242(1) (2014) 576-589.
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.
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
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
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.
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.
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.
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
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.
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.
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
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.
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.
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
T.R. Prabhakar, A singular equation with a generalized Mittag-Leffler function in the kernel, Yokohama Mathematical Journal, 19 (1971) 7-15.
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.
S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, New York, Gordon and Breach, 1993.
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.
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
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.
