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

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

Kaynakça

1. [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. [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. [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. [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. [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. [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. [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. [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. [9] A. Erdelyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher Transcendental Functions, McGraw-Hill, New York. 1955.
10. [10] R. Garra, R. Goreno, F. Polito, Z. Tomovski, Hilfer-Prabhakar Derivative and Some Applications, Appl. Math. Comput.1, 242, (2014), 576-589.
11. [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. [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. [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. [14] R. Hilfer, Threefold introduction to fractional derivatives, Anomalous transport, foundations and application Wiley-VCH, Weinheim, Germany, (2008), 17-73.
15. [15] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 2006.
16. [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. [17] F. Mainardi, Fractional Calculus and Waves in Linear Visco-elasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.
18. [18] K.S. Miller, B. Ross, An Introduction to the Fractional Integrals and Derivatives, Theory and Applications. New York: Willey, 1993.
19. [19] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic, New York, 1974.
20. [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. [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. [22] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego. 1999.
23. [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. [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. [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. [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. [27] A. Wiman, Über den fundamental satz in der teorie der funktionen E α (x), Acta Math. 29, (1905) , 191-201.
28. [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.