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A new approach to solve Cattaneo-Hristov diffusion model and fractional diffusion equations with Hilfer-Prabhakar derivative

1 Amity Institute of information Technology, Amity University, Rajasthan, Jaipur-303002, India
2 Department of Mathematics, University of Rajasthan, Jaipur-302004, Rajasthan, India
3 Department of Mathematics, Amity University, Rajasthan, Jaipur-303002, India
4 Department of Mathematics, Govt. P. G. College, Hisar, Haryana-125001, India

Special Issues: Recent Advances in Fractional Calculus with Real World Applications

In the present article, we investigate complete Cattaneo-Hristov diffusion (CCHD) equation and fractional diffusion equation in one and two dimensional spaces and find their analytic solution by using Elzaki transform technique under the Dirichlet boundary conditions. The fractional diffusion equation describe by the Hilfer-Prabhakar derivative and established the solution in one and two dimensional spaces by using Elzaki and Fourier Sine transform in terms of Mittag-Leffler function. In this paper, we also establish new results such as Elzaki transform of Caputo-Fabrizio and Hilfer-Prabhakar derivative which will be very helpful to find the analytical solution fractional differential equations.
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References

1. R. Garra, R. Gorenflo, F. polito, et al. Hilfer-Prabhakar derivatives and some applications. Appl. Math. Comput., 242 (2014), 576-589.

2. M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl., 1 (2015), 1-13.

3. J. Losada, J. J. Nieto, Properties of a new fractional derivative without singular kernel. Progr. Fract. Differ. Appl., 1 (2015), 87-92.

4. J. Hristov, Approximate solutions to fractional subdiffusion equations. Eur. Phys. J. Spec. Top, 193 (2011), 229-243.

5. J. Hristov, Transient heat diffusion with a non-singular fading memory: From the Cattaneo constitutive equation with Jeffrey's Kernel to the Caputo-Fabrizio time-fractional derivative. Therm. Sci., 20 (2016), 757-762.

6. J. Hristov, Derivation of the fractional dodson equation and beyond: Transient di_usion with a non-singular memory and exponentially fadingout diffusivity. Progr. Fract. Differ. Appl., 3 (2017), 1-16.

7. J. Hristov, On the Atangana-Baleanu derivative and its relation to the fading memory concept: The diffusion equation formulation, Fractional Derivatives with Mittag-Leffler Kernel, Stud. Syst. Decis. Control, Springer, Cham, 194 (2019), 175-193.

8. J. Hristov, Steady-state heat conduction in a medium with spatial non-singular fading memory derivation of caputo-fabrizio space fractional derivative from cattaneo concept with jeffrey's kernel and analytical solutions, Therm. Sci., 21 (2017), 827-839.

9. I. Koca, A. Atangana, Solutions of cattaneo-hristov model of elastic heat diffusion with caputo-fabrizio and atangana-baleanu fractional derivatives, Therm. Sci., 21 (2017), 2299-2305.

10. B. S. T. Alkahtani, A. Atangana, A note on cattaneo-Hristov model with non-singular fading memory. Therm. Sci., 21 (2017), 1-7.

11. J. Hristov, Multiple integral-balance method basic idea and an example with mullins model of thermal grooving. Therm. Sci., 21 (2017), 1555-1560.

12. J. Hristov, The non-linear dodson diffusion equation: Approximate solutions and beyond with formalistic fractionalization. Math. Nat. Sci., 1 (2017), 1-17.

13. J. Hristov, Fourth-order fractional diffusion model of thermal grooving: integral approach to approximate closed form solution of the mullins model. Math. Modell. Nat. Phenom., 13 (2018), 6.

14. J. Hristov, Integral-balance solution to nonlinear subdiffusion equation. Front. Fractional. Calculus., 1 (2017), 71-106.

15. J. Hristov, The heat radiation diffusion equation: Explicit analytical solutions by improved integral-balance method. Therm. Sci., 22 (2018), 777-788.

16. J. Hristov, Integral balance approach to 1-D space-fractional diffusion models, Mathematical Methods in Engineering, Nonlinear Syst. Complex. Springer Cham., 23 (2019), 111-131.

17. J. Hristov, A transient flow of a non-newtonian fluid modelled by a mixed time-space derivative: An improved integral-balance approach, Mathematical Methods in Engineering. Nonlinear Syst. Complex. Springer Cham., 24 (2019), 153-174.

18. T. G. Myers, Optimal exponent heat balance and refined integral methods applied to stefan problems. Int. J. Heat Mass Transfer., 53 (2010), 1119-1127.

19. E. F. D. Goufo, Chaotic processes using the two-parameter derivative with non-singular and non-local kernel: Basic theory and applications. Chaos: An Interdiscip. J. Nonlinear Sci., 26 (2016), 084305.

20. N. Sene, Exponential form for Lyapunov function and stability analysis of the fractional differential equations. J. Math. Comput. Sci., 18 (2018), 388-397.

21. Tarig. M. Elzaki, The new integral transform "ELzaki Transform", Global J. Pure Appl Math., 7 (2011) 57-64.

22. R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, A. Carpinteri and F. Mainardi (Editors): Fractals and Fractional Calculus in Continuum Mechanics, New York: Springer-Verlag Press (1997), 223-276.

23. A. A. Kilbas, M. Saigo, R.K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators. Integr Transfor. Spec Funct., 15 (2004), 31-49.

24. T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the kernel. Yokohama Math. J., 19 (1971), 7-5.

25. I. Podlubny, Matrix approach to discrete fractional calculus ii: Partial fractional differential equations. J. Comput. Phy., 228 (2009), 3131-3153.

26. Y. Ma, F. Zhang, C. Li, The asymptotics of the solutions to the anomalous diffusion equations. Comput. Math. Appl., 66 (2013), 682-692.

27. V. Gill, J. Singh, Y. Singh, Analytical solution of generalized space-time fractional advection-dispersion equation via coupling of Sumudu and Fourier transforms, Frontiers. Phy., 6 (2019), 1-6.

28. N. Sene, Stokes first problem for heated at plate with Atangana-Baleanu fractional derivative. Chaos Solitons Fractals, 117 (2018), 68-75.

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