A numerical method for solving a class of distributed order time-fractional diffusion partial differential equations according to Caputo-Prabhakar fractional derivative

Authors
Alireza Ansari
MohammadHossein Derakhshan
Abstract
In this paper, a time-fractional diffusion equation of distributed order including the Caputo-Prabhakar fractional derivative is studied. We use a numerical method based on the linear B-spline interpolation and finite difference method to study the solutions of these types of fractional equations. Finally, some numerical examples are presented for the performance and accuracy of the proposed numerical method.
Keywords
Fractional derivative
Distributed order time-fractional diffusion equation
Distributed order
Caputo-Prabhakar
Spline approximation
1. Biswas K., Bohannan G., Caponetto R., Lopes A.M., Machado J.A.T., "Fractional-order models of vegetable tissues", Fractional-Order Devices, Springer, (2017) 73-92.

2. Dabiri A., Moghaddam B.P., Machado J.T., "Optimal variable-order fractional PID controllers for dynamical systems", Journal of Computational and Applied Mathematics, 309 (2018) 40-48.

3. Derakhshan M.H., Ansari A., "On Hyers–Ulam stability of fractional differential equations with Prabhakar derivatives", Analysis, 38 (2018) 37-46.

4. Derakhshan M.H., Ansari A., "Fractional Sturm–Liouville problems for Weber fractional derivatives", International Journal of Computer Mathematics, 96 (2019) 217-237.

5. Derakhshan M.H., Ansari A., "Numerical approximation to Prabhakar fractional Sturm–Liouville problem", Computational and Applied Mathematics, 38 (2019) 71.

6. Mainardi F., "Fractional calculus and waves in linear viscoelasticity", an introduction to mathematical models, World Scientific, 2010.

7. Tarasov V.E., Tarasova V.V., "Computation. Macroeconomic models with long dynamic memory: Fractional calculus approach", Applied mathematics and numerical computation, 338 (2018) 466-486.

8. Dehghan M., Abbaszadeh M., "A Legendre spectral element method (SEM) based on the modified bases for solving neutral delay distributed‐order fractional damped diffusion‐wave equation", Mathematical Methods in the Applied Science, 41 (2018) 3476-3494.

9. Sandev T., Tomovski Z., Crnkovic B., "Generalized distributed order diffusion equations with composite time fractional derivative", Computers and mathematics with applications, 73 (2017) 1028-1040.

10. Caputo M., "Linear models of dissipation whose Q is almost frequency independent", Geophysical Journal International, 13 (1967) 529-539.

11. Caputo M., "Elasticita de dissipazione", Zanichelli, Bologna, Italy, 1969.

12. Caputo M., "Distributed order differential equations modelling dielectric induction and diffusion", Fractional Calculus and Applied Analysis, 4 (2001) 421-442.

13. Bagley R., Torvik P., "On the existence of the order domain and the solution of distributed order equations-Part I", International Journal of Applied Mathematics, 2 (2000) 865-882.

14. Hartley T., Lorenzo C., "Fractional system identification: an approach using continuous order-distributions ", Mathematics, 1999.

15. Garra R., Gorenflo R., Polito F., Tomovski Ž., "Hilfer–Prabhakar derivatives and some applications", Applied Mathematics and Computation, 242 (2014) 576-589.

16. Garrappa R., "Numerical evaluation of two and three parameter Mittag-Leffler functions", SIAM Journal on Numerical Analysis, 53 (2015) 1350-1369.

17. Ford N.J., Morgado M.L., Rebelo M., "A numerical method for the distributed order time-fractional diffusion equation", International Conference on Fractional Differentiation and Its Applications, 2014.

18. Kochubei A.N., "Distributed order calculus and equations of ultraslow diffusion", Journal of Mathematical Analysis and Applications, 340 (2008) 252-281.

19. Atanackovic T.M., Oparnica L., Pilipović S., "Semilinear ordinary differential equation coupled with distributed order fractional differential equation", Nonlinear Analysis: Theory, Methods and Applications, 72 (2010) 4101-4114.

20. Umarov S., Gorenflo R., "Cauchy and nonlocal multi-point problems for distributed order pseudo-differential equations: Part one", Journal of Analytical and Applied Pyrolysis, 245 (2005) 449-466.

21. Ye H, Liu F., Anh V., "Compact difference scheme for distributed-order time-fractional diffusion-wave equation on bounded domains", Journal of Computational Physics, 298 (2015) 652-660.

22. Ye H., Liu F., Anh V., Turner I., "Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains", IMA Journal of Applied Mathematics, 80 (2015) 825-838.

23. Ford N.J., Luisa Morgado M., Rebelo M., "An implicit finite difference approximation for the solution of the diffusion equation with distributed order in time", Electronic transactions on numerical analysis, 44 (2015) 289-305.

24. Morgado M.L., Rebelo M., "Numerical approximation of distributed order reaction–diffusion equations", Journal of Computational and Applied Mathematics, 275 (2015) 216-227.

25. Morgado M.L., Rebelo M., Ferras L.L., Ford N.J., "Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method", Applied Numerical Mathematics, 114 (2017) 108-123.

26. Van Bockstal K., "Existence and uniqueness of a weak solution to a non-automonous time-fractional diffusion equation (of distributed order) ", Applied Mathematics Letters, 109 (2020) 106540.

27. Rahimkhani P., Ordokhani Y., Lima P., "An improved composite collocation method for distributed-order fractional differential equations based on fractional Chelyshkov wavelets", Applied Numerical Mathematics, 145 (2019) 1-27.

28. Gorenflo R., Luchko Y., Stojanović M., "Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density", Fractional Calculus and Applied Analysis, 16 (2013) 297-316.

29. Garra R., Garrappa R., Simulation N. "The Prabhakar or three parameter Mittag–Leffler function: Theory and application", Communications in Nonlinear Science and Numerical Simulation, 56 (2018) 314-329.

30. Eshaghi S., Ansari A., "Lyapunov inequality for fractional differential equations with Prabhakar derivative", Mathematical Inequalities and Applications, 19 (2016) 349-358.

31. Garrappa R., Mainardi F., Guido M., "Models of dielectric relaxation based on completely monotone functions", Fractional Calculus and Applied Analysis, 19 (2016) 1105-1160.

32. Stanislavsky A., Weron K., "A typical case of the dielectric relaxation responses and its fractional kinetic equation", Fractional Calculus and Applied Analysis, 19 (2016) 212.

33. Miskinis P., "The Havriliak–Negami susceptibility as a nonlinear and nonlocal process", Physica Scripta, 2009 (2009) 014019.

34. Górska K., Horzela A., Bratek L., Penson K., Dattoli G., "The probability density function for the Havriliak-Negami relaxation", Journal of Physics A Mathematical and Theoretical, 51 (2016) 135202.

35. Giusti A., Colombaro I., "Prabhakar-like fractional viscoelasticity", Communications in Nonlinear Science and Numerical Simulation, 56 (2018) 138-143.

36. Kilbas A.A., Srivastava H.M., Trujillo J.J., "Theory and applications of fractional differential equations", Elsevier Science Limited, 2006.

37. Kilbas A.A., Saigo M., Saxena R.K., "Generalized Mittag-Leffler function and generalized fractional calculus operators", Integral Transforms and Special Functions, 15 (2004) 31-49.