Document Type : Research Paper


1 Department of Applied Mathematics, Faculty of Mathematical Sciences, Lahijan Branch, Islamic Azad University, Lahijan, Iran.

2 Department of Applied Mathematics, Fouman and Shaft Branch, Islamic Azad University, Fouman, Iran.


In this paper, we proposed a numerical approach to solve a distributed order time fractional COVID 19 virus model. The fractional derivatives are shown in the Caputo-Prabhakar contains generalized Mittag-Leffler Kernel. The coronavirus 19 disease model has 8 Inger diets leading to system of 8 nonlinear ordinary differential equations in this sense, we used the midpoint quadrature method and finite different scheme for solving this problem, our approximation method reduce the distributed order time fractional COVID 19 virus equations to a system of algebraic equations. Finally, to confirm the efficiency and accuracy of this method, we presented some numerical experiments for several values of distributed order. Also, all parameters introduced in the given model are positive parameters.


Main Subjects

  1. Podlubny, I. (1998). Fractional differential equations. Academic Press.
  2. Li, C., & Zeng, F. (2015). Numerical methods for fractional calculus(Vol. 24). CRC Press.‏
  3. Kilbas, A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and applications of fractional differential equations(Vol. 204). Elsevier.
  4. Diethelm, K. (2010). The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Springer Science & Business Media. Springer, Berlin.
  5. Herrmann, R. (2011). Fractional calculus: an introduction for physicists.‏ World Scientific
  6. Chechkin, A. V., Gorenflo, R., & Sokolov, I. M. (2002). Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Physical review E66(4), 046129.‏
  7. Sousa, E., & Li, C. (2015). A weighted finite difference method for the fractional diffusion equation based on the riemann–liouville derivative. Applied numerical mathematics90, 22-37.‏
  8. Babolian, E., Vahidi, A. R., & Shoja, A. (2014). An efficient method for nonlinear fractional differential equations: combination of the Adomian decomposition method and spectral method. Indian journal of pure and applied mathematics45(6), 1017-1028.‏
  9. Yang, S., Xiao, A., & Su, H. (2010). Convergence of the variational iteration method for solving multi-order fractional differential equations. Computers & mathematics with applications60(10), 2871-2879.‏
  10. HosseinNia, S. H., Ranjbar, A., & Momani, S. (2008). Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part. Computers & mathematics with applications56(12), 3138-3149.‏
  11. Bhrawy, A. H., Tharwat, M. M., & Yildirim, A. H. M. E. T. (2013). A new formula for fractional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations. Applied mathematical modelling37(6), 4245-4252.‏
  12. Rahimkhani, P., Ordokhani, Y., & Babolian, E. (2017). Fractional-order bernoulli functions and their applications in solving fractional fredholem–volterra integro-differential equations. Applied numerical mathematics122, 66-81.‏
  13. Rahimkhani, P., Ordokhani, Y., & Babolian, E. (2018). Müntz-legendre wavelet operational matrix of fractional-order integration and its applications for solving the fractional pantograph differential equations. Numerical algorithms77(4), 1283-1305.‏
  14. Rahimkhani, P., Ordokhani, Y., & Babolian, E. (2016). Fractional-order bernoulli wavelets and their applications. Applied mathematical modelling40(17-18), 8087-8107.‏
  15. Mahboob Dana, Z., Najafi, H. S., & Refahi Sheikhani, A. H. (2020). An efficient numerical method for solving Benjamin–Bona–Mahony–Burgers equation using difference scheme. Journal of difference equations and applications26(4), 574-585.‏
  16. Mashayekhi, S., & Razzaghi, M. (2016). Numerical solution of distributed order fractional differential equations by hybrid functions. Journal of computational physics315, 169-181.‏
  17. Chen, H., Lü, S., & Chen, W. (2016). Finite difference/spectral approximations for the distributed order time fractional reaction–diffusion equation on an unbounded domain. Journal of computational physics315, 84-97.‏
  18. Aminikhah, H., Sheikhani, A. H. R., & Rezazadeh, H. (2018). Approximate analytical solutions of distributed order fractional Riccati differential equation. Ain shams engineering journal9(4), 581-588.‏
  19. Aminikhah, H., Sheikhani, A. H. R., Houlari, T., & Rezazadeh, H. (2017). Numerical solution of the distributed-order fractional Bagley-Torvik equation. IEEE/CAA journal of automatica sinica6(3), 760-765.‏ DOI: 1109/JAS.2017.7510646
  20. Mashoof, M., & Sheikhani, A. R. (2017). Simulating the solution of the distributed order fractional differential equations by block-pulse wavelets. UPB Sci. Bull., Ser. A, 79, 193-206.‏
  21. Mashoof, M., Refahi Sheikhani, A. H., & Saberi Najafi, H. (2018). Stability analysis of distributed-order hilfer–prabhakar systems based on inertia theory. Mathematical notes104(1), 74-85.‏
  22. Mashoof, M., Sheikhani, A. R., & NAJA, H. S. (2018). Stability analysis of distributed order hilfer-prabhakar differential equations. Hacettepe journal of mathematics and statistics47(2), 299-315.‏
  23. Li, X. Y., & Wu, B. Y. (2016). A numerical method for solving distributed order diffusion equations. Applied mathematics letters53, 92-99.‏
  24. Li, J., Liu, F., Feng, L., & Turner, I. (2017). A novel finite volume method for the Riesz space distributed-order diffusion equation. Computers & mathematics with applications74(4), 772-783.‏
  25. Fan, W., & Liu, F. (2018). A numerical method for solving the two-dimensional distributed order space-fractional diffusion equation on an irregular convex domain. Applied mathematics letters77, 114-121.‏
  26. Jia, J., & Wang, H. (2018). A fast finite difference method for distributed-order space-fractional partial differential equations on convex domains. Computers & mathematics with applications75(6), 2031-2043.‏
  27. Morgado, M. L., Rebelo, M., Ferras, L. L., & Ford, N. J. (2017). Numerical solution for diffusion equations with distributed order in time using a Chebyshev collocation method. Applied numerical mathematics114, 108-123.‏
  28. Zaky, M. A. (2018). A Legendre collocation method for distributed-order fractional optimal control problems. Nonlinear dynamics91(4), 2667-2681.‏
  29. Hu, X., Liu, F., Anh, V., & Turner, I. (2013). A numerical investigation of the time distributed-order diffusion model. ANZIAM journal55, C464-C478.‏
  30. Ye, H., Liu, F., Anh, V., & Turner, I. (2015). Numerical analysis for the time distributed-order and Riesz space fractional diffusions on bounded domains. IMA journal of applied mathematics80(3), 825-838.‏
  31. Ford, N. J., Morgado, M. L., & Rebelo, M. (2014, June). A numerical method for the distributed order time-fractional diffusion equation. In ICFDA'14 international conference on fractional differentiation and its applications 2014(pp. 1-6). IEEE.‏ 1109/ICFDA.2014.6967389
  32. Gao, G. H., & Sun, Z. Z. (2015). Two alternating direction implicit difference schemes with the extrapolation method for the two-dimensional distributed-order differential equations. Computers & mathematics with applications69(9), 926-948..
  33. Jin, B., Lazarov, R., Sheen, D., & Zhou, Z. (2016). Error estimates for approximations of distributed order time fractional diffusion with nonsmooth data. Fractional calculus and applied analysis19(1), 69-93.‏
  34. Attia, Z. I., Kapa, S., Lopez-Jimenez, F., McKie, P. M., Ladewig, D. J., Satam, G., ... & Friedman, P. A. (2019). Screening for cardiac contractile dysfunction using an artificial intelligence–enabled electrocardiogram. Nature medicine25(1), 70-74.‏
  35. Prabhakar, T. R. (1971). A singular integral equation with a generalized mittag leffler function in the kernel. Retrieved from
  36. Srivastava, H. M., Saxena, R. K., Pogany, T. K., & Saxena, R. (2011). Integral transforms and special functions. Applied mathematics and computation22(7), 487-506.‏
  37. Shariffar, F., & Sheikhani, R. (2019). A new two-stage iterative method for linear systems and its application in solving poisson's equation. International journal of industrial mathematics11(4), 283-291.
  38. Sheikhani, A. H. R., & Mashoof, M. (2017). A collocation method for solving fractional order linear system. Journal of the indonesian mathematical society, 27-42.
  39. Garra, R., Gorenflo, R., Polito, F., & Tomovski, Ž. (2014). Hilfer–prabhakar derivatives and some applications. Applied mathematics and computation242, 576-589.