Mohammadhossein Derakhshan Azim Aminataei


We study two numerical techniques based on the homotopy perturba tion transform method (HPTM) and the fractional Adams–Bashforth method (FABM) for solving a class of nonlinear time-fractional differential equations involving the Caputo–Prabhakar fractional derivatives. In this manuscript, the convergence for numerical solutions obtained using HPTM and the con vergence and stability for numerical solutions obtained using FABM are inves tigated. We compare the solutions obtained by the HPTM and the FABM for some nonlinear time-fractional differential equations. Moreover, some numer ical examples are demonstrated in order to show the validity and reliability of the suggested methods.

Article Details


onlinear time-fractional differential equations;, Fractional Homotopy perturbation transform method;, Fractional Adams–Bashforth method;, Caputo–Prabhakar fractional derivative.

1. An, J., Van Hese, E. and Baes, M. Phase-space consistency of stellar dynamical models determined by separable augmented densities, Mon. Not. R. Astron. Soc. 422(1) (2012), 652–664.
2. Atangana, A. and Owolabi, K.M. New numerical approach for fractional differential equations, Math. Model. Nat. Phenom, 13(1) (2018), Paper No. 3, 21 pp.
3. Balcl, E., Ozt¨urk, ¨ I. and Kartal, S. ˙ Dynamical behaviour of fractional order tumor model with Caputo and conformable fractional derivative, Chaos
Solitons Fractals, 123 (2019), 43–51.
4. Derakhshan, M.H., Ahmadi Darani, M., Ansari, A. and Khoshsiar Ghaziani, R. On asymptotic stability of Prabhakar fractional differential systems, Comput. Methods Differ. Equ. 4(4) (2016), 276–284.
5. Derakhshan, M.H. and Ansari, A. On Hyers-Ulam stability of fractional differential equations with Prabhakar derivatives, Anal. 38(1) (2018), 37–46.
6. Derakhshan, M.H. and Ansari, A. Fractional Sturm-Liouville problems for Weber fractional derivatives, Int. J. Comput. Math. 96(2) (2019), 217–237.
7. Derakhshan, M.H. and Ansari, A. Numerical approximation to Prabhakar fractional Sturm-Liouville problem, Comput. Appl. Math, 38(2) (2019),71.
8. Derakhshan, M.H., Ansari, A. and Ahmadi Darani, M. On asymptotic stability of Weber fractional differential systems, Comput. Methods Differ. Equ. 6(1) (2018), 30–39.
9. D’Ovidio, M. and Polito, F. Fractional diffusion-telegraph equations and their associated stochastic solutions, Theory. Probab. Appl. 64(4) (2018), 552–574.
10. Dumitru, B., Kai, D. and Enrico, S. Fractional calculus: Models and numerical methods , World Scientific, 2012.
11. Garra, R., Gorenflo, R., Polito, F. and Tomovski, Z. ˇ Hilfer-Prabhakar derivatives and some applications, Appl. math. comput. 242 (2014), 576–589.
12. Garrappa, R., Mainardi, F. and Guido, M. Models of dielectric relaxation based on completely monotone functions, Fract. Calc. Appl. Anal. 19(5) (2016), 1105–1160.
13. Ghorbani, A. Beyond Adomian polynomials: He polynomials, Chaos. Solitons. Fractals, 39(3) (2009), 1486–1492.
14. Giusti, A. and Colombaro, I. Prabhakar-like fractional viscoelasticity, Commun. Nonlinear. Sci. Numer. Simul. 56 (2018), 138–143.
15. Gorenflo, R., Kilbas, A.A., Mainardi, F. and Rogosin, S.V. Mittag-Leffler functions, related topics and applications, Springer Monographs in Math ematics. Springer, Heidelberg, 2014.
16. G´orska, K., Horzela, A., Bratek, L., Penson, K.A. and Dattoli, G. The probability density function for the Havriliak-Negami relaxation, J. Phys. A, Math. Theor. 51(13) (2018) Article ID 135202, 15 pp.
17. Hamarsheh, M., Ismail, A.I. and Odibat, Z. An analytic solution for fractional order Riccati equations by using optimal homotopy asymptotic method, Appl. Math. Sci. 10(23) (2016), 1131–1150.
18. Heydari, M., Loghmani, G.B. and Hosseini, S.M. Operational matrices of Chebyshev cardinal functions and their application for solving delay differential equations arising in electrodynamics with error estimation, Appl. Math. Model, 37(14–15) (2013), 7789–7809.
19. Hou, T. and Leng, H. Numerical analysis of a stabilized Crank Nicolson/Adams–Bashforth finite difference scheme for Allen-Cahn equations, Appl. Math. Lett. 102 (2020), 106–150.
20. Karaagac, B. Two step Adams Bashforth method for time-fractional Tricomi equation with non-local and non-singular Kernel, Chaos. Solitons. Fractals, 128 (2019), 234–241.
21. Karunakar, P. and Chakraverty, S. Solutions of time-fractional third and fifth-order Korteweg-de-Vries equations using homotopy perturbation transform method, Eng. Comput. 36(7) (2019), 2309–2326.
22. Kilbas, A.A., Saigo, M. and Saxena, R.K. Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral. Transform. Spec. Funct. 15(1) (2004), 31–49.
23. Kumar, S., Kumar, A. and Odibat, Z.M. A nonlinear fractional model to describe the population dynamics of two interacting species, Math. Meth ods. Appl. Sci. 40(11) (2017), 4134–4148.
24. Li, C., Kumar, A., Kumar, S. and Yang, X.J. On the approximate solution of nonlinear time-fractional KdV equation via modified homotopy analysis Laplace transform method, J. Nonlinear. Sci. Appl. 9 (2016), 5463–5470.
25. Liemert, A., Sandev, T. and Kantz, H. Generalized Langevin equation with tempered memory kernel, Physica. A. 466 (2017), 356–369.
26. Miskinis, P. The Havriliak-Negami susceptibility as a nonlinear and non local process, Phys. Scr. 2009.
27. Owolabi, K.M. and Atangana, A. Analysis and application of new fractional Adams–Bashforth scheme with Caputo-Fabrizio derivative, Chaos. Solitons. Fractals, 105 (2017), 111–119.
28. Pan, Y., He, Y. and Mikkola, A. Accurate real-time truck simulation via semirecursive formulation and Adams–Bashforth-Moulton algorithm, Acta. Mech. Sin. 35 (2019), 641–652.
29. Pandey, R. K. and Mishra, H.K. Homotopy analysis Sumudu transform method for time-fractional third order dispersive partial differential equation, Adv. Comput. Math. 43(2) (2017), 365–383.
30. Pirim, N.A. and Ayaz, F. A new technique for solving fractional order systems: Hermite collocation method Appl. Math, 7(18)(2016), 2307–2323.
31. Podlubny, I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
32. Qureshi, S. and Kumar, P. Using Shehu integral transform to solve fractional order Caputo type initial value problems, J. Appl. Math. Comput. Mech. 18(2) (2019), 75–83.
33. Rathore, S., Kumar, D., Singh, J. and Gupta, S. Homotopy analysis Sumudu transform method for nonlinear equations, Int. J. Ind. Math. 4(4) (2012), 301–314.g
34. Ratti, I. Comparative study of nonlinear partial differential equation using homotopy perturbation transform method (HPTM) using He’s polynomial
and mixture of Elzaki transform and partial differential transform method (PDTM), Int. J. Appl. Eng. Res. 13(18) (2018), 14037–14040.
35. Saadatmandi, A. and Dehghan, M. A new operational matrix for solving fractional-order differential equations, Comput. Math. Appl. 59(3) (2010),
36. Singh, J., Kumar, D., Swroop, R. and Kumar, S. An efficient computational approach for time-fractional Rosenau–Hyman equation, Neural. Comput. Appl. 30(10) (2018), 3063–3070.
37. Stanislavsky, A. and Weron, K. A typical case of the dielectric relaxation responses and its fractional kinetic equation, Fract. Calc. Appl. Anal. 19(1) (2016), 212–228.
38. Subashini, R., Ravichandran, C., Jothimani, K. and Baskonus, H.M. Existence results of Hilfer integro-differential equations with fractional order, Discrete. Contin. Dyn. Syst. 13(3) (2020), 911–923.
39. Touchent, K.A., Hammouch, Z., Mekkaoui, T. and Belgacem, F. Implementation and convergence analysis of homotopy perturbation coupled with
sumudu transform to construct solutions of local-fractional pdes, Fractal. Fract. 2(3) (2018), 22.
40. Vahidi, J. The combined Laplace-homotopy analysis method for partial differential equations, J. Math. Comput. Sci. 16(1) (2016), 88–102.
41. Y´epez-Mart´ınez, H., and G´omez-Aguilar, J.F. Numerical and analytical solutions of nonlinear differential equations involving fractional operators with power and Mittag-Leffler kernel, Math. Model. Nat. Phenom. 13(1) (2018), Paper No. 13, 17 pp.
How to Cite
DerakhshanM., & AminataeiA. (2020). Comparison of homotopy perturbation transform method and fractional Adams–Bashforth method for the Caputo–Prabhakar nonlinear fractional differential equations. Iranian Journal of Numerical Analysis and Optimization, 10(2), 63-85. https://doi.org/10.22067/ijnao.v10i2.68360
Research Article