Homotopy perturbation transform method for time-fractional Newell-Whitehead Segel equation containing Caputo-Prabhakar fractional derivative

Document Type : Original Article

Authors

1 Faculty of Mathematics, K. N. Toosi University of Technology

2 Faculty of Mathematics, Khajeh Nasir Toosi University of Technology

10.22060/ajmc.2020.18012.1028

Abstract

The main aim of the current article is to find the solution for Newell Whitehead-Segel equations with constant coefficients containing Caputo-Prabhakar fractional derivative using the homotopy perturbation transform method. The convergence analysis of the obtained solution for the proposed fractional order model is presented. Four examples are presented to illustrate the efficiency and applicability and accurateness of the proposed numerical technique

Keywords

Main Subjects


[1] N. Anjum, J. H. He, Laplace transform: making the variational iteration method easier, Applied Mathematics Letters, 92 (2019) 134-138.
[2] L. Beghin, E. Orsingher, Fractional Poisson processes and related planar random motions, Electronic Journal of Probability, 14 (2009) 1790-1826.
[3] J. An, E. Van Hese, M. Baes, Phase-space consistency of stellar dynamical models determined by separable augmented densities, Monthly Notices of the Royal Astronomical Society, 422(1) (2012) 652-664.
[4] V. Daftardar-Gejji, H. Jafari, An iterative method for solving nonlinear functional equations, Journal of Mathematical Analysis and Applications, 316(2) (2006) 753-763.
[5] M. H. Derakhshan, A. Ansari, Fractional Sturm-Liouville problems for Weber fractional derivatives, International Journal of Computer Mathematics, 96(2) (2019) 217-237.
[6] M. H. Derakhshan, A. Ansari, On Hyers-Ulam stability of fractional differential equations with Prabhakar derivatives, Analysis, 38(1), (2018) 37-46.
[7] M. H. Derakhshan, A. Ansari, Numerical approximation to Prabhakar fractional Sturm-Liouville problem, Computational and Applied Mathematics, 38(2) (2019) 71-90. https://doi.org/10.1007/s40314-019-0826-4.
[8] M. H. Derakhshan, A. Ansari, M. Ahmadi Darani, On asymptotic stability of Weber fractional differential systems, Computational Methods for Differential Equations, 6(1) (2018) 30-39.
[9] M. H. Derakhshan, M. Ahmadi Darani, A. Ansari, R. Khoshsiar Ghaziani, On asymptotic stability of Prabhakar fractional differential systems, Computational Methods for Differential Equations, 4(4) (2016) 276-284.
[10] M. D’Ovidio, F. Polito, Fractional diffusion-telegraph equations and their associated stochastic solutions, (2013) arXiv preprint arXiv:1307.1696.
[11] B. Dumitru, D. Kai, S. Enrico, Fractional calculus: models and numerical methods (Vol. 3). World Scientific (2012).
[12] A. Giusti, I. Colombaro, Prabhakar-like fractional viscoelasticity, Communications in Nonlinear Science and Numerical Simulation, 56 (2018) 138-143.
[13] K. G´orska, A. Horzela, L. Bratek, K. A. Penson, G. Dattoli, The probability density function for the Havriliak Negami relaxation, (2016) arXiv preprint arXiv:1611.06433.
[14] R. Garrappa, F. Mainardi, M. Guido, Models of dielectric relaxation based on completely monotone functions, Fractional Calculus and Applied Analysis, 19(5) (2016) 1105-1160.
[15] A. Ghorbani, Beyond Adomian polynomials: he polynomials, Chaos, Solitons & Fractals, 39(3) (2009) 1486- 1492.
[16] R. Gorenflo, A. A. Kilbas, F. Mainardi, S. V. Rogosin, Mittag-Leffler functions, related topics and applications (Vol. 2). Berlin: Springer (2014).
[17] R. Garra, R. Gorenflo, F. Polito, Z. Tomovski, Hilfer-Prabhakar derivatives and some applications, Applied ˇ Mathematics and Computation, 242 (2014) 576-589.
[18] J. H. He, Homotopy perturbation technique, Computer methods in applied mechanics and engineering, 178(3-4) (1999) 257-262.
[19] M. Hamarsheh, A. I. Ismail, Z. Odibat, An analytic solution for fractional order Riccati equations by using optimal homotopy asymptotic method, Applied Mathematical Sciences, 10(23) (2016) 1131-1150.
[20] H. Jafari, Iterative methods for solving system of fractional differential equations, Doctoral dissertation, Pune University, Pune, India, (2006).
[21] B. Karaagac, Two step Adams Bashforth method for time fractional Tricomi equation with non-local and non-singular Kernel, Chaos, Solitons & Fractals, 128 (2019) 234-241.
[22] P. Karunakar, S. Chakraverty, Solutions of time-fractional third-and fifth-order Korteweg-de-Vries equations using homotopy perturbation transform method. Engineering Computations, 36(7) (2019) 2309-2326. https://doi.org/10.1108/EC-01-2019-0012.
[23] A. A. Kilbas, M. Saigo, R. K. Saxena, Generalized Mittag-Leffler function and generalized fractional calculus operators, Integral Transforms and Special Functions, 15(1) (2004) 31-49.
[24] A. Liemert, T. Sandev, H. Kantz, Generalized Langevin equation with tempered memory kernel, Physica A: Statistical Mechanics and its Applications, 466 (2017) 356-369.
[25] P. Miskinis, The Havriliak-Negami susceptibility as a nonlinear and nonlocal process, Physica Scripta, 2009(T136) (2009) 014019.
[26] T,.R. Prabhakar, A singular integral equation with a generalized Mittag Leffler function in the kernel, Yokohama Mathematical Journal, 19 (1971) 7-15.
[27] A. Prakash, M. Goyal, S. Gupta, Fractional variational iteration method for solving time-fractional Newell Whitehead-Segel equation, Nonlinear Engineering, 8(1) (2019) 164-171.
[28] A. Prakash, H. Kaur, A New Numerical Method for a Fractional Model of Non-Linear Zakharov-Kuznetsov Equations via Sumudu Transform Methods of Mathematical Modelling: Fractional Differential Equations, (2019) 189-204.
[29] S. A. Pasha, Y. Nawaz, M. S. Arif, The modified homotopy perturbation method with an auxiliary term for the nonlinear oscillator with discontinuity, Journal of Low Frequency Noise, Vibration and Active Control, 38(3-4) (2019) 1363-1373.
[30] I. Podlubny, Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Vol. 198). Elsevier (1998).
[31] N. A. Pirim, F. Ayaz, A new technique for solving fractional order systems: Hermite collocation method, Applied Mathematics, 7(18) (2016) 2307-2323.
[32] R. K. Pandey, H. K. Mishra, Homotopy analysis Sumudu transform method for time-fractional third order dispersive partial differential equation, Advances in Computational Mathematics, 43(2) (2017) 365-383.
[33] H. C. Rosu, O. Cornejo-P´erez, Supersymmetric pairing of kinks for polynomial nonlinearities, Physical Review E, 71(4) (2005) 046607. doi: 10.1103/PhysRevE.71.046607.
[34] A. Stanislavsky, K. Weron, Atypical case of the dielectric relaxation responses and its fractional kinetic equation, Fractional Calculus and Applied Analysis, 19(1) (2016) 212-228.
[35] M. Tatari, M. Dehghan, On the convergence of He’s variational iteration method, Journal of Computational and Applied Mathematics, 207(1) (2007) 121-128.
[36] J. Vahidi, The combined Laplace-homotopy analysis method for partial differential equations, J. Math. Comput. Sci.JMCS, 16(1) (2016) 88-102.
[37] H. Y´epez-Mart´ınez, J. F. G´omez-Aguilar, A new modified definition of Caputo-Fabrizio fractional-order derivative and their applications to the Multi Step Homotopy Analysis Method (MHAM), Journal of Computational and Applied Mathematics, 346 (2019) 247-260.
[38] D. N. Yu, J. H. He, A. G. Garcıa, Homotopy perturbation method with an auxiliary parameter for nonlinear oscillators, Journal of Low Frequency Noise, Vibration and Active Control, 38(3-4) (2019) 1540-1554.