An application of the iterative method to study multi-dimensional fractional order Navier-Stokes equations

Year 2022, , 202 - 216, 30.06.2022

Lokesh Kumar Yadav , Garima Agarwal

https://doi.org/10.31197/atnaa.954736

Abstract

In this article, a hybrid method called iteration Shehu transform method has been implemented to solve fractional-order Navier–Stokes equation. Atangana-Balenu operator describes fractional-order derivatives. The analytical solutions of three distinct examples of the time- fractional Navier-Stokes equations are determined by using Iterative shehu transform method. Further, we present the effectiveness and accuracy of the proposed method by comparison of analytical solutions to the exact solutions and the results are represented graphically and numerically.

Keywords

Fractional order Navier-Stokes equations, Iterative Shehu transform method, Atangana-Balenu derivative.

References

• [1] A. Ali, K. Shah, R.A. Khan, Numerical treatment for traveling wave solutions of fractional Whitham-Broer-Kaup equations, Alexandria Eng. J. 57 (3) (2018) 1991-1998.
• [2] F. Haq, K. Shah, G. ur Rahman, M. Shahzad, Numerical solution of fractional order smoking model via laplace Adomian decomposition method, Alexandria Eng. J. 57 (2) (2018) 1061- 1069.
• [3] R. Shah, H. Khan, D. Baleanu, P. Kumam, M. Arif, A novel method for the analytical solution of fractional Zakharov- Kuznetsov equations, Adv. Diff. Eqs. 2019 (1) (2019) 1-14.
• [4] J. Singh, D. Kumar, R. Swroop, Numerical solution of time-and space-fractional coupled Burgers' equations via homotopy algorithm, Alexandria Eng. J. 55 (2) (2016) 1753-1763.
• [5] A. Kilicman, R. Shokhanda, P. Goswami,On the solution of (n+1)-dimensional fractional M-Burgers equation, Alexandria Eng. J. 60 (2014) 1165-1172.
• [6] Y. Meng, Y. Zhang, Numerical analysis on gas lubrication of microsliders with a modi?ed Navier-Stokes equation ,Digest APMRC. IEEE (2012) 1-2.
• [7] P.D. Christo?des, A. Armaou, Nonlinear control of navier-stokes equations, American Control Conference. ACC (IEEE Cat. No. 98CH36207) 3 (2012) 1355-1359.
• [8] A. Goswami, J. Singh, D. Kumar, S. Gupta, An eficient analytical technique for fractional partial differential equations occurring in ion acoustic waves in plasma, J. Ocean Eng. Sci. 4 (2) (2019) 85-99.
• [9] A. Goswami, J. Singh, D. Kumar, An efficient analytical approach for fractional equal width equations describing hydro- magnetic waves in cold plasma, Physica A. 524 (2019) 563-575.
• [10] M. El-Shahed, A. Salem, On the generalized Navier-Stokes equations, Appl. Math. Comput. 156 (1) (2005) 287-293.
• [11] A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and nonsingular kernel: theory and application to heat transfer model, Therm. Sci. 20 (2) (2016) 763-769.
• [12] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65-70.
• [13] F. Jarad, T. Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete Continuous Dyn. Syst. Ser. 13 (3) (2020) 709-722.
• [14] I. Malyk, M.M.A. Shrahili, A.R. Shafay, P. Goswami, S. Sharma, R.S. Dubey, Analytical solution of non-linear fractional Burger's equation in the framework of different fractional derivative operators, Results in Physics 19 (103397) (2020).
• [15] M.A. Almuqrin, P. Goswami, S. Sharma, I. Khan, R.S. Dubey, A. Khan, Fractional model of Ebola virus in population of bats in frame of Atangana-Baleanu fractional derivative, Results in Physics 26 (104295) (2021).
• [16] G.I. Taylor, On the decay of vortices in a viscous fluid, Philos. Mag. 46 (1923) 671-674.
• [17] R. Shah, H. Khan, D. Baleanu, P. Kumam, M. Arif, The analytical investigation of time-fractional multi-dimensional Navier-Stokes equation, Alexandria Eng. J. 59 (2000) 2941-2956.
• [18] B.K. Singh, P. Kumar, FRDTM for numerical simulation of multi-dimensional, time-fractional model of Navier-Stokes equation, Ain Shams Eng. J. 9 (2018) 827-834.
• [19] A. Prakash, P. Veeresha, D.G. Prakasha, M. Goyal, A new efficient technique for solving fractional coupled Navier-Stokes equations using q-homotopy analysis transform method, Pramana -J. Phys. 93 (6) (2019).
• [20] Y.M. Chu, N.A. Shah, P. Agarwal, J.D. Chung, Analysis of fractional multi-dimensional Navier-Stokes equation, Adv. Differ. Equ., 2021 (91) (2020).
• [21] S. Mahmood, R. Shah, H. Khan, M. Arif, Laplace Adomian Decomposition Method for Multi Dimensional Time Fractional Model of Navier-Stokes Equation, Symmetry 149 (11) (2019).
• [22] Hajira1, H. Khan, A. Khan, P. Kumam, D. Baleanu, M. Arif, An approximate analytical solution of the Navier-Stokes equations within Caputo operator and Elzaki transform decomposition method, Adv. Di?er. Equ. 2020 (622) (2020).
• [23] V. Daftardar-Gejji, H. Jafari, An iterative method for solving non linear functional equations. J. Math. Anal. Appl. 316 (2006) 753-763.
• [24] Maitama, S. and Zhao, W., New integral transform: Shehu transform a generalization of Sumudu and Laplace transform for solving differential equations, arXiv (2019) arXiv:1904 11370.
• [25] A. Atangana, I. Koca, Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order, Chaos Solitons Fractals 89 (2016) 447-454.
• [26] A. Bokharia, D. Baleanu, R. Belgacema, Application of Shehu transform to Atangana-Baleanu derivatives, J. Math. Computer Sci. 20 (2020) 101-107.