Effective Computational Methods for Solving the Jeffery-Hamel Flow Problem
DOI:
https://doi.org/10.21123/bsj.2022.7326Keywords:
Approximate solution, Bernstein polynomials, Chebyshev polynomials, Hermite polynomials, Legendre polynomialsAbstract
In this paper, the effective computational method (ECM) based on the standard monomial polynomial has been implemented to solve the nonlinear Jeffery-Hamel flow problem. Moreover, novel effective computational methods have been developed and suggested in this study by suitable base functions, namely Chebyshev, Bernstein, Legendre, and Hermite polynomials. The utilization of the base functions converts the nonlinear problem to a nonlinear algebraic system of equations, which is then resolved using the Mathematica®12 program. The development of effective computational methods (D-ECM) has been applied to solve the nonlinear Jeffery-Hamel flow problem, then a comparison between the methods has been shown. Furthermore, the maximum error remainder ( ) has been calculated to exhibit the reliability of the suggested methods. The results persuasively prove that ECM and D-ECM are accurate, effective, and reliable in getting approximate solutions to the problem.
Received 15/4/2022
Accepted 9/6/2022
Published Online First 20/11/2022
References
Hermann M, Saravi M. Nonlinear Ordinary Differential Equations: Analytical Approximation and Numerical Methods. India: Springer; 2016. 320 p.
AL-Jawary MA, Salih OM. Reliable iterative methods for 1D Swift–Hohenberg equation. Arab J Basic Appl Sci. 2020; 27(1): 56-66.
Abed AM, Younis MF, Hamoud AA. Numerical solutions of nonlinear Volterra-Fredholm integro-differential equations by using MADM and VIM. Nonlinear Funct Anal Appl. 2022; 27(1): 189-201.
Hasan PM, Sulaiman NA. Convergence Analysis for the Homotopy Perturbation Method for a Linear System of Mixed Volterra-Fredholm Integral Equations. Baghdad Sci J. 2020; 17(3(Suppl.): 1010-1018.
Hamoud AA, Hussain KH, Mohammed NM, Ghadle KP. Solving Fredholm integro-differential equations by using numerical techniques. Nonlinear Funct. Anal Appl. 2019; 24(3): 533-542.
Adel W, Sabir Z. Solving a new design of nonlinear second-order Lane–Emden pantograph delay differential model via Bernoulli collocation method. Eur Phys J Plus. 2020; 135(5): 1-12.
Yari A, Mirnia MK. Solving optimal control problems by using Hermite polynomials. Comput methods differ equ. 2020; 8(2): 314-329.
Laib H, Bellour A, Boulmerka A. Taylor collocation method for a system of nonlinear Volterra delay integro-differential equations with application to COVID-19 epidemic. Int J Comput Math. 2022; 99(4): 852-876.
Srivastava HM, Shah FA, Abass R. An application of the Gegenbauer wavelet method for the numerical solution of the fractional Bagley-Torvik equation. Russ. J Math Phys. 2019; 26(1): 77-93.
Singh H. Jacobi collocation method for the fractional advection-dispersion equation arising in porous media. Numer. Methods Partial Differ Equ. 2022; 38(3): 636-653.
Ganji RM, Jafari H, Baleanu D. A new approach for solving multi variable orders differential equations with Mittag–Leffler kernel. Chaos Solit Fractals. 2020; 130: 109405.
Singha N, Nahak C. A numerical method for solving a class of fractional optimal control problems using Boubaker polynomial expansion scheme. Filomat. 2018; 32(13): 4485-4502.
Yuttanan B, Razzaghi M. Legendre wavelets approach for numerical solutions of distributed order fractional differential equations. Appl Math Model. 2019; 70: 350-364.
Salih OM, AL-Jawary MA. Reliable iterative methods for solving 1D, 2D and 3D Fisher’s equation. IIUM Eng J. 2021; 22(1): 138-166.
Hamoud AA, Mohammed NM, Ghadle KP. Some powerful techniques for solving nonlinear Volterra-Fredholm integral equations. J Appl Nonlinear Dyn. 2021; 10(3): 461-469.
Hussin CHC, Azmi A, Ismail AIM, Kilicman A, Hashim I. Approximate Analytical Solutions of Bright Optical Soliton for Nonlinear Schrödinger Equation of Power Law Nonlinearity. Baghdad Sci J. 2021 March 30; 18(1(Suppl.)): 836-845.
Bougoffa L, Mziou S, Rach RC. Exact and approximate analytic solutions of the Jeffery-Hamel flow problem by the Duan-Rach modified Adomian decomposition method. Math Model Anal. 2016; 21(2): 174-187.
Jeffery GBL. The two-dimensional steady motion of a viscous fluid. Lond. Edinb. Dublin philos. Mag J sci. 1915; 29(172): 455-465.
Li Z, Khan I, Shafee A, Tlili I, Asifa T. Energy transfer of Jeffery–Hamel nanofluid flow between non-parallel walls using Maxwell–Garnetts (MG) and Brinkman models. Energy Rep. 2018; 4: 393-399.
Bildik N, Deniz S. Optimal iterative perturbation technique for solving Jeffery–Hamel flow with high magnetic field and nanoparticle. J Appl Anal Comput. 2020; 10(6): 2476-2490.
Bataineh AS, Isik OR, Hashim I. Bernstein Collocation Method for Solving MHD Jeffery–Hamel Blood Flow Problem with Error Estimations. Int J Differ Equ. 2022; 2022: 9 pages.
Nourazar SS, Nazari-Golshan A, Soleymanpour F. On the expedient solution of the magneto-hydrodynamic Jeffery-Hamel flow of Casson fluid. Sci Rep. 2018; 8(1): 1-16.
Patel HS, Meher R. Analytical Investigation of Jeffery–Hamel Flow by Modified Adomian Decomposition. Ain Shams Eng J. 2018; 9(4): 599-606.
Biswal U, Chakraverty S. Investigation of Jeffery-Hamel flow for nanofluid in the presence of magnetic field by a new approach in the optimal homotopy analysis method. J Appl Comput Mech. 2022; 8(1): 48-59.
Ahmad I, Ilyas H. Homotopy Perturbation Method for the nonlinear MHD Jeffery–Hamel blood flows problem. Appl Numer Math. 2019; 141: 124-132.
Adel W, Biçer KE, Sezer M. A Novel Numerical Approach for Simulating the Nonlinear MHD Jeffery–Hamel Flow Problem. Int J Appl Comput Math. 2021; 7(3): 1-15.
Kumbinarasaiah S, Raghunatha KR. Numerical solution of the Jeffery–Hamel flow through the wavelet technique. Heat Transf. 2022; 51(2): 1568-1584.
Meher R, Patel ND. Analytical Investigation of MHD Jeffery–Hamel flow problem with heat transfer by differential transform method. SN. Appl Sci. 2019; 1(7): 1-12.
AL-Jawary MA, Abdul Nabi AJ. Three Iterative Methods for Solving Jeffery-Hamel Flow Problem. Kuwait J Sci. 2020; 47(1): 1-13.
AL-Jawary MA, Ibraheem GH. Two meshless methods for solving nonlinear ordinary differential equations in engineering and applied sciences. Nonlinear Eng. 2020; 9(1): 244-255.
Turkyilmazoglu M. Effective computation of exact and analytic approximate solutions to singular nonlinear equations of Lane–Emden–Fowler type. Appl Math Model. 2013; 37(14-15): 7539-7548.
Turkyilmazoglu M. An effective approach for numerical solutions of high-order Fredholm integro-differential equations. Appl Math Comput. 2014; 227(15): 384-398.
Turkyilmazoglu M. High-order nonlinear Volterra–Fredholm-Hammerstein integro-differential equations and their effective computation. Appl Math Comput. 2014; 247: 410-416.
Turkyilmazoglu M. Effective Computation of Solutions for Nonlinear Heat Transfer Problems in Fins. J Heat Transfer. 2014; 136(9): 091901.
Turkyilmazoglu M. Solution of initial and boundary value problems by an effective accurate method. Int J Comput Methods. 2017; 14(06): 1750069.
Secer A, Ozdemir N, Bayram M. A Hermite Polynomial Approach for Solving the SIR Model of Epidemics. Mathematics. 2018; 6(12): 305.
Al-Humedi HO, Al-Saadawi FA. The Numerical Technique Based on Shifted Jacobi-Gauss-Lobatto Polynomials for Solving Two Dimensional Multi-Space Fractional Bioheat Equations. Baghdad Sci J. 2020; 17(4): 1271-1282.
Talaei Y, Asgari M. An operational matrix based on Chelyshkov polynomials for solving multi-order fractional differential equations. Neural Comput Appl. 2018; 30(5): 1369-1376.
Sharma B, Kumar S, Paswan MK, Mahato D. Chebyshev Operational Matrix Method for Lane-Emden Problem. Nonlinear Eng. 2019; 8(1): 1-9.
Khataybeh SN, Hashim I, Alshbool M. Solving directly third-order ODEs using operational matrices of Bernstein polynomials method with applications to fluid flow equations. J King Saud Univ Sci. 2019; 31(4): 822-826.
Pourbabaee M, Saadatmandi A. A novel Legendre operational matrix for distributed order fractional differential equations. Appl Math Comput. 2019; 361: 215-231.
Ibraheem GH, AL-Jawary MA. The operational matrix of Legendre polynomials for solving nonlinear thin film flow problems. Alexandria Eng J. 2020; 59(5): 4027-4033.
Kosunalp HY, Gülsu M. Operational Matrix by Hermite Polynomials for Solving Nonlinear Riccati Differential Equations. Int J Math Comput Sci. 2021; 16(2): 525-536.
Kadkhoda N. A numerical approach for solving variable order differential equations using Bernstein polynomials. Alexandria Eng J. 2020; 59(5): 3041-3047.
Gülsu M, Yalman H, Öztürk Y, Sezer M. A New Hermite Collocation Method for Solving Differential Difference Equations. Appl Appl Math. 2011; 6(1): 116-129.
Downloads
Published
Issue
Section
License
Copyright (c) 2022 Baghdad Science Journal
This work is licensed under a Creative Commons Attribution 4.0 International License.
How to Cite
