Novel Approximate Solutions for Nonlinear Blasius Equations

Amna M. Mahdi; Majeed A.  AL-Jawary; Mustafa Turkyilmazoglu

doi:10.30526/37.1.3292

Authors

Amna M. Mahdi Department of Mathematics, College of Education for Pure Sciences Ibn AL-Haitham, University of Baghdad, Baghdad, Iraq.
Majeed A. AL-Jawary Department of Mathematics, College of Education for Pure Sciences Ibn AL-Haitham, University of Baghdad, Baghdad, Iraq.
Mustafa Turkyilmazoglu Department of Mathematics, Hacettepe University, Ankara, Turkey. Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan.

DOI:

https://doi.org/10.30526/37.1.3292

Keywords:

Blasius equations; Bernstein polynomial; Legendre polynomial; Bernoulli polynomial; operational matrices.

Abstract

The method of operational matrices based on different types of polynomials such as Bernstein, shifted Legendre and Bernoulli polynomials will be presented and implemented to solve the nonlinear Blasius equations approximately. The nonlinear differential equation will be converted into a system of nonlinear algebraic equations that can be solved using Mathematica^®12. The efficiency of these methods has been studied by calculating the maximum error remainder ( ), and it was found that their efficiency increases as the polynomial degree (n) increases, since the errors decrease. Moreover, the approximate solutions obtained by the proposed methods are compared with the solution of the 4^th order Runge-Kutta method (RK4), which gives very good agreement. In addition, the convergence of the proposed approximate methods is given based on one of the Banach fixed point theorem results.

References

Murphy, G.M. Ordinary Differential Equations and Their Solutions; Dover Publications, Inc., New York, 1960.

Boyce, W.E.; DiPrima, R.C. Elementary differential equations and boundary value problems, Ed.;9th ed. 2009; John Wiley & Sons, Inc., United States of America, 2009; ISBN 9780470383346.

Yousefi, S.A.; Behroozifar, M.; Operational matrices of Bernstein polynomials and their applications, International Journal of Systems Science 2010, 41(6), 709-716. DOI: https://doi.org/10.1080/00207720903154783 DOI: https://doi.org/10.1080/00207720903154783

Pirabaharan, P.; Chandrakumar, R.D.; A computational method for solving a class of singular boundary value problems arising in science and engineering, Egyptian Journal of Basic and Applied Sciences 2016, 3(4), 383-391. DOI: https://doi.org/10.1016/j.ejbas.2016.09.004 DOI: https://doi.org/10.1016/j.ejbas.2016.09.004

Khataybeh, S.; Hashim, I.; Alshbool, M.; Solving directly third-order ODEs using operational matrices of Bernstein polynomials method with applications to fluid flow equations, Journal of King Saud University-Science 2019, 31(4), 822-826. DOI: https://doi.org/10.1016/j.jksus.2018.05.002 DOI: https://doi.org/10.1016/j.jksus.2018.05.002

Asgari, M.; Ezzati, R.; Using operational matrix of two-dimensional Bernstein polynomials for solving two-dimensional integral equations of fractional order, Applied Mathematics and Computation 2017, 307(C), 290-298. DOI: https://doi.org/10.1016/j.amc.2017.03.012 DOI: https://doi.org/10.1016/j.amc.2017.03.012

Hesameddini, E.; Shahbazi, M.; Two-dimensional shifted Legendre polynomials operational matrix method for solving the two- dimensional integral equations of fractional order, Applied Mathematics and Computation 2018, 322(C), 40-54. DOI: https://doi.org/10.1016/j.amc.2017.11.024 DOI: https://doi.org/10.1016/j.amc.2017.11.024

Sharma, B.; Kumar, S.; Paswan, M.K.; Mahato, D.; Chebyshev operational matrix method for Lane-Emden problem, Nonlinear Engineering 2019, 8(1), 1-9. DOI: https://doi.org/10.1515/nleng-2017-0157 DOI: https://doi.org/10.1515/nleng-2017-0157

Bazm, S.; Bernoulli polynomials for the numerical solution of some classes of linear and nonlinear integral equations, Journal of Computational and Applied Mathematics 2015, 275(C), 44-60. DOI: https://doi.org/10.1016/j.cam.2014.07.018 DOI: https://doi.org/10.1016/j.cam.2014.07.018

Al-Jawary, M.A.; Ibraheem, G.H.; Tow meshless methods for solving nonlinear ordinary differential equations in engineering and applied sciences, Nonlinear Engineering 2020, 9(1), 244-255. DOI: https://doi.org/10.1515/nleng-2020-0012

Ibraheem, G.H.; Al-Jawary, M.A.; The operational matrix of Legendre polynomials for solving nonlinear thin film flow problems, Alexandria Engineering Journal, 2020, 59(5), 4027-4033. DOI: https://doi.org/10.1016/j.aej.2020.07.008

Talib, I.; Tunc, C.; Noor, Z.A.; New operational matrices of orthogonal Legendre polynomials and their operational, Journal of Taibah University for Science 2019, 13(1), 377-389. DOI: https://doi.org/10.1080/16583655.2019.1580662

Bani-Ahmad, F.; Alomari, A.K.; Bataineh, A.S.; Sulaiman, J.; Hashim, I.; On the approximate solutions of systems of ODEs by Legendre operational matrix of differentiation, Italian Journal of Pure and Applied Mathematics 2016, 36, 483-494.https://ijpam.uniud.it/online_issue/201636/42

Kumar, S.; Pandey, P.; Das, S.; Craciun, E.-M.; Numerical solution of two dimensional reaction-diffusion equation using operational matrix method based on Genocchi polynomial-Part I: Genocchi polynomial and opperatorial matrix, Proceedings of the Romanian Academy, Series A 2019, 20(4), 393-399. https://acad.ro/sectii2002/proceedings/doc2019-4

Loh, J.R.; Phang, C.; Numerical solution of Fredholm fractional integro-differential equation with right-sided Caputo’s derivative using Bernoulli polynomials operational matrix of fractional derivative, Mediterranean Journal of Mathematics 2019, 16(2), 1-25.

DOI: https://doi.org/10.1007/s00009-019-1300-7

Zeghdane, R.; Numerical solution of stochastic integral equations by using Bernoulli operational matrix, Mathematics and Computers in Simulation 2019, 165(C), 238-254. DOI: https://doi.org/10.1016/j.matcom.2019.03.005

Jasim, S.M.N.; Ibraheem, G.H.; Fractional Pantograph Delay Equations Solving by the Meshless Methods, Ibn AL-Haitham Journal For Pure and Applied Sciences 2023, 36, 382-397. DOI: https://doi.org/10.30526/36.3.3076 DOI: https://doi.org/10.30526/36.3.3076

Alshbool, M.H.T.; Mohammad, M.; Isik, O.; Hashim, I.; Fractional Bernstein operational matrices for solving integro-differential equations involved by Caputo fractional derivative, Results in Applied Mathematics 2022, 14, 100258. DOI: https://doi.org/10.1016/j.rinam.2022.100258

Salih, A.A.; Shihab, S.; New operational matrices approach for optimal control based on modified Chebyshev polynomials, Samarra Journal of Pure and Applied Science 2020, 2(2), 68–78. DOI: http://dx.doi.org/10.54153/sjpas.2020.v2i2.115

Jalal, R.; Shihab, S.; Abed Alhadi, M.; Rasheed, M.; Spectral Numerical Algorithm for Solving Optimal Control Using Boubaker-Turki Operational Matrices, Journal of Physics: Conference Series, IOP Publishing 2020, 1660(1), 012090. DOI: https://doi.org/10.1088/1742-6596/1660/1/012090

Rani, D.; Mishra, V.; Numerical inverse Laplace transform based on Bernoulli polynomials operational matrix for solving nonlinear differential equations, Results in Physics, 2020, 16, 102836. DOI: https://doi.org/10.1016/j.rinp.2019.102836

Kaur, H.; Mishra, V.; Mittal, R.C.; Numerical solution of a laminar viscous flow boundary layer equation using uniform Haar wavelet quasi-linearization method, International Journal of Mathematical and Computational Sciences 2013, 79, 1410-1415. DOI: http://dx.doi.org/10.5281/zenodo.1087368

Cortell, R.; Numerical solutions of the classical Blasius flat-plate problem, Applied Mathematics and Computation 2005, 170(1), 706-710. DOI: https://doi.org/10.1016/j.amc.2004.12.037 DOI: https://doi.org/10.1016/j.amc.2004.12.037

[24] He, J.; Approximate analytical solution of Blasius’ equation, Communications in Nonlinear Science and Numerical Simulation 1999, 4(1), 75-78. DOI: https://doi.org/10.1016/S1007-5704(99)90063-1 DOI: https://doi.org/10.1016/S1007-5704(99)90063-1

Abbasbandy, S.; A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method, Chaos, Solitons & Fractals 2007, 31(1), 257-260. DOI: https://doi.org/10.1016/j.chaos.2005.10.071 DOI: https://doi.org/10.1016/j.chaos.2005.10.071

Aminikhah, H.; An analytical approximation for solving nonlinear Blasius equation by NHPM, Numerical Methods for Partial Differential Equations 2010, 26(6), 1291-1299. DOI: https://doi.org/10.1002/num.20490 DOI: https://doi.org/10.1002/num.20490

Liao, S.; An optimal homotopy-analysis approach for strongly nonlinear differential equations, Communications in Nonlinear Science and Numerical Simulation 2010, 15(8), 2003-2016. DOI: https://doi.org/10.1016/j.cnsns.2009.09.002 DOI: https://doi.org/10.1016/j.cnsns.2009.09.002

Mohammed Ali, M.N.; A New operational matrix of derivative for orthonormal Bernstein polynomial’s, Baghdad Science Journal 2014, 11(3), 1295-1300. DOI:https://doi.org/10.21123/bsj.2014.11.3.1295-1300

Al-A’asam, J.A.; Deriving the composite Simpson rule by using Bernstein polynomials for solving Volterra integral equations, Baghdad Science Journal 2014, 11(3), 1274-1281. DOI: https://doi.org/10.21123/bsj.2014.11.3.1274-1281

Turkyilmazoglu, M.; Convergent optimal variational iteration method and applications to heat and fluid flow problems, International Journal of Numerical Methods for Heat & Fluid Flow 2016, 26(3/4), 790–804. DOI: https://doi.org/10.1108/HFF-09-2015-0353 DOI: https://doi.org/10.1108/HFF-09-2015-0353