# Analytical approximation solution of nonlinear Blasius problem

## Keywords:

Blasius equation, Optimal Auxiliary Functions Method, Nonlinear differential equation, Optimal parameters## Abstract

A new alternative technique of the Optimal Auxiliary Functions Methods (OAFM) is proposed and applied to solve nonlinear differential equations of the Blasius problem. The proposed procedure is very effective and convenient and does not require linearization or small parameters. The main advantage of this approach consists in that it provides a convenient way to control convergence of the approximate solution in a very rigorous way. The solution obtained using the present procedure is in a very good agreement with numerical results and some well-known results, which prove that OAFM is a power tool for nonlinear problems, very efficient and accurate.

## References

NAYFEH, A., MOOK, D., Nonlinear Oscillations, John Wiley and Sons, New York, 1979.

POPESCU, Mihai, Periodic solution for differential equation with small parameters, Rev. Roum. Sci. Techn.-Mec. Appl., 54, 1, pp. 35-43, 2009.

WU, B.S., LI, P., A method for obtaining approximate analytical periods for a class of nonlinear oscillations, Meccanica, 36, 2, pp. 167-176, 2001; doi:10.1023/A:1013067311749.

HE, J.H., WU, G,C., AUSTIN F., The variational iteration method which should be followed, Nonlinear Science Letters A, 1, 1, pp. 1-32, 2010.

MARINCA, V., HERISANU, N., Nonlinear Dynamical Systems in Engineering-Some Approximate Approaches, Springer, 2011.

MARINCA, V., HERISANU, N., Optimal Homotopy Asymptotic Method Engineering Application, Springer, 2015.

YILDIRIM, A., Determination of periodic solutions for nonlinear oscillations with fractional powers by He’s modified Lindstedt-Poincare method, Mecanica 45, 1, pp. 111-121, 2010; doi:10.1007/s11012-009-9212-4.

CRANDALL, S.H., Engineering Analysis, McGraw-Hill, New-York, 1976.

WANG, L., A New Algorithm for Solving Clasiccal Blasius Equation, Appl.Math.Comp., 157, 1, pp. 1-9, 2004; https://doi.org/10.1016/j.amc.2003.06.011.

FAZIO, R., Transformation methods for the Blasius problem and its recent variants, Proceed. of the World Congress on Engineering, vol. I, London, 2008.

BORSA, E., Thin film flow driven by gravity and surface tension gradient, Rev. Roum. Sci. Techn,-Mec. Appl. 54, 12, pp. 81-86, 2009.

AHMAD, F., AL-BARAKATI, W.H., An approximate analytic solution of the Blasius problem, Communications in Nonlinear Science and Numerical Simulation, 14, 4, pp. 1021-1024, 2009; https://doi.org/10.1016/j.cnsns.2007.12.010.

CORTELLl, R., BATALLER, Numerical Comparisons of Blasius and Sakiadis flow, Matematika, 26, 2, pp. 187-196, 2010.

PARAND, K., NIKARAYA, M., RAD, J.A., BAHARIFARD, F., A new reliable numerical algorithm based on the first kind of Bessel functions to solve Prandtl-Blasius laminar viscous flow over a semi-infinite flat plate, Z. Naturforsch, pp. 885-893, 2012.

MARINCA, Vasile, HERISANU, Nicolae, An application of the optimal auxiliary functions to Blasius problem, Ro. .J. Techn. Sci.–Appl. Mechanics, 60, 13, pp. 206-215, 2015.

ROBIN, W., Some new uniform approximate analytical representation of the Blasius function, Global J. of Mathematics, 2, 2, pp. 150-155, 2015.

AKGUL, A., A novel method for the solution Blasius equation in semi-infinite domains, An International Journal of Optimisation and Control Theories and Applications, 7, 2, pp. 225-233, 2017; doi:10.11121/ijocta.01.2017.00363.

NAJAFI, E., Numerical quasilinearization scheme for the integral equation form of the Blasius equation, Computational Methods for Differential Equations, 6, 2, pp. 141-156, 2010.

ELSGOLTS, Lev, Differential Equations and the Calculus of variations, Mir Publishes, Moscov, 1977.

MARINCA, V., HERISANU, N., The nonlinear thermomechanical vibration of a functionally graded beam on Winkler-Pasternak foundation, MATEC Web of Conf. 1481(2018), 13004.

MARINCA, Vasile, HERISANU, Nicolae, Vibration of nonlinear nonlocal elastic column with initial imperfection, Acoustics and Vibration of Mechanical Structures (AVMS), 2017, Springer; Procced in Physics, 198, pp. 49-56, 2018.

HOWARTH, L., On the relation of the laminar boundary layer equations, Proc.London.Math.Soc.A, 164, pp. 547-579, 1938.

GANJI, D.D., BABAZADECH, H., NOORI, F., PIROUZ, M.M., JAMPOUR, M., An application of Homotopy Perturbation method for nonlinear Blasius equation to boundary layer flow plate, Int. J. Nonl. Sci., 7, pp. 399-403, 2009.

ESMAEILPOUR, M., GANJI, D.D., Application of He’s homotopy perturbation method to boundary layers flow and convection heat transfer over a flat plate, Physics Letters A, 372, 1, pp. 35-38, 2007; https://doi.org/10.1016/j.physleta.2007.07.002.

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