A novel method for the solution of Blasius equation in semi-infinite domains

Ali Akgül

Abstract


Many known methods fail in the attempt to get analytic solutions of Blasius-type equations. In this work, we apply the reproducing kernel method for ivestigating Blasius equations with two different boundary conditions in semi-infinite domains. Convergence analysis of the reproducing kernel method is given. The numerical approximations are presented and compared with some other techniques, Howarth's numerical solution and Runge-Kutta Fehlberg method.

Keywords


Reproducing kernel method, Blasius equations, reproducing kernel functions.

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References


Omar Abu Arqub, Mohammed Al-Smadi, and Shaher Momani. Application of reproducing kernel method for solving nonlinear Fredholm-Volterra integrodifferential equations. Abstr. Appl. Anal., pages Art. ID 839836, 16, 2012.

Desmond Adair and Martin Jaeger. Simulation of tapered rotating beams with centrifugal stiffening using the Adomian decomposition method. Appl. Math. Model., 40(4):3230–3241, 2016.

M. Aghakhani, M. Suhatril, M. Mohammadhassani, M. Daie, and A. Toghroli. A simple modification of homotopy perturbation method for the solution of Blasius equation in semi-infinite domains. Math. Probl. Eng., pages Art. ID 671527, 7, 2015.

Ali Akgül. A new method for approximate solutions of fractional order boundary value problems. Neural Parallel Sci. Comput., 22(1-2):223–237, 2014.

Ali Akgül. New reproducing kernel functions. Math. Probl. Eng., pages Art. ID 158134, 10, 2015.

Ali Akgül, Mustafa Inc, and Esra Karatas. Reproducing kernel functions for difference equations. Discrete Contin. Dyn. Syst. Ser. S, 8(6):1055–1064, 2015.

Hossein Aminikhah. An analytical approximation for solving nonlinear Blasius equation by NHPM. Numer. Methods Partial Differential Equations, 26(6):1291– 1299, 2010.

N. Aronszajn. Theory of reproducing kernels. Trans. Amer. Math. Soc., 68:337– 404, 1950.

Asai Asaithambi. Solution of the Falkner-Skan equation by recursive evaluation of Taylor coefficients. J. Comput. Appl. Math., 176(1):203–214, 2005.

Samia Bushnaq, Banan Maayah, Shaher Momani, and Ahmed Alsaedi. A reproducing kernel Hilbert space method for solving systems of fractional integrodifferential equations. Abstr. Appl. Anal., pages Art. ID 103016, 6, 2014.

Samia Bushnaq, Shaher Momani, and Yong Zhou. A reproducing kernel Hilbert space method for solving integro-differential equations of fractional order. J. Optim. Theory Appl., 156(1):96–105, 2013.

Minggen Cui and Yingzhen Lin. Nonlinear numerical analysis in the reproducing kernel space. Nova Science Publishers, Inc., New York, 2009.

Barun Kumar Datta. Analytic solution for the Blasius equation. Indian J. Pure Appl. Math., 34(2):237–240, 2003.

Vedat Suat Ertürk and Shaher Momani. Numerical solutions of two forms of Blasius equation on a half-infinite domain. J. Algorithms Comput. Technol., 2(3):359– 370, 2008.

Tiegang Fang, Wei Liang, and Chia-fon F. Lee. A new solution branch for the Blasius equation—a shrinking sheet problem. Comput. Math. Appl., 56(12):3088– 3095, 2008.

Mojtaba Fardi, Reza Khoshsiar Ghaziani, and Mehdi Ghasemi. The Reproducing Kernel Method for Some Variational Problems Depending on Indefinite Integrals. Math. Model. Anal., 21(3):412–429, 2016.

Riccardo Fazio. Numerical transformation methods: Blasius problem and its variants. Appl. Math. Comput., 215(4):1513–1521, 2009.

D. D. Ganji, H. Babazadeh, F. Noori, M. M. Pirouz, and M. Janipour. An application of homotopy perturbation method for non-linear Blasius equation to boundary layer flow over a flat plate. Int. J. Nonlinear Sci., 7(4):399–404, 2009.

Fazhan Geng and Minggen Cui. Solving singular nonlinear second-order periodic boundary value problems in the reproducing kernel space. Appl. Math. Comput., 192(2):389–398, 2007.

H. Ghaneai and M. M. Hosseini. Solving differential-algebraic equations through variational iteration method with an auxiliary parameter. Appl. Math. Model.,40(5-6):3991–4001, 2016.

Ji-Huan He. A simple perturbation approach to Blasius equation. Appl. Math. Comput., 140(2-3):217–222, 2003.

L. Howarth. Laminar boundary layers. In Handbuch der Physik (herausgegeben von S. Fl¨ugge), Bd. 8/1, Str¨omungsmechanik I (Mitherausgeber C. Truesdell), pages 264–350. Springer-Verlag, Berlin-G¨ottingen-Heidelberg, 1959.

Mustafa Inc and Ali Akgül. Approximate solutions for MHD squeezing fluid flow by a novel method. Bound. Value Probl., pages 2014:18, 17, 2014.

Mustafa Inc, Ali Akgül, and Fazhan Geng. Reproducing kernel Hilbert space method for solving Bratu’s problem. Bull. Malays. Math. Sci. Soc., 38(1):271–287, 2015.

Ernest D. Kennedy. Application of a new method of approximation in the solution of ordinary differential equations to the Blasius equation. Trans. ASME Ser. E. J. Appl. Mech., 31:112–114, 1964.

Shi-Jun Liao. An explicit, totally analytic approximate solution for Blasius’ viscous flow problems. Internat. J. Non-Linear Mech., 34(4):759–778, 1999.

Jianguo Lin. A new approximate iteration solution of Blasius equation. Commun. Nonlinear Sci. Numer. Simul., 4(2):91–99, 1999.

Chin-Chia Liu. Numerical study of mixed convection MHD flow in vertical channels using differential transformation method. Appl. Math. Inf. Sci., 9(1L):105– 110, 2015.

Banan Maayah, Samia Bushnaq, Shaher Momani, and Omar Abu Arqub. Iterative multistep reproducing kernel Hilbert space method for solving strongly nonlinear oscillators. Adv. Math. Phys., pages Art. ID 758195, 7, 2014.

Vasile Marinca and Nicolae Heris¸anu. The optimal homotopy asymptotic method for solving Blasius equation. Appl. Math. Comput., 231:134–139, 2014.

M. O. Miansari, M. E. Miansari, A. Barari, and G. Domairry. Analysis of Blasius equation for flat-plate flow with infinite boundary value. Int. J. Comput. Methods Eng. Sci. Mech., 11(2):79–84, 2010.

D. E. Panayotounakos, N. B. Sotiropoulos, A. B. Sotiropoulou, and N. D. Panayotounakou. Exact analytic solutions of nonlinear boundary value problems in fluid mechanics (Blasius equations). J. Math. Phys., 46(3):033101, 26, 2005.

K. Parand, Mehdi Dehghan, and A. Pirkhedri. Sinc-collocation method for solving the Blasius equation. Phys. Lett. A, 373(44):4060–4065, 2009.

Haldun Alpaslan Peker, Onur Karao˘glu, and Galip Oturanc¸. The differential transformation method and Pade approximant for a form of Blasius equation. Math. Comput. Appl., 16(2):507–513, 2011.

Arjuna I. Ranasinghe and Fayequa B. Majid. Solution of Blasius equation by decomposition. Appl. Math. Sci. (Ruse), 3(13-16):605–611, 2009.

W. Robin. Some remarks on the homotopy-analysis method and series solutions to the Blasius equation. Int. Math. Forum, 8(25-28):1205–1213, 2013.

M. Sajid, Z. Abbas, N. Ali, and T. Javed. A hybrid variational iteration method for Blasius equation. Appl. Appl. Math., 10(1):223–229, 2015.

Mehmet Giyas Sakar, Fatih Uludag, and Fevzi Erdogan. Numerical solution of time-fractional nonlinear PDEs with proportional delays by homotopy perturbation method. Appl. Math. Model., 40(13-14):6639–6649, 2016.

Nabil Shawagfeh, Omar Abu Arqub, and Shaher Momani. Analytical solution of nonlinear second-order periodic boundary value problem using reproducing kernel method. J. Comput. Anal. Appl., 16(4):750–762, 2014.

G. I. Shishkin. Grid approximation of the solution of the Blasius equation and of its derivatives. Zh. Vychisl. Mat. Mat. Fiz., 41(1):39–56, 2001.

Z. Q. Tang and F. Z. Geng. Fitted reproducing kernel method for singularly perturbed delay initial value problems. Appl. Math. Comput., 284:169–174, 2016.

Inayat Ullah, Hamid Khan, and M. T. Rahim. Approximation of first grade MHD squeezing fluid flow with slip boundary condition using DTM and OHAM. Math. Probl. Eng., pages Art. ID 816262, 9, 2013.

Hector Vazquez-Leal. Generalized homotopy method for solving nonlinear differential equations. Comput. Appl. Math., 33(1):275–288, 2014.

Abdul-Majid Wazwaz. The variational iteration method for solving two forms of Blasius equation on a half-infinite domain. Appl. Math. Comput., 188(1):485–491, 2007.

Min-Qiang Xu and Ying-Zhen Lin. Simplified reproducing kernel method for fractional differential equations with delay. Appl. Math. Lett., 52:156–161, 2016.

Baoheng Yao and Jianping Chen. A new analytical solution branch for the Blasius equation with a shrinking sheet. Appl. Math. Comput., 215(3):1146–1153, 2009.

Lien-Tsai Yu and Cha’o-Kuang Chen. The solution of the Blasius equation by the differential transformation method. Math. Comput. Modelling, 28(1):101–111, 1998.

Zhihong Zhao, Yingzhen Lin, and Jing Niu. Convergence Order of the Reproducing Kernel Method for Solving Boundary Value Problems. Math. Model. Anal.,

(4):466–477, 2016.




DOI: http://dx.doi.org/10.11121/ijocta.01.2017.00363

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