Numerical Solution of a Laminar Viscous Flow Boundary Layer Equation Using Uniform Haar Wavelet Quasi-linearization Method
In this paper, we have proposed a Haar wavelet quasilinearization
method to solve the well known Blasius equation. The
method is based on the uniform Haar wavelet operational matrix
defined over the interval [0, 1]. In this method, we have proposed the
transformation for converting the problem on a fixed computational
domain. The Blasius equation arises in the various boundary layer
problems of hydrodynamics and in fluid mechanics of laminar
viscous flows. Quasi-linearization is iterative process but our
proposed technique gives excellent numerical results with quasilinearization
for solving nonlinear differential equations without any
iteration on selecting collocation points by Haar wavelets. We have
solved Blasius equation for 1≤α ≤ 2 and the numerical results are
compared with the available results in literature. Finally, we
conclude that proposed method is a promising tool for solving the
well known nonlinear Blasius equation.
<p>[1] F.K. Tsou, E. M. Sparrow and R. J. Goldstein, “Flow and heat transfer
in the boundary layer on a continuous moving surface,” Int. J. Heat.
Mass Trans., vol.10, no. 2, pp. 219-235, February 1967.
[2] J. P. Boyd, “Pad´e approximant algorithm for solving nonlinear ODE
boundary value problems on an unbounded domain,” Comput. Phys.,
vol. 11, pp. 299-303, 1997.
[3] J. P. Boyd, “The Blasius function in the complex plane,” J. Experi.
Math., vol. 8, pp. 381-394, 1999.
[4] R. Cortell, “Numerical solution of the classical Blasius flat-plate
problem,” Appl. Math. Comput. vol 170, pp. 706-10, 2005.
[5] L. Howarth ,”On the solution of the laminar boundary layer equation,”
Proc Roy Soc London, vol. 164, pp. 547-79,1938.
[6] A. Asaithambi, “Solution of the Falkner-Skan equation by recursive
evaluation of Taylor coefficients,” J. Comput. Appl. Math., vol. 176, pp.
203-14, 2005.
[7] T. Fang, F. Guo and C.F. Lee, “A note on the extended Blasius
Problem,” Appl. Math. Lett. vol. 19, pp. 613-17, 2004.
[8] J. H. He , “Comparison of homotopy perturbation method and
homotopy analysis method,” Appl. Math. Comput, vol. 156, pp. 527-39,
2004.
[9] F. Ahmad, ”Degeneracy in the Blasius problem,” Electron J Differ
Equations, vol. 92, pp. 1-8, 1998.
[10] F. Ahmad, Al-Barakati W.H., “An approximate analytic solution of the
Blasius problem,” Commun. Nonlinear Sci. Numer. Simul, vol. 14, pp.
1021-24, 2009.
[11] S.J. Liao,”An explicit, totally analytic solution of laminar viscous flow
over a semi-infinite flat plate,” Commun. Nonlinear Sci. Numer. Simul.
vol. 3 no. 2, pp. 53-57, 1998.
[12] S. J. Liao, “A an explicit, totally analytical approximate solution for
blasius viscous flow problem,” Int. J. Non-Linear Mech., vol. 34, 1999.
[13] J. He. “Approximate analytical solution of Blasius equation,” Commun
Nonlinear Sci Numer Simul., vol. 4 no. 1, pp. 75-80, 1999.
[14] S. Abbasbandy, “A numerical solution of blasius equation by adomians
decomposition method and comparison with homotopy perturbation
method,” Chaos, Solitons and Fractals, vol. 31, pp. 257-260, 2007.
[15] I. Daubechies, “Orthonormal bases of compactly supported wavelets,”
Comm. Pure Appl. Math. , vol. 41, pp. 909-996, 1998.
[16] A. Haar, Zur theorie der orthogonalen Funktionsysteme. Math Annal.,
vol. 69, pp. 331-71, 1910.
[17] C.H. Hsiao, “State analysis of linear time delayed system via Haar
wavelets,” Math. Comput. Simu. vol. 44, pp. 457-470, 1997.
[18] R. E. Bellman and Kalaba, Quasilinearization and nonlinear boundary
value problems, Elsevier, New York, 1965.
[19] H. Kaur, R.C. Mittal and V. Mishra,” Haar wavelet quasilinearization
approach for solving nonlinear boundary value problems,” Amer. J.
Comput. Math., vol. 1, pp. 176-182, 2011.
[20] V. Mishra, H. Kaur and R.C. Mittal, “Haar wavelet algorithm for
solving certain differential, integral and integro-differential equations,”
Int. J. Appl. Math and Mech., vol. 8, pp. 1-15, 2012.
[21] H. Kaur, R.C. Mittal and V. Mishra, “Haar wavelet approximate
solutions for the generalized Lane Emden equations arising in
astrophysics,” Comput. Phys. Commun. (2013). DOI:
http://dx.doi.org/10.1016/j.cpc.2013.04.013)</p>
<p>[1] F.K. Tsou, E. M. Sparrow and R. J. Goldstein, “Flow and heat transfer
in the boundary layer on a continuous moving surface,” Int. J. Heat.
Mass Trans., vol.10, no. 2, pp. 219-235, February 1967.
[2] J. P. Boyd, “Pad´e approximant algorithm for solving nonlinear ODE
boundary value problems on an unbounded domain,” Comput. Phys.,
vol. 11, pp. 299-303, 1997.
[3] J. P. Boyd, “The Blasius function in the complex plane,” J. Experi.
Math., vol. 8, pp. 381-394, 1999.
[4] R. Cortell, “Numerical solution of the classical Blasius flat-plate
problem,” Appl. Math. Comput. vol 170, pp. 706-10, 2005.
[5] L. Howarth ,”On the solution of the laminar boundary layer equation,”
Proc Roy Soc London, vol. 164, pp. 547-79,1938.
[6] A. Asaithambi, “Solution of the Falkner-Skan equation by recursive
evaluation of Taylor coefficients,” J. Comput. Appl. Math., vol. 176, pp.
203-14, 2005.
[7] T. Fang, F. Guo and C.F. Lee, “A note on the extended Blasius
Problem,” Appl. Math. Lett. vol. 19, pp. 613-17, 2004.
[8] J. H. He , “Comparison of homotopy perturbation method and
homotopy analysis method,” Appl. Math. Comput, vol. 156, pp. 527-39,
2004.
[9] F. Ahmad, ”Degeneracy in the Blasius problem,” Electron J Differ
Equations, vol. 92, pp. 1-8, 1998.
[10] F. Ahmad, Al-Barakati W.H., “An approximate analytic solution of the
Blasius problem,” Commun. Nonlinear Sci. Numer. Simul, vol. 14, pp.
1021-24, 2009.
[11] S.J. Liao,”An explicit, totally analytic solution of laminar viscous flow
over a semi-infinite flat plate,” Commun. Nonlinear Sci. Numer. Simul.
vol. 3 no. 2, pp. 53-57, 1998.
[12] S. J. Liao, “A an explicit, totally analytical approximate solution for
blasius viscous flow problem,” Int. J. Non-Linear Mech., vol. 34, 1999.
[13] J. He. “Approximate analytical solution of Blasius equation,” Commun
Nonlinear Sci Numer Simul., vol. 4 no. 1, pp. 75-80, 1999.
[14] S. Abbasbandy, “A numerical solution of blasius equation by adomians
decomposition method and comparison with homotopy perturbation
method,” Chaos, Solitons and Fractals, vol. 31, pp. 257-260, 2007.
[15] I. Daubechies, “Orthonormal bases of compactly supported wavelets,”
Comm. Pure Appl. Math. , vol. 41, pp. 909-996, 1998.
[16] A. Haar, Zur theorie der orthogonalen Funktionsysteme. Math Annal.,
vol. 69, pp. 331-71, 1910.
[17] C.H. Hsiao, “State analysis of linear time delayed system via Haar
wavelets,” Math. Comput. Simu. vol. 44, pp. 457-470, 1997.
[18] R. E. Bellman and Kalaba, Quasilinearization and nonlinear boundary
value problems, Elsevier, New York, 1965.
[19] H. Kaur, R.C. Mittal and V. Mishra,” Haar wavelet quasilinearization
approach for solving nonlinear boundary value problems,” Amer. J.
Comput. Math., vol. 1, pp. 176-182, 2011.
[20] V. Mishra, H. Kaur and R.C. Mittal, “Haar wavelet algorithm for
solving certain differential, integral and integro-differential equations,”
Int. J. Appl. Math and Mech., vol. 8, pp. 1-15, 2012.
[21] H. Kaur, R.C. Mittal and V. Mishra, “Haar wavelet approximate
solutions for the generalized Lane Emden equations arising in
astrophysics,” Comput. Phys. Commun. (2013). DOI:
http://dx.doi.org/10.1016/j.cpc.2013.04.013)</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:49798", author = "Harpreet Kaur and Vinod Mishra and R. C. Mittal", title = "Numerical Solution of a Laminar Viscous Flow Boundary Layer Equation Using Uniform Haar Wavelet Quasi-linearization Method", abstract = "In this paper, we have proposed a Haar wavelet quasilinearization
method to solve the well known Blasius equation. The
method is based on the uniform Haar wavelet operational matrix
defined over the interval [0, 1]. In this method, we have proposed the
transformation for converting the problem on a fixed computational
domain. The Blasius equation arises in the various boundary layer
problems of hydrodynamics and in fluid mechanics of laminar
viscous flows. Quasi-linearization is iterative process but our
proposed technique gives excellent numerical results with quasilinearization
for solving nonlinear differential equations without any
iteration on selecting collocation points by Haar wavelets. We have
solved Blasius equation for 1≤α ≤ 2 and the numerical results are
compared with the available results in literature. Finally, we
conclude that proposed method is a promising tool for solving the
well known nonlinear Blasius equation.
", keywords = "Boundary layer Blasius equation, collocation points,
quasi-linearization process, uniform haar wavelets.", volume = "7", number = "7", pages = "1072-6", }
