Modified rational Legendre approach to laminar viscous flow over a semi-infinite flat plate

A numerical method for solving the classical Blasius’ equation is proposed. The Blasius’ equation is a third order nonlinear ordinary differential equation , which arises in the problem of the two-dimensional laminar viscous flow over a semi-infinite flat plane. The approach is based on a modified rational Legendre tau method. The operational matrices for the derivative and product of the modified rational Legendre functions are presented. These matrices together with the tau method are utilized to reduce the solution of Blasius’ equation to the solution of a system of algebraic equations. A numerical evaluation is included to demonstrate the validity and applicability of the method and a comparison is made with existing results.

[1]  Ji-Huan He.  Approximate analytical solution of Blasius' equation , 1998 .

[2]  Saeid Abbasbandy,et al.  A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method , 2007 .

[3]  Jie Shen,et al.  A Rational Approximation and Its Applications to Differential Equations on the Half Line , 2000, J. Sci. Comput..

[4]  L. Howarth,et al.  On the solution of the laminar boundary layer equations. , 1938, Proceedings of the Royal Society of London. Series A - Mathematical and Physical Sciences.

[5]  D. Funaro,et al.  Approximation of some diffusion evolution equations in unbounded domains by hermite functions , 1991 .

[6]  C. Lanczos.  Applied Analysis , 1961 .

[7]  J. Boyd.  The Optimization of Convergence for Chebyshev Polynomial Methods in an Unbounded Domain , 1982 .

[8]  S. Liao.  A kind of approximate solution technique which does not depend upon small parameters — II. An application in fluid mechanics , 1997 .

[9]  G. Ben-yu.  Error estimation of Hermite spectral method for nonlinear partial differential equations , 1999 .

[10]  S. Momani,et al.  Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order , 2006 .

[11]  Shijun Liao,et al.  On the homotopy analysis method for nonlinear problems , 2004, Appl. Math. Comput..

[12]  Ji-Huan He,et al.  A simple perturbation approach to Blasius equation , 2003, Appl. Math. Comput..

[13]  J. Boyd.  Spectral methods using rational basis functions on an infinite interval , 1987 .

[14]  T. A. Zang,et al.  Spectral methods for fluid dynamics , 1987 .

[15]  J. Boyd.  Orthogonal rational functions on a semi-infinite interval , 1987 .

[16]  Jie Shen,et al.  Stable and Efficient Spectral Methods in Unbounded Domains Using Laguerre Functions , 2000, SIAM J. Numer. Anal..

[17]  A. Wazwaz.  A NOTE ON USING ADOMIAN DECOMPOSITION METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS , 2000 .

[18]  Shaher Momani,et al.  Application of He’s variational iteration method to Helmholtz equation , 2006 .

[19]  H. Blasius.  Grenzschichten in Flüssigkeiten mit kleiner Reibung , 1907 .

[20]  John P. Boyd,et al.  The Blasius Function in the Complex Plane , 1999, Exp. Math..

[21]  Sana’a A. Zarea,et al.  The Decomposition Method Applied to Solve High-order Linear Volterra-Fredholm Integro-differential Equations , 2004 .

[22]  Mehdi Dehghan,et al.  He’s variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation , 2007 .

[23]  Shijun Liao,et al.  An explicit, totally analytic solution of laminar viscous flow over a semi-infinite flat plate , 1998 .

[24]  Jie Shen,et al.  Laguerre-Galerkin method for nonlinear partial differential equations on a semi-infinite interval , 2000, Numerische Mathematik.

[25]  C. Christov.  A Complete Orthonormal System of Functions in $L^2 ( - \infty ,\infty )$ Space , 1982 .

[26]  Error analysis of one-leg methods for differential-algebraic equations of index 2 , 1999 .

[27]  A. A. Soliman,et al.  Variational iteration method for solving Burger's and coupled Burger's equations , 2005 .

[28]  Magdy A. El-Tawil,et al.  The Solution of KdV and mKdV Equations Using Adomian Pade Approximation , 2004 .

[29]  Ji-Huan He.  Approximate solution of nonlinear differential equations with convolution product nonlinearities , 1998 .

[30]  Ji-Huan He.  A new approach to nonlinear partial differential equations , 1997 .

[31]  Ji-Huan He.  HOMOTOPY PERTURBATION METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS , 2006 .

[32]  Ji-Huan He.  Approximate analytical solution for seepage flow with fractional derivatives in porous media , 1998 .

[33]  M. Ramadan,et al.  The use of adomian decomposition method for solving the regularized long-wave equation , 2005 .

[34]  Ji-Huan He,et al.  Variational iteration method for autonomous ordinary differential systems , 2000, Appl. Math. Comput..

[35]  Ji-Huan He,et al.  Comparison of homotopy perturbation method and homotopy analysis method , 2004, Appl. Math. Comput..

[36]  D. F. Hays,et al.  Table of Integrals, Series, and Products , 1966 .

[37]  B. Guo,et al.  On spectral approximations using modified Legendre rational functions: Application to the Korteweg-de Vries equation on the half line , 2001 .

[38]  Ji-Huan He.  Variational iteration method – a kind of non-linear analytical technique: some examples , 1999 .