A numerical solution of a singular boundary value problem arising in boundary layer theory

In this paper, a second-order nonlinear singular boundary value problem is presented, which is equivalent to the well-known Falkner–Skan equation. And the one-dimensional third-order boundary value problem on interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,\infty )$$\end{document}[0,∞) is equivalently transformed into a second-order boundary value problem on finite interval \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[\beta , 1]$$\end{document}[β,1]. The finite difference method is utilized to solve the singular boundary value problem, in which the amount of computational effort is significantly less than the other numerical methods. The numerical solutions obtained by the finite difference method are in agreement with those obtained by previous authors.

Current numerical analysis is an important technique for the solution of the Falkner-Skan equation. One key problem for numerical technique is how to deal with the infinite boundary. Early approaches have mainly used shooting or invariant imbedding (Cebeci and Keller 1971;Na 1979). Asaithambi presented an asymptotic condition and truncated the infinite boundary condition by an unknown η ∞ (Asaithambi 1998(Asaithambi , 2004(Asaithambi , 2005. Adomian decomposition method was developed to obtain series solutions instead of truncating the infinite boundary (Elgazery 2005;Alizadeh et al. 2009). Yang and Hu (2008) transformed the problem to a singular boundary value problem on finite interval and proposed Galerkin finite element method.
Based on ideas (Yang 2003;Lan and Yang 2008), the purpose of this paper is to transform the problem mentioned above to a singular boundary value problem on a finite interval and develop a finite difference method which is much more effective and simpler than the other existing methods for BVP (1-2), and which requires much less computational effort. Lan and Yang (2008) established the equivalence between the Falkner-Skan equation and a singular integral equation. In this paper, the BVP (1-2) is transformed to a secondorder singular boundary value problem, and the solution of BVP (1-2) is characterized by f ′′ (0).

Numerical solutions of boundary value problem
Equation (10) can be changed to the following equivalent form subject to the boundary conditions In this paper, the numerical solution of Eq. (13) with boundary conditions (14, 15) is based on the the finite difference method. The interval [β, 1] is divided into N subintervals with step size h = 1−β N , and define t j = β + jh for j = 0, 1, . . . , N. Let w j denotes the values of w(t j ) for j = 0, 1, . . . , N. Let t = t j , the finite difference formulation of Eq. (13) writes as for j = 1, 2, . . . , N − 1. The boundary condition (14) corresponds to And the discretization of boundary condition (15) reads as The discretization formulation (16-18) is a nonlinear equation system, so Newton iteration method is recommended to solve approximate solutions. We now proceed to describe the iterative process for the solution of the nonlinear system (16-18). Let w T = [w 0 · · · w N ], and where and for j = 1, 2, . . . , N − 1.
The solving Eqs. (16-18) is equivalent to solving the system described by Newton's iteration method is recommended to solve nonlinear system (22). Given and initial values w 0 j , j = 0, 1, 2, . . . , N, the k-th Newton's iterates w k = [w k 0 , w k 1 , . . . w k N ] T , k = 1, 2, . . . , can be obtained by solving system (22). Newton's method for the solution of Eq. (22) proceeds to yield subsequent iterates for w as where △w k satisfies the equation The iterative process described by Eqs. (23, 24) may be repeated in succession until �△w k � ∞ < ε for some prescribed error tolerance ε.

Results and discussion
The Falkner-Skan equation has two parameters β and , and Aly et al. (2003) obtained some numerical solution for various β and . Also, the numerical solutions of the equation have been simulated by using Galerkin finite element methods for various values of β and (Yang and Hu 2008). In order to demonstrate the reliability and efficiency of the proposed theory. The numerical results have been obtained by solving the boundary value problems (13-15) with different parameters and β. And comparison of the . . .
It can be seen from Fig. 1, where f ′′ (0)(= w(β)) is plotted as a function of β in the range of 0 ≤ β ≤ 1, curves are drawn for value = −0.30, −0.25, −0.20, −0.18, −0.15, −0.10. It is also shown that f ′′ (0)(= w(β)) changes smoothly with β. As increases, the results also increase in the range of 0 ≤ β ≤ 1. Figure 2 shows the characteristics of numerical solutions f ′′ (0)(= w(β)) for β = 0.0-0.9 by solving the boundary value problems (13-15). The solutions indicate that f ′′ (0)(= w(β)) decreases with increasing of the parameter β, i.e., f ′′ (0)(= w(β)) is a decrease function of parameter β. For each fixed value of , solution of f ′′ (0)(= w(β)) decreases with increase of β in the range of [0, 1 ], and especially, when β = 0 and = 0, the classical Balasis solution is obtained (Aly et al. 2003). Because the Eq. (13) is a second-order boundary value problem, the amount of computational effort used by finite difference method is significantly less than the other numerical methods of the third-order differential equation which essentially solve two or more initial value problems during each iteration (Asaithambi 2004). In general, the numerical simulation shows that the initial guess for w 0 could be far away from the exact value. For each fixed value of w 0 , the method in this paper required 2-6 iterations in order to solve system (22) to the desired accuracy.

Conclusions
In this work, we have demonstrated the effectiveness of the finite difference method to Falkner-Skan equation. Applying equivalent transformation to Falkner-Skan equation, a third-order boundary value problem in infinite interval is transformed into a secondorder boundary value problem in finite interval. By using finite difference method and Newton's iteration approximation, the numerical solution have been calculated.
The results of comparison studied in this paper indicate that, the values of the Newton's iteration for f ′′ (0)(= w(β)) are in excellent agreement with those results obtained by previous authors. Therefore, the method presented in this work shows its validity and great potential for the solution of Falkner-Skan equations arising in science and engineering.