falkner skan
An iterative method for solving the Falkner-Skan equation Jiawei Zhang∗
Binghe Chen
Department of Mathematics, Zhejiang University
Hangzhou, Zhejiang 310027, China
Abstract
We propose a new iterative method to solve the boundary value problems (BVPs) of the Falkner-Skan equation over a semi-infinite interval. In our approach, we use the free boundary formulation to truncate the semi-infinite interval into a finite one. Then we use the shooting method to transform the BVP into initial value problems (IVPs).
In order to find the “shooting angle” and the unknown free boundary, a modification of the classical Newton’s method is used where the Jacobian matrix can be accurately obtained by solving another two IVPs. To illustrate the effectiveness of our method, we compare our numerical results with those obtained by previous methods under various instances of the Falkner-Skan equation.
Keywords: Nonlinear boundary value problems; Semi-infinite intervals; Newton’s method; Shooting; Free boundary formulation.
1 Introduction
Finding the numerical solution of nonlinear BVPs on infinite intervals is one of the problems attracting many scientists. The nonlinear third-order Falkner-Skan equation is a famous example of the BVPs on infinite intervals arisen in many branches of sciences, e.g. applied mathematics, physics, fluid dynamics and biology.
We are to solve the following Falkner-Skan equation d3 f d2 f df + β0 f 2 + β 1 −
3
dη dη dη
2
= 0,
0 < η < ∞,
(1)
with the boundary condition f = 0,
∗ Email:
at η = 0,
jiaweiz@ZJU.edu.cn
1
(2)
df
= 0, dη df
= 1, dη at η = 0,
(3)
at η → ∞,
(4)
in which β0 and β are constants.
Current numerical methods are mainly based on shooting [1], finite differences [2], finite elements [3], and the Adomian decomposition method [4]. In this paper, we aim at providing an easy and efficient method based on shooting and the free boundary formulation [5], where the shooting
Binghe Chen
Department of Mathematics, Zhejiang University
Hangzhou, Zhejiang 310027, China
Abstract
We propose a new iterative method to solve the boundary value problems (BVPs) of the Falkner-Skan equation over a semi-infinite interval. In our approach, we use the free boundary formulation to truncate the semi-infinite interval into a finite one. Then we use the shooting method to transform the BVP into initial value problems (IVPs).
In order to find the “shooting angle” and the unknown free boundary, a modification of the classical Newton’s method is used where the Jacobian matrix can be accurately obtained by solving another two IVPs. To illustrate the effectiveness of our method, we compare our numerical results with those obtained by previous methods under various instances of the Falkner-Skan equation.
Keywords: Nonlinear boundary value problems; Semi-infinite intervals; Newton’s method; Shooting; Free boundary formulation.
1 Introduction
Finding the numerical solution of nonlinear BVPs on infinite intervals is one of the problems attracting many scientists. The nonlinear third-order Falkner-Skan equation is a famous example of the BVPs on infinite intervals arisen in many branches of sciences, e.g. applied mathematics, physics, fluid dynamics and biology.
We are to solve the following Falkner-Skan equation d3 f d2 f df + β0 f 2 + β 1 −
3
dη dη dη
2
= 0,
0 < η < ∞,
(1)
with the boundary condition f = 0,
∗ Email:
at η = 0,
jiaweiz@ZJU.edu.cn
1
(2)
df
= 0, dη df
= 1, dη at η = 0,
(3)
at η → ∞,
(4)
in which β0 and β are constants.
Current numerical methods are mainly based on shooting [1], finite differences [2], finite elements [3], and the Adomian decomposition method [4]. In this paper, we aim at providing an easy and efficient method based on shooting and the free boundary formulation [5], where the shooting
References: Fract. 35:738–746 (2008). [7] D. F. Rogers, Laminar Flow Analysis, Cambridge University Press, Cambridge, 1992.