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

Jiancheng Hu

doi:10.1186/s40064-016-1835-z

Abstract

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 [0,infty ) is equivalently transformed into a second-order boundary value problem on finite interval [beta , 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.

Highlights

The well-known nonlinear third-order Falkner–Skan equation is one of the nonlinear two-point boundary value problem (BVP) on infinite intervals
The solutions of the one-dimensional third-order boundary-value problem described by the Falkner–Skan equation are the similarity solutions of the two-dimensional incompressible laminar boundary layer equations (Cheng 1977; Merkin 1980; Salama 2004; Postelnicu and Pop 2011; Mosayebidorcheh 2013)
In this work, we have demonstrated the effectiveness of the finite difference method to Falkner–Skan equation

Summary

Introduction

The well-known nonlinear third-order Falkner–Skan equation is one of the nonlinear two-point boundary value problem (BVP) on infinite intervals. This problem arises in the study of laminar boundary layers exhibiting similarity in fluid mechanics. The solutions of the one-dimensional third-order boundary-value problem described by the Falkner–Skan equation are the similarity solutions of the two-dimensional incompressible laminar boundary layer equations (Cheng 1977; Merkin 1980; Salama 2004; Postelnicu and Pop 2011; Mosayebidorcheh 2013). The nonlinear BVP (1–2) with β = 0 is studied (Aly et al 2003; Nazar et al 2004) and comes from the study of a plane mixed convection boundary-layer flow near a semiinfinite vertical surface, with a prescribed power law of the distance from the leading edge for the temperature. About BVP (1–2), there have existed some interesting results

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Conclusion