The variational iteration method for solving two forms of Blasius equation on a half-infinite domain

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Abstract

The variational iteration method is applied for a reliable treatment of two forms of the third order nonlinear Blasius equation which comes from boundary layer equations. The study shows that the series solution is obtained without restrictions on the nonlinearity behavior. The obtained series solution is combined with the diagonal Padé approximants to handle the boundary condition at infinity for only one of these forms.

Introduction

Blasius equation is one of the basic equations of fluid dynamics. Blasius equation describes the velocity profile of the fluid in the boundary layer theory [1], [2] on a half-infinite interval. A broad class of analytical solutions methods and numerical solutions methods in [1], [2], [3], [4], [5] were used to handle this problem.

Two forms of Blasius equation appear in the fluid mechanic theory, where each is subjected to specific physical conditions. The equation has the formsu(x)+12u(x)u(x)=0,u(0)=0,u(0)=1,u()=0,andu(x)+12u(x)u(x)=0,u(0)=0,u(0)=0,u()=1.It is obvious that the differential equations are the same, but differ in boundary conditions. For more details about the appearance of the two forms, see reference [1]. It is well known that the Blasius equation is the mother of all boundary-layer equations in fluid mechanics. Many different, but related, equations have been derived for a multitude of fluid-mechanical situations, for instance, the Falkner–Skan equation [1].

An analytic treatment will be approached to find the numerical values of u″(0) for both boundary value problems. The goal will be achieved by using the reliable variational iteration method developed by He in [5], [6], [7], [8], [9], [10] and used in [11], [12], [13], [14] and the references therein. The obtained series is best manipulated by Padé approximants for numerical approximations [15], [16], [17], [18], [19] for the first form (1). Using the power series, isolated from other concepts, is not always useful because the radius of convergence of the series may not contain the two boundaries.

The variational iteration method (VIM) established in (1999) by He in [5], [6], [7], [8], [9], [10] is thoroughly used by many researchers to handle linear and nonlinear models. The method has been used by many authors in [11], [12], [13], [14] and the references therein to handle a wide variety of scientific and engineering applications: linear and nonlinear, and homogeneous and inhomogeneous as well. It was shown by many authors that this method provides improvements over existing numerical techniques. The method gives successive approximations of high accuracy of the solution. The VIM does not require specific treatments as in Adomian method, and perturbation techniques for nonlinear terms.

In this paper, only a brief discussion of the variational iteration method will be emphasized, because the complete details of the method are found in [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] and in many related works. The objectives of this work are twofold. Firstly, we aim to apply the variational iteration method (VIM) in a direct manner to establish series solutions for Eqs. (1), (2). Secondly, we seek to show the power of the method in handling linear and nonlinear equations in a unified manner without requiring any additional restriction. The VIM method is capable of greatly reducing the size of calculations while still maintaining high accuracy of the numerical solution. In what follows, we highlight the main steps of the He’s variational iteration method.

Section snippets

The He’s variational iteration method

Consider the differential equationLu+Nu=g(x,t),where L and N are linear and nonlinear operators respectively, and g(x, t) is the source inhomogeneous term. In [5], [6], [7], [8], [9], [10], the variational iteration method was proposed by He where a correction functional for Eq. (3) can be written asun+1(x,t)=un(x,t)+0tλLun(ξ)+Nu˜n(ξ)-g(ξ)dξ,n0.It is obvious that the successive approximations uj, j  0 can be established by determining λ, a general Lagrange’s multiplier, which can be identified

The first form of Blasius equation

We first consider the first form of Blasius equationu(x)+12u(x)u(x)=0,u(0)=0,u(0)=1,u()=0,The correction functional for (6) readsun+1(x)=un(x)+0xλ(ξ)3un(ξ)ξ3+12u˜n(ξ)2u˜n(ξ)ξ2dξ.Following [5] the stationary conditionsλ(ξ)=0,1+λ|ξ=x=0,λ|ξ=x=0,λξ=x=0.This in turn givesλ=-12(ξ-x)2.Substituting this value of the Lagrangian multiplier into the functional (7) gives the iteration formulaun+1(x)=un(x)-120x(ξ-x)23un(ξ)ξ3+12u˜n(ξ)2u˜n(ξ)ξ2dξ,n0.We select the initial value u0=x+12Ax2

The second form of Blasius equation

We next consider the second form of Blasius equationu(x)+12u(x)u(x)=0,u(0)=0,u(0)=0,u()=1.Following [20], we reformulate (13) by introducing a new dependent variabley(x)=Bu(Bx),that is equivalentu(x)=1By1Bx,where B is a parameter to be determined. As a result, Eq. (13) will be the same as the equation for u:y(x)+12y(x)y(x)=0.Consequently, we have the conditionsy(0)=0,y(0)=0,y(x)=B2u(Bx)B2asx.This also means that we can impose the condition y″(0) = 1, which means that u(0)=1B3.

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