The variational iteration method for solving two forms of Blasius equation on a half-infinite domain
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 formsandIt 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 equationwhere 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 asIt 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 equationThe correction functional for (6) readsFollowing [5] the stationary conditionsThis in turn givesSubstituting this value of the Lagrangian multiplier into the functional (7) gives the iteration formulaWe select the initial value
The second form of Blasius equation
We next consider the second form of Blasius equationFollowing [20], we reformulate (13) by introducing a new dependent variablethat is equivalentwhere B is a parameter to be determined. As a result, Eq. (13) will be the same as the equation for u:Consequently, we have the conditionsThis also means that we can impose the condition y″(0) = 1, which means that .
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