The modified decomposition method and Padé approximants for a boundary layer equation in unbounded domain

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Abstract

The modified decomposition method is applied for analytic treatment of nonlinear differential equations that appear on boundary layers in fluid mechanics. The modified method accelerates the rapid convergence of the series solution, dramatically reduces the size of work. The obtained series solution is combined with the diagonal Padé approximants to handle the boundary condition at infinity.

Introduction

Nonlinear phenomena, that appear in many areas of scientific fields such as solid state physics, plasma physics, fluid mechanics, population models and chemical kinetics, can be modelled by nonlinear differential equations. In particular, the nonlinear differential equations that characterize boundary layers in unbounded domain, that will be examined here, is of much interest. A broad class of analytical solutions methods and numerical solutions methods were used to handle these problems. The Adomian decomposition method has been proved to be effective and reliable for handling differential equations, linear or nonlinear.

In this work we aim to apply the modified decomposition method to handle the nonlinear differential equationf+(n-1)ff-2n(f)2=0,f(0)=0,f(0)=1,f()=0,n>0,that appear in boundary layers in fluid mechanics [1], [2], where f″(0) < 0. An analytic treatment will be approached to find the numerical values of f″(0) for several values of n. The goal will be achieved by combining the series obtained by the modified decomposition method with the diagonal Padé approximants.

It is interesting to point out that Kuiken [1], [2] investigated this problem for three cases of n, namely for 0 < n < 1, n = 1 and for n > 1. It was shown in [1] that the algebraic behavior at the outer edge of a boundary layer in fluid mechanics can sometimes be allowed in regions of non-vanishing extent.

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 [3]. Eq. (1) used by Kuiken [1] does not include the Blasius equation for a particular choice of n. Indeed, this equation is for backward boundary layers, that is, boundary layers originating at −∞.

Although the convergence provided by Adomian decomposition method is rapid [4], the modified decomposition method developed by Wazwaz in [5] accelerates this rapid convergence of the series solution. The method provides the series solution by using few iterations; therefore it facilitates the computational work and minimizes its volume.

Although the modified form introduces a slight change in the formulation of Adomian recursive relation, but it provides a qualitative improvement over standard Adomian method. While this slight variation is rather simple, it does demonstrate the reliability, efficiency, and power of the method. It is well known that polynomials are used to approximate truncated power series. Moreover, polynomials can never blow up in a finite plane and this makes the singularities not apparent. To overcome these difficulties, the obtained series is best manipulated by Padé approximants for numerical approximations. 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.

In this paper, only a brief discussion of the modified decomposition analysis will be emphasized, because the complete details of the method are found in [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16] and in many related works.

Section snippets

The modified decomposition method

In this section, we will briefly discuss the use of the modified decomposition method for nonlinear differential equations where details can be found in [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16].

Without loss of generality, we consider the differential equationLu+Ru+Nu=g(x)with prescribed conditions, where u(x) is the unknown function, L is the highest order derivative which is assumed to be easily invertible, R is a linear differential operator of less order than L, Nu

The boundary layer problem

The nonlinear third order boundary layer problemf+(n-1)ff-2n(f)2=0,n>0,f(0)=0,f(0)=1,f()=0,n>0,that appear in boundary layers in fluid mechanics [1], [2], where f″(0) = α < 0, will be examined in this work.

To apply the modified decomposition method, we first rewrite (8) in an operator formLf=-(n-1)ff+2n(f)2,where L is a third order differential operator, and hence L−1 is a three-fold integration operator defined byL-1(·)=0t0t0t(·)dtdtdt.Operating with L−1 on both sides of (9) and using

Discussion

The modified decomposition method was applied to compute an approximation for the function f(x) for several values of n, n > 0. This advantage demonstrates the reliability and the efficiency of the method. Further, the accuracy level can be dramatically enhanced by computing further components of f(x). Moreover, combining the series obtained with the Padé approximants provides a promising tool to handle problems on an unbounded domain.

An important conclusion, that was thoroughly investigated by

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