Modified Variational Iteration Method for Free-Convective Boundary-Layer Equation Using Padé Approximation

This paper is devoted to the study of a free-convective boundary-layer flow modeled by a system of nonlinear ordinary differential equations. We apply a modified variational iteration method MVIM coupled with He’s polynomials and Padé approximation to solve freeconvective boundary-layer equation. It is observed that the combination of MVIM and the Padé approximation improves the accuracy and enlarges the convergence domain.


Introduction
The boundary-layer flows of viscous fluids are of utmost importance for industry and applied sciences. These flows can be modeled by systems of nonlinear ordinary differential equations on an unbounded domain, see 1-4 and the references therein. Keeping in view the physical importance of such problems, there is a dire need of extension of some reliable and efficient technique for the solution of such problems. He 1, 2, 5-15 developed the variational iteration VIM and homotopy perturbation HPM methods which are very efficient and accurate and are 1, 2, 4-42 being used very frequently for finding the appropriate solutions of nonlinear problems of physical nature. In a later work, Ghorbani and Nadjfi 24 introduced He's polynomials which are calculated for He's homotopy perturbation method. It is also established 24 that He's polynomials are compatible with Adomian's polynomials but are easier to implement and are more user friendly. Recently, Mohyud-Din, Noor and Noor 4, 33-36 made the elegant coupling of He's polynomials and the correction functional of variational iteration method VIM and found the solutions of number of nonlinear singular and nonsingular problems. It is observed that 4, 33-36 the modified version of VIM is very efficient in solving nonlinear problems. The basic motivation of this paper is the extension of the modified variational iteration method MVIM coupled with Padé approximation to solve a free-convective boundary-layer flow modeled by a system of nonlinear ordinary differential equations. Numerical and figurative illustrations show that it is a promising tool to solve nonlinear problems. It needs to be highlighted that Herisanu and Marinca 41 suggested an optimal variational iteration algorithm. It needs to be highlighted that He in his latest article "The variational iteration method which should be followed" 42 presented a very comprehensive and detailed study on various aspects of variational iteration method in connection with partial differential equations, ordinary differential equations, fractional differential equations, fractal-differential equations, and difference-differential equations.

Modified Variational Iteration Method (MVIM)
To illustrate the basic concept of the modified variational iteration method MVIM , we consider the following general differential equation: where L is a linear operator, N is a nonlinear operator, and g x is the forcing term. According to variational iteration method 1, 2, 4, 10-23, 28, 33-39, 41, 42 , we can construct a correction functional as follows: where λ is a Lagrange multiplier 1, 2, 10-15, 42 , which can be identified optimally via variational iteration method. The subscripts n denote the nth approximation; u n is considered as a restricted variation. That is, δ u n 0; 2.2 is called a correction functional. Now, we apply He's polynomials 24

Mathematical Model
Let us consider the problem of cooling of a low-heat-resistance sheet that moves downwards in a viscous fluid 3 : where u and v are the velocity components in the x-and y-directions, respectively. ψT is the temperature, T 0 is the temperature of the surrounding fluid, ν is the kinematic viscosity, κ is the thermal diffusivity, g is the acceleration due to gravity, and β is the coefficient of thermal expansion. Using the similarity variables where ψ is the stream function defined by u ∂ψ/∂y and v −∂ψ/∂x, f and θ are the similarity functions dependent on η, T 0, 0 subject to the boundary conditions where the primes denote differentiation with respect to η, and σ is the Prandtl number.

The Padé Approximation
We denote L, M Padé approximants to f z by where P L z is polynomial of degree at most L and Q M z Q M z / 0 is a polynomial of degree at most M. The former power series is And we write the P L z and Q M z as We can write out 4.5 as . . ., c L c L−1 · q 1 · · · c 0 · q L p L .

Solution Procedure
Consider problems 3.4 -3.5 formulated in Section 3 and is related to the free-convective boundary-layer flow.
The correction functional is given by

5.1
Making the correction functional stationary, the Lagrange multipliers can easily be identified Consequently, df n dη θ n η ds.

5.3
Applying the modified variational iteration method MVIM , we get

8 Mathematical Problems in Engineering
Comparing the coefficient of like powers of p, we get  The series solution is given by

5.6
It is observed in Figures 1 and 2 that the flow has a boundary-layer structure and the thickness of this boundary-layer decreases with increase in the Prandtl number, σ as expected. This is due to the inhibiting influence of the viscous forces. Figure 3 shows the increase of the Prandtl number, σ, that results in the decrease, as expected, of temperature distribution at a particular point of the flow region, that is, there would be a decrease of the thermal boundary-layer thickness with the increase of values of σ implying a slow rate of thermal diffusion. Thus higher Prandtl number σ leads to faster cooling of the plane sheet.

Conclusions
In this study, we employed modified variational iteration method MVIM coupled with Padé approximation to solve a system of two nonlinear ordinary differential equations that describes a free-convective boundary-layer in glass-fiber production process. The results show strong effects of the Prandtl number on the velocity and temperature profiles since the two model equations are coupled.