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 free-convective boundary-layer equation. It is observed that the combination of MVIM and the Padé approximation improves the accuracy and enlarges the convergence domain.
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  introduced He’s polynomials which are calculated for He’s homotopy perturbation method. It is also established  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  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”  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.
2. 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 is a linear operator, is a nonlinear operator, and 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 is a Lagrange multiplier [1, 2, 10–15, 42], which can be identified optimally via variational iteration method. The subscripts denote the th approximation; is considered as a restricted variation. That is, (2.2) is called a correction functional. Now, we apply He’s polynomials 
which is the coupling of variational iteration method and He’s polynomials and is called the modified variational iteration method (MVIM) [4, 33–36]. The comparison of like powers of gives solutions of various orders.
3. Mathematical Model
Let us consider the problem of cooling of a low-heat-resistance sheet that moves downwards in a viscous fluid :
where and are the velocity components in the - and -directions, respectively.is the temperature, is the temperature of the surrounding fluid, is the kinematic viscosity, is the thermal diffusivity, is the acceleration due to gravity, and is the coefficient of thermal expansion. Using the similarity variables
where is the stream function defined by and , and are the similarity functions dependent on and , (3.1) is transformed to
subject to the boundary conditions
where the primes denote differentiation with respect to , and is the Prandtl number.
4. The Padé Approximation
We denote Padé approximants to by
where is polynomial of degree at most and is a polynomial of degree at most . The former power series is
And we write the and as
and the coefficients of and are determined by the equation. From (4.4), we have
which system of homogeneous equations with unknown quantities. We impose the normalization condition
We can write out (4.5) as
From (4.7), we can obtain . Once the values of are all known (4.8) gives an explicit formula for the unknown quantities . For the diagonal approximants like have the most accurate approximants by built-in utilities of Maple.
5. Solution Procedure
The correction functional is given by
Making the correction functional stationary, the Lagrange multipliers can easily be identified
Applying the modified variational iteration method (MVIM), we get
Comparing the coefficient of like powers of , we get
The series solution is given by
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, 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 σimplying a slow rate of thermal diffusion. Thus higher Prandtl number σleads to faster cooling of the plane sheet.
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.
The authors are highly grateful to the referee for his/her very constructive comments.
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