Application of He’s Variational Iterative Method for Solving Thin Film Flow Problem Arising in Non-Newtonian Fluid Mechanics

Abdul M Siddiqui,Tahira Haroon,Muhammad A Rana,Ali A Farooq,Bruce S Babcock

doi:10.4236/wjm.2012.23016

Abstract

In this paper, He’s variational iteration method is successfully employed to solve a nonlinear boundary value problem arising in the study of thin film flow of a third grade fluid down an inclined plane. For comparison, the same problem is solved by the Adomian decomposition method. The results show that the difference between the two solutions is negligible. The conclusion is that this technique may be considered an alternative and efficient method for finding approximate solutions of both linear and nonlinear boundary value problems. Furthermore, the variational iteration method has an advantage over the decomposition method in that it solves the nonlinear problems without using the Adomian polynomials.

Highlights

Many approximate analytical and numerical methods have been suggested for solving linear and nonlinear boundary value problems arising in different branches of science and engineering
He’s variational iteration method is successfully employed to solve a nonlinear boundary value problem arising in the study of thin film flow of a third grade fluid down an inclined plane
The variational iteration method has an advantage over the decomposition method in that it solves the nonlinear problems without using the Adomian polynomials

Summary

Introduction

Many approximate analytical and numerical methods have been suggested for solving linear and nonlinear boundary value problems arising in different branches of science and engineering. It is not difficult to solve a linear problem because of the availability of high performance digital computers, but finding solutions of nonlinear problems is still not easy. We have applied the VIM and the ADM to find the approximate solutions of nonlinear and inhomogeneous differential equation governing the thin film flow of a third grade fluid down an inclined plane, and have made a graphical comparison of the numerical results from these two methods. Mustafa Inc and Ebru Cavlak [10] have provided a comparative study of ADM and VIM in solving a new coupled MKdV system of equations. These methods generate the solution in a convergent series with components that are elegantly computed.

Governing Equation

Solution by Variational Iteration Method

Solution by Adomian Decomposition Method

Conclusion