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


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.

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A. Siddiqui, A. Farooq, T. Haroon, M. Rana and B. Babcock, "Application of He’s Variational Iterative Method for Solving Thin Film Flow Problem Arising in Non-Newtonian Fluid Mechanics," World Journal of Mechanics, Vol. 2 No. 3, 2012, pp. 138-142. doi: 10.4236/wjm.2012.23016.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] S. J. Liao, “Beyond Perturbation: Introduction to Homo-topy Analysis Method,” Chapman & Hall/CRC Press, Boca Raton, 2004.
[2] J. H. He, “Homotopy Perturbation Technique,” Computer Methods in Applied Mechanics and Engineering, Vol. 178, No. 3-4, 1999, pp. 257-262. doi:10.1016/S0045-7825(99)00018-3
[3] J. H. He, “A Coupling Method of a Homotopy Technique and a Perturbation Technique for Non-Linear Problems,” International Journal of Non-Linear Mechanics, Vol. 35, No. 1, 2000, pp. 37-43.
[4] J. H. He, “Variational Iteration Method—A Kind of Non- Linear Analytical Technique: Some Examples,” International Journal of Non-Linear Mechanics, Vol. 34, No. 4, 1999, pp. 699-708. doi:10.1016/S0020-7462(98)00048-1
[5] J. H. He, “Variational Iteration Method—Some Recent Results and New Interpretations,” Journal of Computational and Applied Mathematics, Vol. 207, No. 1, 2007, pp. 3-17.
[6] G. Adomian, “Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer Academic Publishers, Boston, 1994.
[7] G. Adomian, “A Review of the Decomposition Method in Applied Mathematics,” Journal of Mathematical Analysis and Applications, Vol. 135, No. 2, 1988, pp. 501-544. doi:10.1016/0022-247X(88)90170-9
[8] V. Marinca and N. Herisanu, “Application of Optimal Homotopy Asymptotic Method for Solving Nonlinear Equations Arising in Heat Transfer,” International Communications in Heat and Mass Transfer, Vol. 35, No. 6, 2008, pp. 710-715. doi:10.1016/j.icheatmasstransfer.2008.02.010
[9] S. Islam, Z. Bano, I. Siddique and A. M. Siddiqui, “The Optimal Solution for the Flow of a Fourth-Grade Fluid with Partial Slip,” Journal Computers & Mathematics with Applications, Vol. 61, No. 6, 2011, pp. 1507-1516. doi:10.1016/j.camwa.2011.01.014
[10] M. Inc and E. Cavlak, “On Numerical Solutions of a New Coupled MKdV System by Using the Adomian Decomposition Method and He’s Variational Iteration Method,” Physica Scripta, Vol. 78, No. 4, 2008, pp. 1-7. doi:10.1088/0031-8949/78/04/045008
[11] N. Bildik and A. Konuralp, “Two-Dimensional Differential Transform Method, Adomian’s Decomposition Method, and Variational Iteration Method for Partial Differential Equations,” International Journal of Computer Mathematics, Vol. 83, No. 12, 2006, pp. 973-987. doi:10.1080/00207160601173407
[12] J. I. Ramos, “On the Variational Iteration Method and Other Iterative Techniques for Nonlinear Differential Equations,” Applied Mathematics and Computation, Vol. 199, No. 1, 2008, pp. 39-69. doi:10.1016/j.amc.2007.09.024
[13] A. M. Siddiqui, R. Mahmood and Q. K. Ghori, “Homotopy Perturbation Method for Thin Film Flow of a Third Grade Fluid Down an Inclined Plane,” Chaos, Solitons & Fractals, Vol. 35, No. 1, 2008, pp. 140-147. doi:10.1016/j.chaos.2006.05.026
[14] M. Tari and M Dehghan, “On the Convergence of He’s Variational Iteration Method,” Journal of Computational and Applied Mathematics, Vol. 207, No. 1, 2007, pp. 121-128. doi:10.1016/
[15] A. M. Siddiqui, M. Hameed, B. M. Siddiqui and Q. K. Ghori, “Use of Adomian Decomposition Method in the Study of Parallel Plate Flow of a Third Grade Fluid,” Communication in Nonlinear Science and Numerical Simula- tion, Vol. 15, No. 9, pp. 2388-2399, 2010. doi:10.1016/j.cnsns.2009.05.073

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