0
Technical Brief

Optimal Perturbation Iteration Method for Solving Nonlinear Heat Transfer Equations

[+] Author and Article Information
Sinan Deniz

Department of Mathematics,
Faculty of Art and Sciences,
Manisa Celal Bayar University,
Manisa 45140, Turkey
e-mail: sinan.deniz@cbu.edu.tr

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received May 16, 2016; final manuscript received February 21, 2017; published online April 4, 2017. Assoc. Editor: George S. Dulikravich.

J. Heat Transfer 139(7), 074503 (Apr 04, 2017) (4 pages) Paper No: HT-16-1277; doi: 10.1115/1.4036085 History: Received May 16, 2016; Revised February 21, 2017

In this paper, the new optimal perturbation iteration method (OPIM) is introduced and applied for solving nonlinear differential equations arising in heat transfer. The effectiveness of the proposed method will be tested by considering two specific applications: the temperature distribution equation in a thick rectangular fin radiation to free space and cooling of a lumped system with variable specific heat. Comparing different methods shows that the results obtained by optimal perturbation iteration method are very good agreement with the numerical solutions and perform better than the most existing analytic methods.

FIGURES IN THIS ARTICLE
<>
Copyright © 2017 by ASME
Your Session has timed out. Please sign back in to continue.

References

Doğan, N. , Ertürk, V. S., Momani, S., Akın, Ö., and Yıldırım, A., 2011, “ Differential Transform Method for Solving Singularly Perturbed Volterra Integral Equations,” J. King Saud Univ., Sci., 23(2), pp. 223–228. [CrossRef]
Liao, S. , 2004, “ On the Homotopy Analysis Method for Nonlinear Problems,” Appl. Math. Comput., 147(2), pp. 499–513.
Liao, S. , 2003, Beyond Perturbation: Introduction to the Homotopy Analysis Method, CRC Press, Boca Raton, FL.
He, J.-H. , 2005, “ Homotopy Perturbation Method for Bifurcation of Nonlinear Problems,” Int. J. Nonlinear Sci. Numer. Simul., 6(2), pp. 207–208.
He, J.-H. , 1999, “ Homotopy Perturbation Technique,” Comput. Methods Appl. Mech. Eng., 178(3), pp. 257–262. [CrossRef]
He, J.-H. , 2000, “ A Coupling Method of a Homotopy Technique and a Perturbation Technique for Non-Linear Problems,” Int. J. Non-Linear Mech., 35(1), pp. 37–43. [CrossRef]
He, J.-H. , 2006, “ Homotopy Perturbation Method for Solving Boundary Value Problems,” Phys. Lett. A, 350(1), pp. 87–88. [CrossRef]
Öziş, T. , and Ağirseven, D. , 2008, “ He's Homotopy Perturbation Method for Solving Heat-Like and Wave-Like Equations With Variable Coefficients,” Phys. Lett. A, 372(38), pp. 5944–5950. [CrossRef]
Evans, D. J. , and Raslan, K. R. , 2005, “ The Adomian Decomposition Method for Solving Delay Differential Equation,” Int. J. Comput. Math., 82(1), pp. 49–54. [CrossRef]
Daftardar-Gejji, V. , and Jafari, H. , 2006, “ An Iterative Method for Solving Nonlinear Functional Equations,” J. Math. Anal. Appl., 316(2), pp. 753–763. [CrossRef]
Aksoy, Y. , Pakdemirli, M., Abbasbandy, S., Boyacı, H., 2012, “ New Perturbation-Iteration Solutions for Nonlinear Heat Transfer Equations,” Int. J. Numer. Methods Heat Fluid Flow, 22(7), pp. 814–828. [CrossRef]
Marinca, V. , and Herişanu, N. , 2008, “ Application of Optimal Homotopy Asymptotic Method for Solving Nonlinear Equations Arising in Heat Transfer,” Int. Commun. Heat Mass Transfer, 35(6), pp. 710–715. [CrossRef]
Aksoy, Y. , and Pakdemirli, M. , 2010, “ New Perturbation–Iteration Solutions for Bratu-Type Equations,” Comput. Math. Appl., 59(8), pp. 2802–2808. [CrossRef]
Şenol, M. , Dolapçı, İ. T., Aksoy, Y., and Pakdemirli, M., 2013, “ Perturbation-Iteration Method for First-Order Differential Equations and Systems,” Abstract and Applied Analysis, Vol. 2013, Hindawi Publishing Corporation, London, UK.
Khalid, M. , Sultana, M., Zaidi, F., and Khan, F. S., 2015, “ Solving Polluted Lakes System by Using Perturbation-Iteration Method,” Int. J. Comput. Appl., 114(4), pp. 1–7.
Abbasbandy, S. , 2006, “ The Application of Homotopy Analysis Method to Nonlinear Equations Arising in Heat Transfer,” Phys. Lett. A, 360(1), pp. 109–113. [CrossRef]
Abbasbandy, S. , 2007, “ Homotopy Analysis Method for Heat Radiation Equations,” Int. Commun. Heat Mass Transfer, 34(3), pp. 380–387. [CrossRef]
Ganji, D. D. , 2006, “ The Application of He's Homotopy Perturbation Method to Nonlinear Equations Arising in Heat Transfer,” Phys. Lett. A, 355(4), pp. 337–341. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Comparison between the results obtained by OPIM (•, ▲) and the numerical results (–) for example 1: (a) OPIA-1 (•) and OPIA-2 (▲) solutions for ε = 1 and (b) OPIA-1 (•) and OPIA-2 (▲) solutions for ε = 2

Grahic Jump Location
Fig. 2

Absolute error of OPIM by third-order approximation and comparison between the results obtained by OPIM and the numerical results (–) for example 2: (a) second (•) and third-order (▲) approximate solutions for ε = 1 and (b) absolute error for ε = 1

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In