Afrika Matematika

, Volume 24, Issue 1, pp 103–116 | Cite as

Optimal homotopy asymptotic method for convective–radiative cooling of a lumped system, and convective straight fin with temperature-dependent thermal conductivity

  • Saeed DinarvandEmail author
  • Reza Hosseini


Here, the optimal homotopy asymptotic method (OHAM), one of the newest analytical methods which is powerful and easy-to-use, is applied to solve heat transfer problems with high nonlinearity order. In this research, the OHAM is used to investigate two problems: the temperature distribution equation in a convective straight fin with temperature-dependent thermal conductivity and the convective–radiative cooling of a lumped system with variable specific heat. The validity of our results is verified by numerical results. The OHAM provides us with a convenient way to control the convergence of approximation series and adjust convergence regions when necessary. This realizes using a number of auxiliary constants which are optimally determined. Thus, unlike perturbation methods, the OHAM does not depend on any small physical parameters and it is valid for both weakly and strongly nonlinear problems. It has been attempted to show the capabilities and wide-range applications of the OHAM in comparison with the previous methods in solving heat transfer problems.


Nonlinear heat transfer equations Extended surface Convective–radiative cooling Temperature-dependent thermal conductivity Optimal homotopy asymptotic method (OHAM) Homotopy perturbation method (HPM) 

List of symbols


Area (m2)


Cross-sectional area of fin (m2)


Fin length (m)


Specific heat (J kg−1 K−1)


Specific heat at ambient temperature (J kg−1 K−1)


Surface emissivity (W)


Convection heat transfer coefficient (W m−2 K−1)


Thermal conductivity of fin material (W m−1 K−1)


Thermal conductivity at ambient temperature (W m−1 K−1)


Thermal conductivity at base temperature (W m−1 K−1)


Fin perimeter (m)


Temperature (K)


Ambient temperature (K)


Base temperature of fin (K)


Initial temperature (K)


Effective sink temperature (K)


Volume (m3)


Distance measured from fin tip (m)

Greek symbols


Volumetric thermal expansion coefficient (K−1)


Dimensionless temperature (–)


Slope of thermal conductivity-temperature curve (K−1)


Dimensionless length (–)


Density (kg m−3)


Stefan–Boltzman constant (–)


Dimensionless time (–)


Thermo-geometric fin parameter (–)

Mathematics Subject Classification (2000)

35A24 (Methods of ordinary differential equations) 49K15 (Problems involving ordinary differential equations) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aziz A., Na T.Y.: Perturbation Method in Heat Transfer. Hemisphere Publishing Corporation, Washington (1984)Google Scholar
  2. 2.
    Nayfeh AH.: Introduction to Perturbation Techniques. Wiley, New York (1979)Google Scholar
  3. 3.
    Rand R.H., Armbruster D.: Perturbation Methods, Bifurcation Theory and Computer Algebraic. Springer, Berlin (1987)CrossRefGoogle Scholar
  4. 4.
    He, J.H.: Non-perturbative methods for strongly nonlinear problems. Dissertation, de-Verlag im Internet GmbH, Berlin (2006)Google Scholar
  5. 5.
    Bildik N., Konuralp A.: The use of variational iteration method, differential transform method and adomian decomposition method for solving different types of nonlinear partial differential equations. Int. J. Nonlinear Sci. Numer. Simul. 7, 65–70 (2006)CrossRefGoogle Scholar
  6. 6.
    Lyapunov A.M.: General Problem on Stability of Motion (English translation). Taylor and Francis, London (1992)Google Scholar
  7. 7.
    Karmishin A.V., Zhukov A.I., Kolosov V.G.: Methods of Dynamics Calculation and Testing for Thin-Walled Structures. Mashinostroyenie, Moscow (1990)Google Scholar
  8. 8.
    Adomian G.: Solving Frontier Problems of Physics: The Decomposition Method. Kluwer, Dordrecht (1994)zbMATHGoogle Scholar
  9. 9.
    Liao, S.J.: The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiao Tong University (1992)Google Scholar
  10. 10.
    Liao S.J.: Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and Hall/CRC Press, Boca Raton (2003)CrossRefGoogle Scholar
  11. 11.
    Liao S.J.: On the homotopy analysis method for nonlinear problems. Appl. Math. Comput. 147, 499–513 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    He J.H.: Variational iteration method for autonomous ordinary differential systems. Appl. Math. Comput. 114(2–3), 115–123 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Marinca V., Herişanu N., Nemeş I.: Optimal homotopy asymptotic method with application to thin film flow. Cent. Eur. J. Phys. 6, 648–653 (2008)CrossRefGoogle Scholar
  14. 14.
    Marinca V., Herişanu N.: Application of optimal homotopy asymptotic method for solving nonlinear equations arising in heat transfer. Int. Commun. Heat Mass Transf. 35, 710–715 (2008)CrossRefGoogle Scholar
  15. 15.
    Marinca V., Herişanu N., Bota C., Marinca B.: Optimal homotopy asymptotic method to the steady flow of a fourth grade fluid past a porous plate. Appl. Math. Lett. 22, 245–251 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Herişanu N., Marinca V., Dordea T., Madescu G.: A new analytical approach to nonlinear vibration of an electric machine. Proc. Romanian Acad. Ser. A 9(3), 229–236 (2008)Google Scholar
  17. 17.
    Marinca V., Herişanu N.: Determination of periodic solutions for the motion of a particle on a rotating parabola by means of the optimal homotopy asymptotic method. J. Sound Vib. 359, 1450–1459 (2010)CrossRefGoogle Scholar
  18. 18.
    He J.H.: A coupling method for homotopy technique and perturbation technique for nonlinear problem. Int. J. NonLinear Mech. 35, 37–43 (2000)zbMATHCrossRefGoogle Scholar
  19. 19.
    He J.H.: Asymptotology by homotopy perturbation method. Appl. Math. Comput. 156, 591–596 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    He J.H.: Homotopy perturbation method for solving boundary problems. Phys. Lett. A 350, 87–88 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Kern D.Q., Kraus D.A.: Extended Surface Heat Transfer. McGraw-Hill, New York (1972)Google Scholar
  22. 22.
    Aziz A., Enamul Hug S.M.: Perturbation solution for convecting fin with variable thermal conductivity. J. Heat Transf. 97, 300–310 (1975)CrossRefGoogle Scholar
  23. 23.
    Razelos P., Imre K.: The optimum dimension of circular fins with variable thermal parameters. J. Heat Transf. 102, 420–425 (1980)CrossRefGoogle Scholar
  24. 24.
    Laor K., Kalman H.: Performance and optimum dimensions of different cooling fins with a temperature-dependent heat transfer coefficient. Int. J. Heat Mass Transf. 39, 1993–2003 (1996)CrossRefGoogle Scholar
  25. 25.
    Yu L.T., Chen C.K.: Optimization of circular fins with variable thermal parameters. J. Franklin Inst. B 336, 77–95 (1999)zbMATHCrossRefGoogle Scholar
  26. 26.
    Incropera F.P., Dewitt D.P.: Introduction to Heat Transfer, 3rd edn, pp. 114. Wiley, New York (1996)Google Scholar
  27. 27.
    Herisanu N., Marinca V.: Accurate analytical solutions to oscillators with discontinuities and fractional-power restoring force by means of the optimal homotopy asymptotic method. Comput. Math. Appl. 60, 1607–1615 (2010)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© African Mathematical Union and Springer-Verlag 2011

Authors and Affiliations

  1. 1.Mechanical Engineering DepartmentAmirkabir University of TechnologyTehranIran

Personalised recommendations