## THERMAL SCIENCE

International Scientific Journal

### ASSESSMENT OF HOMOTOPY PERTURBATION METHOD IN NONLINEAR CONVECTIVE-RADIATIVE NON-FOURIER CONDUCTION HEAT TRANSFER EQUATION WITH VARIABLE COEFFICIENT

**ABSTRACT**

Analytical solutions play a very important role in heat transfer. In this paper, the He's homotopy perturbation method (HPM) has been applied to nonlinear convective-radiative non-Fourier conduction heat transfer equation with variable specific heat coefficient. The concept of the He's homotopy perturbation method are introduced briefly for applying this method for problem solving. The results of HPM as an analytical solution are then compared with those derived from the established numerical solution obtained by the fourth order Runge-Kutta method in order to verify the accuracy of the proposed method. The results reveal that the HPM is very effective and convenient in predicting the solution of such problems, and it is predicted that HPM can find a wide application in new engineering problems.

**KEYWORDS**

PAPER SUBMITTED: 2011-01-09

PAPER REVISED: 2011-07-01

PAPER ACCEPTED: 2011-08-18

**THERMAL SCIENCE** YEAR

**2011**, VOLUME

**15**, ISSUE

**Supplement 2**, PAGES [S263 - S274]

- He, J. H., Homotopy Perturbation Technique, Comput. Methods Appl. Mech. Eng., 178 (1999), 3-4, pp. 257–262
- He, J. H., A Note on the Homotopy Perturbation Method, Thermal Science, 14 (2010), 2, pp. 565-568
- He, J. H., Homotopy Perturbation Method for Solving Boundary Value Problems, Phys. Lett. A, 350 (2006), 1-2, pp. 87-88
- Ganji, D. D., The Application of He’s Homotopy-Perturbation Method to Nonlinear Equations Arising in Heat Transfer, Phys. Lett. A, 355 (2006), 4-5, pp. 337-341
- Ganji, D. D., Rajabi, A., Assessment of Homotopy-Perturbation and Perturbation Methods in Heat Radiation Equations, Int. Commun. Heat Mass Transfer, 33 (2006), 3, pp. 391-400
- Ganji, D. D., Rafei, M., Solitary Wave Solutions for a Generalized Hirota-Satsuma Coupled KdV Equation by Homotopy Perturbation Method, Phys. Lett. A, 356 (2006), 2, pp. 131-137
- Ganji, D. D., Sadighi, A., Application of He’s Homotopy-Perturbation Method to Nonlinear Coupled Systems of Reaction-Diffusion Equations, Int. J. Nonlinear Sci. Numer. Simul., 7 (2006), 4, pp. 411-418
- Ganji, D. D., Hosseini, M. J., Shayegh, J., Some Nonlinear Heat Transfer Equations Solved by Three Approximate Methods, Int. Comm. Heat Mass Transf., 34 (2007), 8, pp. 1003-1016
- Ganji, D. D., Afrouzi, G. A., Talarposhti, R. A., Application of Variational Iteration Method and Homotopy Perturbation Method for Nonlinear Heat Diffusion and Heat Transfer Equations, Phys. Lett. A, 368 (2007), 6, pp. 450-457
- Khaleghi, H., Ganji, D. D., Sadighi, A., Application of Variational Iteration and Homotopy Perturbation Methods to Nonlinear Heat Transfer Equations with Variable Coefficients, Numer. Heat Transfer A, 52 (2007), 1, pp. 25-42
- Rajabi, A., Ganji, D. D., Taherian, H., Application of Homotopy Perturbation Method in Nonlinear Heat Conduction and Convection Equations, Phys. Lett. A, 360 (2007), 4-5, pp. 570-573
- Sajid, M., Hayat, T., Comparison of HAM and HPM Methods in Nonlinear Heat Conduction and Convection Equations, Nonlinear Analysis: Real World Application, 9 (2008), 5, pp. 2296-2301
- Sajid, M., Hayat, T., Comparison of HAM and HPM Solutions in Heat Radiation Equations, Int. Commun. Heat Mass Transfer, 36 (2009), 1, pp. 59-62
- Chowdhury, M. S. H., Hashim, I., Abdulaziz, O., Comparison of Homotopy Analysis Method and Homotopy-Perturbation Method for Purely Nonlinear Fin-Type Problems, Commun. Nonlinear Sci. Numer. Simul., 14 (2009), 2, pp. 371-378
- Fathizadeh, M., Rashidi, F., Boundary Layer Convective Heat Transfer with Pressure Gradient Using Homotopy Perturbation Method (HPM) over a Flat Plate, Chaos, Solitons Fractals, 42 (2009), 4, pp. 2413-2419
- Marinca, V., Herişanu, N., Optimal Homotopy Perturbation Method for Strongly Nonlinear Differential Equations, Nonlinear Science Letters A, 1 (2010), 3, pp. 273-280
- He, J. H., Analytical Methods for Thermal Science – An Elementary Introduction, Thermal Science, 15 (2011), Suppl. 1, pp. S1-S3
- Ganji, D. D., Ganji, Z. Z., Ganji, H. D., Determination of Temperature Distribution for Annular Fins with Temperature Dependent Thermal Conductivity by HPM, Thermal Science, 15 (2011), Suppl. 1, pp. S111-S115
- Jackson, H. E., Walker, C. T., Thermal Conductivity, Second Sound, and Phonon-Phonon Interactions in NaF, Physical Review B, 3 (1971), 4, pp. 1428-1439
- Narayanamurti, V., Dynes, R. C., Observation of Second Sound in Bismuth, Physical Review Letters, 28 (1972), 22, pp. 1461-1465
- He, J. H., A Coupling Method of a Homotopy Technique and a Perturbation Technique for Non-Linear Problems, Int. J. Non-linear Mech., 35 (2000), 1, pp. 37-43