## The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer.(English)Zbl 1255.80026

Summary: The homotopy perturbation method (HPM), which does not need small parameters in the equations, is compared with the perturbation and numerical methods in the heat transfer field. The perturbation method depends on small parameter assumption, and the obtained results, in most cases, end up with a non-physical result, the numerical method leads to inaccurate results when the equation is intensively dependent on time, while He’s homotopy perturbation method (HPM) overcomes completely the above shortcomings, revealing that the HPM is very convenient and effective. Comparing different methods shows that, when the effect of the nonlinear term is negligible, homotopy perturbation method and the common perturbation method have got nearly the same answers but when the nonlinear term in the heat equation is more effective, there will be a considerable difference between the results. As the homotopy perturbation method does not need a small parameter, the answer will be nearer to the exact solution and also to the numerical one.

### MSC:

 80M25 Other numerical methods (thermodynamics) (MSC2010) 80A20 Heat and mass transfer, heat flow (MSC2010)
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### References:

 [1] Bellman, R., Perturbation techniques in mathematics, physics and engineering, (1964), Holt, Rinehart & Winston New York [2] Cole, J.D., Perturbation methods in applied mathematics, (1968), Blaisedell Waltham, MA · Zbl 0162.12602 [3] L. Cveticanin, Homotopy-perturbation method for pure nonlinear differential equation, Chaos Solitons Fractals, 2005, in press · Zbl 1238.65085 [4] El-Shahed, M., Int. J. nonlinear sci. numer. simul., 6, 2, 163, (2005) [5] Ganji, D.D.; Rajabi, A., Int. commun. heat mass transfer, 33, 3, 391, (2006) [6] He, J.H., J. comput. methods appl. mech. eng., 167, 1-2, 57, (1998) [7] He, J.H., J. comput. methods appl. mech. eng., 167, 1-2, 69, (1998) [8] He, J.H., Int. J. non-linear mech., 34, 4, 699, (1999) [9] He, J.H., J. comput. methods appl. mech. eng., 178, 3-4, 257, (1999) [10] He, J.H., Int. J. non-linear mech., 35, 1, 37, (2000) [11] He, J.H., J. appl. math. comput., 114, 2-3, 115, (2000) [12] He, J.H., J. appl. math. comput., 135, 1, 73, (2000) [13] He, J.H.; Wan, Y.Q.; Guo, Q., Int. J. circuit theory appl., 32, 6, 629, (2004) [14] He, J.H., J. appl. math. comput., 151, 1, 287, (2004) [15] He, J.H., J. appl. math. comput., 156, 3, 591, (2004) [16] He, J.H., Int. J. nonlinear sci. numer. simul., 6, 2, 207, (2005) [17] He, J.H., Chaos solitons fractals, 26, 3, 695, (2005) [18] He, J.H., Chaos solitons fractals, 26, 3, 827, (2005) [19] He, J.H., Phys. lett. A, 347, 4-6, 228, (2005) [20] He, J.H.; Wu, X.H., Chaos solitons fractals, 29, 1, 108, (2006) [21] He, J.H., Phys. lett. A, 350, 1-2, 87, (2006) [22] G.L. Liu, New research directions in singular perturbation theory: artificial parameter approach and inverse-perturbation technique, in: Conference of 7th Modern Mathematics and Mechanics, Shanghai, 1997 [23] Nayfeh, A.H., Perturbation methods, (1973), Wiley New York · Zbl 0375.35005 [24] O’Malley, R.E., Introduction to singular perturbation, (1974), Academic Press New York [25] Van Dyke, M., Perturbation methods in fluid mechanics, (1975), Parabolic Press Stanford, CA, annotated edition · Zbl 0329.76002 [26] Y’aziz, A.; Hamad, G., Int. J. mech. eng. educ., 5, 167, (1977)
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