Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations. (English) Zbl 1156.76436

Summary: Recently, A. Rajabi et al. [Phys. Lett., A 360, No. 4–5, 570–573 (2006; Zbl 1236.65059)] discussed the solutions of temperature distribution in lumped system of combined convection-radiation. They solved a nonlinear equation of the steady conduction in a slab with variable thermal conductivity using both perturbation and homotopy perturbation methods. They claim that homotopy perturbation method (HPM) does not require any small parameter. However, this statement is not true always. Moreover, HPM have no criteria for establishing the convergence of the series solution. In this letter we have explicitly shown that the results of the problem considered in example 2 of [loc. cit.] are valid only for \(0\leqslant \varepsilon \leqslant 1\). We have used the homotopy analysis method for finding the more meaningful solution.


76N25 Flow control and optimization for compressible fluids and gas dynamics
76R10 Free convection
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
65L99 Numerical methods for ordinary differential equations


Zbl 1236.65059
Full Text: DOI


[1] Abbas, Z.; Sajid, M.; Hayat, T., MHD boundary layer flow of an upper-convected Maxwell fluid in a porous channel, Theor. comput. fluid dyn., 20, 229-238, (2006) · Zbl 1109.76065
[2] Abbasbandy, S., The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys. lett. A, 360, 109-113, (2006) · Zbl 1236.80010
[3] S. Abbasbandy, Solitary wave solutions to the Kuramoto-Sivashinsky by means of homotopy analysis method, Nonlinear Dyn., in press. · Zbl 1173.35646
[4] S. Abbasbandy, Y. Tan, S.J. Liao, Newton-homotopy analysis method for nonlinear equations, Appl. Math. Comput., in press. · Zbl 1119.65032
[5] P.M. Fitzpatrick, Advanced Calculus, PWS Publishing Company, 1996. · Zbl 1229.26001
[6] Hayat, T.; Abbas, Z.; Sajid, M., Series solution for the upper-convected Maxwell fluid over a porous stretching plate, Phys. lett. A, 358, 396-403, (2006) · Zbl 1142.76511
[7] Hayat, T.; Khan, M.; Asghar, S., Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid, Acta mech., 168, 213-232, (2004) · Zbl 1063.76108
[8] Hayat, T.; Khan, M.; Ayub, M., On the explicit analytic solutions of an Oldroyd 6-constant fluid, Int. J. eng. sci., 42, 123-135, (2004) · Zbl 1211.76009
[9] Hayat, T.; Khan, M.; Ayub, M., Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field, J. math. anal. appl., 298, 225-244, (2004) · Zbl 1067.35074
[10] Hayat, T.; Sajid, M., On analytic solution of thin film flow of a fourth grade fluid down a vertical cylinder, Phys. lett. A, 361, 316-322, (2007) · Zbl 1170.76307
[11] He, J.H., Homotopy perturbation technique, Comput. meth. appl. mech. eng., 178, 257-262, (1999) · Zbl 0956.70017
[12] He, J.H., A coupling method of homotopy technique and a perturbation technique for nonlinear problems, Int. J. non-linear mech., 35, 37-43, (2000) · Zbl 1068.74618
[13] He, J.H., Homotopy perturbation method a new nonlinear analytical technique, Appl. math. comput., 135, 73-79, (2003) · Zbl 1030.34013
[14] S.J. Liao, On the proposed homotopy analysis technique for nonlinear problems and its applications, Ph.D. dissertation, Shanghai Jio Tong University, 1992.
[15] Liao, S.J., General boundary element method for non-linear heat transfer problems governed by hyperbolic heat conduction equation, Comput. mech., 20, 397-406, (1997) · Zbl 0890.65121
[16] Liao, S.J., Numerically solving nonlinear problems by the homotopy analysis method, Comput. mech., 20, 530-540, (1997) · Zbl 0923.73076
[17] Liao, S.J., A kind of approximate solution technique which does not depend upon small parameters (II): an application in fluid mechanics, Int. J. non-linear mech., 32, 815-822, (1997) · Zbl 1031.76542
[18] Liao, S.J., A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate, J. fluid mech., 385, 101-128, (1999) · Zbl 0931.76017
[19] S.J. Liao, Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman & Hall/CRC Press, London/Boca Raton, 2003.
[20] Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl. math. comput., 147, 499-513, (2004) · Zbl 1086.35005
[21] Liao, S.J., A new branch of solutions of boundary-layer flows over an impermeable stretched plate, Int. J. heat mass transfer, 48, 2529-2539, (2005) · Zbl 1189.76142
[22] Liao, S.J., Series solutions of unsteady boundary layer flows over a stretching flat plate, Stud. appl. math., 117, 239-264, (2006) · Zbl 1145.76352
[23] Liao, S.J., An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate, Comm. non-linear sci. numer. simm., 11, 326-339, (2006) · Zbl 1078.76022
[24] Rajabi, A.; Ganji, D.D.; Therian, H., Application of homotopy perturbation method in nonlinear heat conduction and convection equations, Phys. lett. A, 360, 570-573, (2007) · Zbl 1236.65059
[25] M. Sajid, T. Hayat, The application of homotopy analysis method for thin film flow of a third order fluid, Chaos Solitons Fractal, in press. · Zbl 1146.76588
[26] Sajid, M.; Hayat, T.; Asghar, S., On the analytic solution of the steady flow of a fourth grade fluid, Phys. lett. A, 355, 18-24, (2006)
[27] Sajid, M.; Hayat, T.; Asghar, S., Comparison between HAM and HPM solutions of thin film flows of non-Newtonian fluids on a moving belt, Nonlinear dyn., 50, 27-35, (2007) · Zbl 1181.76031
[28] Wu, W.; Liao, S.J., Solving solitary waves with discontinuity by means of the homotopy analysis method, Chaos solitons fractals., 26, 177-185, (2005) · Zbl 1071.76009
[29] Wu, Y.Y.; Wang, C.; Liao, S.J., Solving the one loop solution of the Vakhnenko equation by means of the homotopy analysis method, Chaos solitons fractals, 23, 1733-1740, (2005) · Zbl 1069.35060
[30] Yang, C.; Liao, S.J., On the explicit purely analytic solution of von karman swirling viscous flow, Comm. non-linear sci. numer. simm., 11, 83-93, (2006) · Zbl 1075.35059
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.