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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 $0⩽\epsilon ⩽1$. We have used the homotopy analysis method for finding the more meaningful solution.
MSC:
 76N25 Flow control and optimization (compressible fluids and gas dynamics) 76R10 Free convection (fluid mechanics) 34A25 Analytical theory of ODE (series, transformations, transforms, operational calculus, etc.) 65L99 Numerical methods for ODE