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<p>A comparison of the one-parameter optimal method of homotopy analysis and the analogous Liao method with several control parameters, performed by the example of approximate solution of the problem on nonlinear heat transfer, is presented. It was established that the one-parameter approach to the solution of such problems offers a certain advantage over the analogous multiparameter approach. It is shown that the use of two parameters (<i>c</i><sub>0</sub>, <i>c</i><sub>1</sub>) or three parameters (<i>c</i><sub>0</sub>, <i>c</i><sub>1</sub>, <i>c</i><sub>2</sub>) controlling the convergence of solution of the indicated problem gives no advantages, as compared to the classical one-parameter method with parameter c0. All the determined optimal parameters {<i>c</i><sub>0</sub>, <i>c</i><sub>1</sub>} and {<i>c</i><sub>0</sub>, <i>c</i><sub>1</sub>, <i>c</i><sub>2</sub>} of the multiparameter approach, unlike those of the one-parameter approach, are complex quantities. The presence of two or three control parameters in the homotopy equation with the zero-order deformation substantially complicates the process of obtaining approximate solutions and calls for more processor time compared to that of the “classical” variant with one parameter c0 controlling the convergence of solution of the problem.</p>

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On the Problem of the Two- and Three-Parameter Approaches in the Homotopy Analysis Method

  • 期刊论文

  • 发表日期：

    2026年1月1日

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  • V. A. Kot

摘要

A comparison of the one-parameter optimal method of homotopy analysis and the analogous Liao method with several control parameters, performed by the example of approximate solution of the problem on nonlinear heat transfer, is presented. It was established that the one-parameter approach to the solution of such problems offers a certain advantage over the analogous multiparameter approach. It is shown that the use of two parameters (c0, c1) or three parameters (c0, c1, c2) controlling the convergence of solution of the indicated problem gives no advantages, as compared to the classical one-parameter method with parameter c0. All the determined optimal parameters {c0, c1} and {c0, c1, c2} of the multiparameter approach, unlike those of the one-parameter approach, are complex quantities. The presence of two or three control parameters in the homotopy equation with the zero-order deformation substantially complicates the process of obtaining approximate solutions and calls for more processor time compared to that of the “classical” variant with one parameter c0 controlling the convergence of solution of the problem.