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<p>By the example of the nonlinear problem on the radiation cooling of a thin rectangular fin the nonuniform convergence of the approximate solutions obtained using the homotopy analysis method is demonstrated. It was established that the solutions of the problem in all the even-order approximations not only do not improve the approximation quality compared to that of the corresponding uneven-order approximations, but even markedly worsen it. It is shown that a significant decrease in the approximation error is achieved only in each successive pair of approximations, and exclusively in the case where these approximations are of uneven order. When the "small" parameter ε &gt; 1, to obtain sufficiently good approximate solutions, it is necessary to use a relatively large approximation order, which is associated first of all with a slow nonuniform convergence of the sought-for solution.</p>

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On the Problem of Convergence of Solutions in the Homotopy Analysis Method: Nonlinear Problem on Radiation Heat Transfer

  • 期刊论文

  • 发表日期：

    2026年6月29日

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  • V. A. Kot

摘要

By the example of the nonlinear problem on the radiation cooling of a thin rectangular fin the nonuniform convergence of the approximate solutions obtained using the homotopy analysis method is demonstrated. It was established that the solutions of the problem in all the even-order approximations not only do not improve the approximation quality compared to that of the corresponding uneven-order approximations, but even markedly worsen it. It is shown that a significant decrease in the approximation error is achieved only in each successive pair of approximations, and exclusively in the case where these approximations are of uneven order. When the "small" parameter ε > 1, to obtain sufficiently good approximate solutions, it is necessary to use a relatively large approximation order, which is associated first of all with a slow nonuniform convergence of the sought-for solution.