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首页> 外文期刊>Physica Scripta: An International Journal for Experimental and Theoretical Physics >Higher-order approximate solutions to the relativistic and Duffing-harmonic oscillators by modified He's homotopy methods
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Higher-order approximate solutions to the relativistic and Duffing-harmonic oscillators by modified He's homotopy methods

机译：修正的He's同伦方法对相对论和Duffing谐振子的高阶近似解

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摘要

A modified He's homotopy perturbation method is used to calculate higher-order analytical approximate solutions to the relativistic and Duffing-harmonic oscillators. The He's homotopy perturbation method is modified by truncating the infinite series corresponding to the first-order approximate solution before introducing this solution in the second-order linear differential equation, and so on. We find this modified homotopy perturbation method works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. The approximate formulae obtained show excellent agreement with the exact solutions, and are valid for small as well as large amplitudes of oscillation, including the limiting cases of amplitude approaching zero and infinity. For the relativistic oscillator, only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate frequency of less than 1.6% for small and large values of oscillation amplitude, while this relative error is 0.65% for two iterations with two harmonics and as low as 0.18% when three harmonics are considered in the second approximation. For the Duffing-harmonic oscillator the relative error is as low as 0.078% when the second approximation is considered. Comparison of the result obtained using this method with those obtained by the harmonic balance methods reveals that the former is very effective and convenient.
机译：改良的He's同伦摄动方法用于计算相对论和Duffing谐振子的高阶解析近似解。通过将与一阶近似解相对应的无穷级数截断，然后将其引入到二阶线性微分方程中，来修改He's同伦摄动方法。我们发现，这种改进的同伦摄动方法在整个初始振幅范围内都非常有效，并且已经证明并讨论了近似频率和周期解与精确解的极佳一致性。所获得的近似公式显示出与精确解的极佳一致性，并且对于小振幅和大振幅的振动都有效，包括振幅接近零和无穷大的极限情况。对于相对论振荡器，只有一次迭代会导致解的高精度，对于较小和较大的振荡幅度值，近似频率的最大相对误差小于1.6％，而两次迭代的相对误差为0.65％，其中两次如果在第二近似中考虑了三个谐波，则该谐波可低至0.18％。对于Duffing谐波振荡器，当考虑第二近似时，相对误差低至0.078％。将使用该方法获得的结果与通过谐波平衡法获得的结果进行比较表明，前者非常有效且方便。

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