Purpose

The purpose of this paper is to describe the Lane–Emden equation by the fractal derivative and establish its variational principle by using the semi-inverse method. The variational principle is helpful to research the structure of the solution. The approximate analytical solution of the fractal Lane–Emden equation is obtained by the variational iteration method. The example illustrates that the suggested scheme is efficient and accurate for fractal models.

Design/methodology/approach

The author establishes the variational principle for fractal Lane–Emden equation, and its approximate analytical solution is obtained by the variational iteration method.

Findings

The variational iteration method is very fascinating in solving fractal differential equation.

Originality/value

The author first proposes the variational iteration method for solving fractal differential equation. The example shows the efficiency and accuracy of the proposed method. The variational iteration method is valid for other nonlinear fractal models as well.

中文翻译：

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分形Lane-Emden方程的变分原理及其分形近似解

目的

本文的目的是用分形导数描述Lane-Emden方程，并用半逆方法建立其变分原理。变分原理有助于研究解的结构。分形Lane-Emden方程的近似解析解是通过变分迭代法获得的。该示例说明所建议的方案对于分形模型是有效且准确的。

设计/方法/方法

作者建立了分形Lane-Emden方程的变分原理，并通过变分迭代法得到其近似解析解。

发现

变分迭代法在求解分形微分方程方面非常有吸引力。

原创性/价值

作者首先提出了求解分形微分方程的变分迭代法。该示例显示了所提出方法的效率和准确性。变分迭代法也适用于其他非线性分形模型。