In this paper, the new iterative method with $$\rho $$ -Laplace transform of getting the approximate solution of fractional differential equations was proposed with Caputo generalized fractional derivative. The effect of the various value of order $$\alpha $$ and parameter $$\rho $$ in the solution of certain well known fractional differential equation with Caputo generalized fractional derivative. Applications to the certain fractional differential equation in various systems demonstrate that the proposed method is more reliable and powerful. The graphical representations of the approximate analytic solutions of the fractional differential equations described by the Caputo generalized fractional derivative were provided and the nature of the achieved solution in terms of plots for distinct arbitrary order is captured.

中文翻译：

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用$$\rho $$-Laplace变换求解带有Caputo广义分数阶导数的分数阶微分方程的一种新方法

本文提出了一种用Caputo广义分数阶导数求分数阶微分方程近似解的带有$$\rho $$-Laplace变换的迭代方法。阶数$$\alpha $$ 和参数$$\rho $$ 的各种值对某些著名的带有Caputo 广义分数阶导数的分数阶微分方程的影响。应用于各种系统中的某些分数阶微分方程表明所提出的方法更加可靠和强大。提供了由 Caputo 广义分数阶导数描述的分数阶微分方程的近似解析解的图形表示，并根据不同任意阶的图捕获了所获得解的性质。