The main purpose of this paper is to solve the variational problem containing real- and complex-order fractional derivatives. We define a new version of the complex-order derivative based on the ψ-Riemann-Liouville fractional derivative, and get the Euler–Lagrange equation for the variational problem. By introducing the approximated expansion formula of the complex-order fractional derivative to the variational problem, we derive the corresponding approximated Euler–Lagrange equation. It is proved that the approximated Euler–Lagrange equation converges to the original one in the weak sense. At the same time, the minimal value of the approximated action integral tends to the minimal value of the original one. We also conduct a stress relaxation experiment and discuss the feasibility of the complex-order derivative in real problem modeling.

中文翻译：

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包含 psi-RL 复阶分数阶导数的变分问题

本文的主要目的是解决包含实阶和复阶分数阶导数的变分问题。我们基于 ψ-Riemann-Liouville 分数阶导数定义了复阶导数的新版本，并得到了变分问题的 Euler-Lagrange 方程。通过将复阶分数阶导数的近似展开公式引入变分问题，我们推导出相应的近似欧拉-拉格朗日方程。证明了近似的欧拉-拉格朗日方程在弱意义上收敛于原方程。同时，近似动作积分的最小值趋向于原始积分的最小值。我们还进行了应力松弛实验并讨论了复阶导数在实际问题建模中的可行性。