Variational problem containing psi-RL complex-order fractional derivatives

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.