Fundamental solutions to electrical circuits of non-integer order via fractional derivatives with and without singular kernels

Abstract

This paper deals with constructing analytical solutions of electrical circuits RC and RLC of non-integer order involving fractional time derivatives of type Liouville-Caputo, Caputo-Fabrizio-Caputo, Atangana-Baleanu-Caputo, Atangana-Koca-Caputo and fractional conformable derivative in the Liouville-Caputo sense. These fractional derivatives involve power law, exponential decay or Mittag-Leffler function. For each equation with order $ \alpha \in (0;1]$ , we presented the exact solution using the properties of Laplace transform operator together with the convolution theorem. Numerical simulations are presented for evaluating the difference between these operators. Based on the power-law, exponential-decay or Mittag-Leffler function new behaviors for the voltage and charge were obtained.